NOTE:
----
1. See NOTE section at the end of Section III.D.
2. I recommend you not to forget to study the last section (IV) of
this article even as an independent part separate from the other
sections. There you can see interesting material about the
experiments for determination of charge and mass of the electron.
We use the following special terminology in this article:
{} indicates superscript (including the power).
[] indicates subscript.
~A means the vector A.
^a means the unit vector a.
<four> means 4.
We show integral around a closed space as <circulation>.
In a capital Greek letter, the word "cap." is written.
6 Mistakes in Electrostatics;
Ed 01.12.31 ---------------------------
Dreadful consequences in Modern Physics
---------------------------------------
Abstract
--------
It is shown that there exists a uniqueness theorem, stating that the
charges given to a constant configuration of conductors take a unique
distribution, which contrary to what is believed does not have any
relation to the uniqueness theorem of electrostatic potential. Using
this thorem we obtain coefficients of potential analytically. We show
that a simple carelessness has caused the famous formula for the
electrostatic potential to be written as U=1/2<integral>~D.~Edv while
its correct form is U=1/2<integral>~D.~E[<rho>]dv in which ~E[<rho>]
is the electrostatic field arising only from the external charges not
also from the polarization charges.
Considering the above-mentioned material it is shown that, contrary
to the current belief, capacitance of a capacitor does not at all
depend on the dielectric used in it and depends only on the
configuration of its conductors. We proceed to correct some current
mistakes resulted from the above-mentioned mistakes, eg electrostatic
potential energy of and the inward force exerted on a dielectric
block entering into a parallel-plate capacitor are obtained and
compared with the wrong current ones.
It is shown that existence of dielectric in the capacitor of a
circuit causes attraction of more charges onto the capacitor because
of the polarization of the dielectric. Then, in electric circuits we
should consider the capacitor's dielectric as a source of potential
not think wrongly that existence of dielectric changes the
capacitor's capacitance. Difference between these two understandings
are verified completely during some examples, and some experiments
are proposed for testing the theory. For example it is shown that
contrary to what the current theory predicts, resonance frequency of
a circuit of RLC will increase by inserting dielectric into the
capacitor (without any change of the geometry of its conductors).
It is also shown that what is calculated as K (dielectric constsant)
is in fact 2-(1/K).
It is also shown that contrary to this current belief that the
electrostatic potential difference between the two conductors of a
capacitor is the same potential difference between the two poles of
the battery which has charged it, the first is twofold compared with
the second. We see the influence of this in the experiments performed
for determination of charge and mass of the electron.
I. Introduction
---------------
In the current electrostatic discussions it is stated that a solution
of Laplace's equation which fits a set of boundary conditions is
unique, and while this matter has not been proven in the case that
these boundary conditions are the charges on the boundaries, the
known charges on the boundaries are taken as boundary conditions.
First section of this article solves this problem after which obtains
the coefficient of potential, while in the current electromagnetic
books these coefficients are obtained by using the above mentioned
unproven generalization of the boundary conditions which
incorrectness of this way is also shown.
The relation U=1/2<integral over V>~D.~Edv for the electrostatic
potential energy of a system is a quite familiar equation to every
physicist, but a careful scrutiny shows an existent undoubted mistake
in this equation. This mistake is easily arising from this fact that
in the process of obtaining this equation, while accepting that
<del>.~D=<rho> where <rho> is the external electric charge density,
it is forgotten that in the primary equation of the electrostatic
potential energy of the system the potential arising only from this
<rho>, <phi>[<rho>], not also from the polarization charges be taken
into account resulting in considering ~E (obtained from -<del><phi>)
instead of ~E[<rho>] (obtained from -<del><phi>[<rho>]) which is the
electrostatic field arising only from <rho> not also from the
polarization. This careful scrutiny is presented in the third section
of this article. A great part of this section proceeds to some
consequences of this same mistake including this current belief that
the capacitance of a capacitor depends on its dielectric, while we
shall prove that this is not at all the case and it depends only on
the form of the configuration of the conductors of the capacitor.
To another much simple and obvious current mistake is paid in the
last section: We connect a battery, which the potential difference
between its poles is <cap. delta><phi>, to the two plates of an
uncharged capacitor until it will be charged. Then, what is the
electrostatic potential difference between the plates of the charged
capacitor? All the current literature on the subject answer that this
electrostatic potential difference is the same potential difference
between the poles of the battery, <cap. delta><phi>, while this
is not the case and is equal to 2<cap. delta><phi>.
As it is seen, the above current mistakes some of which being
fundamental are totally in bases of the subject of Electromagnetism,
and cannot be ignored, because not only are much widespread and
taught in all the universities but also some of them are basis for
some subsequent deductions in other branches of physics. This matter
shows that in the progress of physics the attention should not be
only to its rapidity but also to its profundity, otherwise, as in the
case of this article, sometimes some of the obvious mistakes remain
hidden from the physicists' view yielding probably very other wrong
consequences.
II. Another uniqueness theorem in Electrostatics
------------------------------------------------
II.A. Uniqueness theorem of charge distribution in conductors
-------------------------------------------------------------
In solving electrostatic problems there is a uniqueness theorem that
distinctly states that when the electrostatic potential or the normal
component of its gradient is given in each point of the bounding
surfaces then if the potential is given in at least one point, the
solution of Laplace's equation is uique, and otherwise we may add any
constant to a solution of this equation. Unfortunately, sometimes
negligence is seen in careful applying of the quite clear stated
above boundary conditions. For instance without any reason the
charges of bounding surfaces are taken as boundary conditions in
terms of which the above theorem is applied in obtaining coefficients
of potential of a system of conductors. The reasoning being used is
this (see Foundations of Electromagnetic Theory by Reitz, Milford and
Christy, Addison-Wesley, 1979): "Suppose there are N conductors in
fixed geometry. Let all the conductors be uncharged except conductor
j, which bears the charge Q[j0]. The appropriate solution to
Laplace's equation in the space exterior to the conductors will be
given the symbol <phi>{(j)}(x,y,z) and the potential of each of the
conductors will be indicated by <phi>{(j)}[1], <phi>{(j)}[2], ....,
<phi>{(j)}[j], ....,<phi>{(j)}[N]. Now let us change the charge of
the jth conductor to <lambda>Q[j0]. The function
<lambda><phi>{(j)}(x,y,z) satisfies Laplace's equation, since
<lambda> is a constant; that the new boundary conditions are
satisfied by this function may be seen from the following argument.
The potential at all points in space is multiplied by <lambda>; thus
all derivatives (and in particular the gradient) of the potential are
multiplied by <lambda>. Because <sigma>=<epsilon>[0]E[n], it follows
that all charge densities are multiplied by <lambda>. Thus the charge
of the jth conductor is <lambda>Q[j0] and all other conductors remain
uncharged. A solution of Laplace's equation which fits a particular
set of boundary conditions is unique; therefore we have found the
correct solution, <lambda><phi>{(j)}(x,y,z) to our modified problem.
The conclusion we draw from this discussion is that the potential of
each conductor is proportional to the charge Q[j] of conductor j,
that is <phi>{(j)}=p[ij]Q[j], (i=1,2,...,N) where p[ij] is a
constant which depends only on the geometry."
The fault may be found in this reasoning is arising from the same
incorrect distinction of boundary conditions. This fault is that a
solution to Laplace's equation other than <lambda><phi>{(j)} can be
found such that it can make the charge of the jth conductor
<lambda>fold retaining all other conductors uncharged. This solution
can be <lambda><phi>{(j)}(x,y,z)+c for a non-zero constant c. It is
obvious that its gradient and therefore <sigma>=<epsilon>[0]E[n]
arising from it compared with before are <lambda>fold and then the
charge of the jth conductor will be <lambda>fold while all other
conductors remain uncharged. But this solution is no longer
proportional to the charge of the jth conductor, Q[j], ie we won't
have <phi>{(j)}(x,y,z)=p[ij]Q[j].
In order to clear obviously that the uniqueness theorem of potential
does not include boundary conditions on charges, suppose that there
is an initially uncharged conductor. We then give it some charge. We
want to see when the given charge is definite whether potential
function outside the conductor will or won't be determined uniquely
by this theorem. We say that the given charge distributes itself onto
the surface of the conductor and remain fixed causing that the
potential of the equipotential surface of the conductor to become
specified. With specifying of the conductor potential, potential
function outside the conductor is determined uniquely according to
the theorem. But important for us is knowing that whether form of the
charge distribution onto the conductor surface is uniquely determined
or not. One can say that maybe the charge can take another form of
distribution on the surface causing another potential for the
equipotential surface of the conductor and according to the theorem
we shall have another unique function for the potential outside the
conductor. In a geometric illustration there is not anything to
prevent the above problem for a sharp conductor being solved with
equipotential surfaces concentrated near either the sharp end or the
other end; the charge is concentrated at the sharp end in the first
and at the other end in the second case. Which occurs really is a
matter that must be determined by another uniqueness theorem,
uniqueness theorem of charge distribution, which has no relation to
the uniqueness theorem of potential.
Analytical proof of this theorem is a problem that must be solved.
That this theorem is valid can be understood by some thinking and
visualizing. Separate from inner parts of the conductors consider
external surfaces of the conductors as some conducting thin shells.
Obviously if some charge is to distribute itself in these shells, the
components of the charge, as a result of the repulsive forces, will
take the most distant possible distances from one another, and even
when for instance uncharged conducting shells are set in the vicinity
of charged conducting shells, their conducting (or valence) charges
will be separated in order that like charges take the most distant
and unlike charges take the most neighboring possible distances from
one another. What is clear is that these "most"s indicate to some
unique situation. Therefore we can say that form of the surface
charge distribution is a function of geometrical form of the
conductors and then will be specified uniquely for a definite
configuration of conductors.
II.B. Proportion of charge density to net charge
-----------------------------------------------
Now suppose that for a particular configuration of and definite
amount of charge given to some conductors we can find two
distributions of charge in the conductors in each of which the
resultant electrostatic force on each infinitesimal partial charge
due to other infinitesimal partial charges is outward normal to the
conductor surface and there exists no tangential component for this
force. (Of course these outward normal forces are balanced by surface
stress in the material of the conductors.) Because there is not any
tangential component for the mentioned forces, existence of these two
charge distributions is possible. But because of the same
configuration for the both, the uniqueness theorem of charge
distribution necessitates that the both distribution be the same.
We shall benefit form this matter soon.
We prove that in a constant configuration of some conductors from
which only one has net charge, Q, change of this net charge form Q to
<lambda>Q causes that the surface density in each point of the
conductors' surfaces becomes <lambda>fold: Visualize the constant
situation existent before that Q becomes <lambda>Q. The charges in
the conductors have a unique distribution according to the uniqueness
theorem of charge distribution. In this distribution there exists a
resultant electrostatic force exerted on each infinitesimal partial
surface charge <sigma>da due to other partial charges which is
outward normal to the conductor surface. Suppose that this
distribution becomes nailed up in some manner, ie each partial charge
becomes fixed in its position and no longer has the state of a
conducting free charge (in order that won't probably change its
position as a result of change of the charge). Now suppose each
partial charge becomes <lambda>fold in its position, ie we have for
the new partial charge <sigma>'da=<lambda><sigma>da. Since the
partial charges are nailed up, they are not free to redistribute
themselves on the conductors' surfaces probably. It is obvious that
resultant electrostatic force exerted on a partial charge <sigma>'da
will be still outward normal to the conductor surface, since firstly
this partial charge is <lambda>fold of previous <sigma>da and
secondly each of other partial charges is <lambda>fold of previous
partial charges and then the only change in the resultant force on
<sigma>da will be in its magnitude which becomes <lambda>{2}fold,
while its direction will remain unchanged. Therefore, by changing
each <sigma>da to <lambda><sigma>da we have found a nailed up
distribution for the charges which exerts a resultant force on each
partial surface charge outward normal to the conductor surface, and
furthermore, the only change in the net charges of the conductors is
in the conductor bearing net charge Q previously which now bears
<lambda>Q, and then it is obvious that if the partial charges get
free from the nailed state will retain this distribution. Therefore,
this distribution is a possible one, and according to what said
previously based on the uniqueness theorem, is the same distribution
that really occurs on the conductors' surfaces when the net charge of
the mentioned conductor changes from Q to <lambda>Q.
II.C. Generalization of the uniqueness theorem and of the charge
----------------------------------------------------------------
density proportion to net charge
--------------------------------
In fact, the uniqueness theorem of charge distribution on the
conductors is true in case of a particular configuration of
conductors and a constant (nailed up) charge distribution and a
constant set of linear dielectrics in the space exterior to the
conductors, ie in such a case a charge given to the conductors causes
a unique charge distribution on their surfaces. The truth of this
theorem can be found out with some indications similar to previous
ones.
Now consider a constant configuration of conductors and a constant
set of linear dielectrics outside the conductors. There is no charge
outside the conductors. We give a net charge to only one of the
conductors. Certainly, according to the above theorem we shall have a
unique charge distribution in the conductors. Suppose that the given
charge of that conductor becomes <lambda>fold. We want to prove that
the surface free charge densties on all of the conductors and also
the dielectrics' polarizations will become <lambda>fold consequently.
Visualize the situation existent before that the given charge becomes
<lambda>fold. An outward resultant force normal to the conductor
surface is exerted on each partial surface charge <sigma>da due to
other nonpolarization and polarization partial charges. Now suppose
that all the nonpolarization (or free) partial charges be nailed up
in their positions and then all the nonpolarization and polarization
partial charges (ie the previous free charges and dielectrics'
polarizations) become <lambda>fold. Obviously, in this case the
resultant electrostatic force on each partial surface charge is
outward normal to the conductor surface too (and only its magnitude
has become <lambda>{2}fold). Furthermore, it is obvious that in each
point of each dielectric the electrostatic field has only become
<lambda>fold (without any change in its direction) and then we see
that this field is propotional to the polarization at that point as
must be so expectedly. Thus, if the charges get free from the nailed
state, they will remain on their positions, and furthermore, the only
change in net charges is in the above mentioned conductor, net charge
of which has now become <lambda>fold. Therefore, this is a possible
distribution and according to the above mentioned uniqueness theorem
of charge distribution is unique and then is the same distribution
that really occurs.
II.D. Superposition principle for the charge densities
------------------------------------------------------
We must also notice another point. We understood that in a
configuration of some conductors that only one of them has net
charge, charge distribution is unique. Suppose that we have N
conductors and only conductor i has net charge (Q). The unique
distribution that charges get, prescribes charge surface density
<sigma>(~r) (and polarization ~P(~r)) for each point of each
conductor (and each point outside the conductors).
Now consider this same configuration of these conductors from which
only conductor j (such that j is not equal to i) has net charge
(Q[j]). The unique distribution that charges get, prescribes charge
surface density <sigma>[j](~r) (and polarization ~P[j](~r)) for each
point of each conductor (and each point outside the conductors).
It is clear intuitively that if we have this same configuration of
the conductors from which only two conductors have net charges, the
ith conductor has the same relevant net charge (Q) and the jth
conductor has the same relevant net charge (Q[j]), then the unique
distribution that charges get, prescribes charge surface density
<sigma>(~r)+<sigma>[j](~r) (and polarization ~P(~r)+~P[j](~r))
for each point of each conductor (and each point outside the
conductors). This fact has generality for when each conductor has a
specified net charge or when there is a fixed distribution of
external charge outside the conductors (ie we can add contribution of
this distribution towards forming charge surface density on the
conductors (and forming polarization) to other contributions). We can
even, when there are linear dielectrics, obtain surface charge
distribution on the conductors by adding the charge surface density
in each point on the conductors related to charge distribution in the
absence of dielectrics to the charge surface density in the same
point produced only by the polarizations of the dielectrics assuming
that there exists no net charge in any conductor but only the
polarizations exist.
Therefore, considering the theorems we have proven so far, we can
conceive that in a system of some charged conductors and some fixed
external charge distribution and some linear dielectrics if the net
charge of a conductor becomes <lambda>fold, free partial charge
surface density arising from that conductor, assuming that other
conductors are uncharged and there are not any dielectrics or other
external charges, will become <lambda>fold in each point on the
conductors. It is evident that, considering the integral definition
of electrostatic potential and assuming that the potential is zero at
infinity, the partial potential arising from that conductor (ie in
fact from its effect on forming the free charges) will become
<lambda>fold in each point, too, and then the partial potential
arising from that conductor will become <lambda>fold in each
conductor which is an equipotential region for this partial
potential. In other words, the free net charge of one of the
conductors is proportional to the partial potential arising from the
(effect of the free net) charge of that conductor (assuming that
there are not any dielectrics or other external charges and that
other conductors are uncharged) in each of the conductors:
(i=1,2,3,...,N) <phi>{(j)}=p[ij]Q[j]. Furthermore, this fact that
each conductor is an equipotential region for this partial potential
proves that p[ij] depends only on the geometry of the configuration
of the conductors and even does not depend on the dielectrics and
their positions (or other external charge distributions outside the
conductors), because, as we mentioned, this constant coefficient of
the proportion, p[ij], is related to when we suppose that there
are not at all any dielectrics (or other external charges) and infer
that the charge surface densities will become <lambda>fold if the net
charge of a conductor (the jth one) becomes <lambda>fold (assuming
that other conductors are uncharged).
Now since the potential of each conductor is the sum of its partial
potentials plus a constant, we have
<phi>=<summation from j=1 to N>p[ij]Q[j]+c. (Adding of c removes
the worry arising from generalization of the necessity of the above
reasoning that the partial potentials must be zero at infinity.)
III. Static potential energy and current mistakes
-------------------------------------------------
III.A. Static potential energy
------------------------------
We know that if a closed surface S contains external electric charge
Q and polarization electric charge Q[P], then we shall have
<circulation over S>~E.^nda=(Q+Q[P])/<epsilon>[0]. In this relation
~E is the partial electrostatic field arising from both an elective
distribuition of external charge, the part of which inside the closed
surface being equal to Q, and an elective distribution of
polarization charge, the part of which inside the closed surface
being equal to Q[P]. (The word "elective" implies that the entire
existent charge distribution is not necessarily taken into
consideration, and similarly the word "partial" implies that maybe
only a part of the existent field is intended. Notice the
superposition principle of field and the linearity of potential.)
On the other hand we have
Q[P]=<integral over S'>~P.^nda+<integral over V>(-<del>.~P)dv in
which V is the volume of the dielectric enclosed by S, and S' is the
surface of the conductors inside the closed surface S. In this
relation ~P.^n and -<del>.~P are the the polarization charge
densities of the elective distribution of polarization charge, and
then we can say that in this relation ~P is an elective (ie not
necessarily entire) distribution of electrostatic polarization. If
using the divergence theorem we change the volume integral into the
surface integral, we finally shall obtain
Q[P]=-<circulation over S>~P.^nda. The comparison of this relation
with the first relation of this section shows that
<circulation over S>(<epsilon>[0]~E+~P).^nda=Q in which ~P is an
elective distribution of polarization, and Q is the total charge of
that part of the elective distribution of external charge which is
inside the closed surface S, and ~E is the partial field arising from
both the totality of the elective distribution of external charge and
the totality of the elective distribution of polarization. On
definition, the electric displacement vector is ~D=<epsilon>[0]~E+~P.
Then <circulation over S>~D.^nda=<integral over V><rho>dv. This
relation says that if ~E is arising from both <rho>, which is an
elective distribution of external electric charge, and ~P, which is
an elective distribution of electrostatic polarization, then the
surface integration of ~D=<epsilon>[0]~E+~P on the closed surface S
is equal to the totality of only that part of our elective external
charge which is inside the closed surface. If we use the divergence
theorem in the recent relation, we shall conclude <del>.~D=<rho>.
Considering the above discussions the following deduction may be
interesting. (In this deduction the expression "the ~E arising from
both <rho> and ~P" is shown as "~E(<rho>,P)".)
<del>.~D[1]=<del>.~D[2] or
<del>.(<epsilon>[0]~E(<rho>,P[1])+~P[1]) =
<del>.(<epsilon>[0]~E(<rho>,P[2])+~P[2])
The electrostatic potential energy of a bounded system of electric
charges (which can exist in various forms of external charge,
polarization charge, etc, eg in the form of canceled charges, from
the macroscopic viewpoint, in a molecule) having the density <rho>,
which is in fact the spent energy for assembling all the fractions of
the charge differentially from infinity, is
U=1/2<integral over V[h]><rho>(~r)<phi>(~r)dv (1)
in which V[h] is the whole space and <phi> is the partial
electrostatic potential due to the distribution of <rho>. The way of
obtaining the relation (1) can be seen in many of the electromagnetic
texts.
As it is so actually in the tridimentional world of matter, we
disburden ourselves from the dualizing the charge density as the
surface and volume ones and say we have only the volume density of
the electrostatic charge that, for instance, can have an excessive
absolute amount on the surface of a charged electric conductor. Now
we take into consideration an elective distribution of the volume
density of the external (ie nonpolarization) electric charge, <rho>.
We want to obtain the electrostatic potential energy of this
distribution. We know that <del>.~D=<rho> so that
~D=<epsilon>[0]~E+~P in which ~P is the elective distribution of the
electrostatic polarization and ~E is the resultant field arising from
both the elective distribution of the external electric charge
density (<rho>) and the polarization charge densities due to the
elective distribution of the electrostatic polarization (~P). Since
the electrostatic potential energy of this elective distribution of
the external electric charge is U=1/2<integral over
V[h]><rho><phi>dv, in which (V[h] is the whole space and) <phi> is
only arising from <rho> (not from both <rho> and ~P), we shall have
U=1/2<integral over V[h]><phi><del>.~Ddv, and since
<integral over V[h]><phi><del>.~Ddv=<integral over V[h]><del>.(<
phi>~D)dv-<integral over V[h]>~D.<del><phi>dv=<integral over S[h]><
phi>~D.^n'da-<integral over V[h]>~D.<del><phi>dv=0-<integral over
V[h]>~D.(-~E[<rho>])dv=<integral over V[h]>~D.~E[<rho>]dv
(V[h] and S[h] being in turn the whole space and the total surfaces
of the problem (which of course there is not any surface)), we shall
have
U=1/2<integral over V[h]>~D.~E[<rho>]dv (2)
in which as we said " U is the electrostatic potential energy of an
elective distribution of the external electric charge with the
density <rho>, and we have <del>.~D=<rho> in which
~D=<epsilon>[0]~E+~P in which ~P is an elective distribution of
electrostatic polarization and ~E is arising from both ~P and <rho>,
while ~E[<rho>] is the field arising only from <rho>." It is obvious
that this electrostatic potential energy has been distributed in the
space with the volume density u=1/2~D.~E[<rho>].
(We saw previously that <del>.~D[1]=<del>.~D[2]. Uniqueness of the
electrostatic potential energy of a definite distribution of external
electric charges with the density <rho> necessitates having
1/2<integral over V[h]>~D[1].~E[<rho>]dv =
1/2<integral over V[h]>~D[2].~E[<rho>]dv ;
but although these total energies are equal to each other this won't
necessarily mean that the energy densities are also the same, ie we
cannot infer ~D[1].~E[<rho>]=~D[2].~E[<rho>] or ~D[1]=~D[2] (although
their divergences are equal).)
It is very opportune to compare the above accurate definition of the
electrostatic potential energy with what is set forth for discussion
under this very title in the present electromagnetic books, and to
pay attention to the existent inaccuracy in the definitions of the
involved terms caused by the omission of the subscript <rho> from the
term ~E[<rho>]. This is a sample of the existent inaccuracies in the
present current electromagnetic theory specially in not correct
distinguishing between different electric fields. This mistake has
caused that, considering relation ~D=<epsilon>~E for linear
dielectrics, wrong relations like
u=1/2<epsilon>E{2}=1/2D{2}/<epsilon> to be current in present
electromagnetic textbooks. We shall pay to some other mistakes soon.
III.B. Independence of capacitance from dielectric
--------------------------------------------------
Consider a system consisting of some fixed perfect conductors and
some linear dielectrics in the space exterior to the conductors and
some fixed distribution of external charge density in this space. We
want to obtain electrostatic potential energy arising from all the
free net charges on these conductors, ie the electrostatic potential
energy of that part of the charge distribution in all of the
conductors which comes into existence as a result of these free net
charges (which of course does not include electrostatic potential
energy of the polarization and distribution of external charges and
that (other) part of the charge distribution in all of the conductors
which comes into existence as a result of these polarization and
external charges). Since each conductor is an equipotential region
for the potential arising from these free net charges, for this
electrostatic potential energy we have U=1/2<summation from j=1 to
N>Q[j]<phi>[j] from the relation (1), in which Q[j] is the net charge
of the conductor j and <phi>[j] is the electrostatic potential on the
conductor j arising from all free net charges of the conductors of
the system (ie one related to free net charges themselves and their
effect on the conductors, not also related to dielectric polarization
and other external charges and their effect on the conductors). What
is necessary to be emphasized again (and is important in the coming
discussion) is that the <phi>[j]'s are arising only from net charges
of the conductors not also from the polarization charges. Using the
coefficients of potential for this system we can also write
<phi>=<summation from j=1 to N>p[ij]Q[j] in which Q[j] is the net
charge of the conductor j, and <phi> is the electrostatic
potential on the conductor i arising from all (Q[j]'s ie all) net
charges of the conductors of the system (ie one related to free net
charges themselves and their effect in the conductors, not also
related to dielectric polarization and other external charges and
their effect on the conductors). Combining the two recent relations
yields
U=1/2<summation from i=1 to N><summation from j=1 to N>p[ij]QQ[j]
for the electrostatic potential energy arising from free net charges
of the conductors of a system consisting of some perfect conductors
and probably some linear dielectrics and external charge distribution
outside the conductors.
A capacitor is defined as two conductors (denoted by 1 and 2), from
among a definite configuration of some conductors, that one of them
bears net charge Q (Q being greater than or equal to zero) and the
other one bears -Q. (Existence of net charges on other conductors in
the configuration or of linear dielectrics or external charges
outside the conductors and the effect which each has on these two
conductors (ie 1 and 2) are not important at all. We shall find out
this soon.)
By using the relation <phi>=<summation from j=1 to N>p[ij]Q[j] for
the above capacitor we have:
<cap. delta><phi>=<phi>[1]-<phi>[2]=(p[11]+p[22]-2p[12])Q=Q/C
(We know that p[12]=p[21] proof of which can be seen in many of the
electromagnetic books.) We have attention that in the relation
<cap. delta><phi>=Q/C, <cap. delta><phi> is the potential difference
between the potential arising from net charges of the conductors 1
and 2 (related to themselves and their effect in other conductors) on
the conductor 1 and the potential arising from these charges (related
to themselves and their effect in other conductors) on the conductor
2. Therefore, since the potential of other charges is not considered
and considering linearity of potentials and that C, which is called
as the capacitance of the capacitor, depends only on the form of the
configuration of all (and not only two) of the conductors, it is
obvious that existence of net charges on the conductors other than
the conductors 1 and 2 and existence of any linear dielectrics or
external charges in the space exterior to the conductors, so far as
the configuration of the conductors is constant, are unimportant (and
there is no need that one of the conductors 1 and 2 be shielded by
the other, the way presented in some electromagnetic books for the
potential difference independence of whether other conductors are
charged). We specially emphasize again that so, we have proven that
the capacitance (C) of a capacitor does not depend on whether there
exist any dielectrics at all and only depends on the configuration of
the conductors introduced in the definition of the capacitor.
Using the relation
U=1/2<summation from i=1 to N><summation from j=1 to N>p[ij]QQ[j]
we obtain
U=1/2Q{2}/C=1/2Q<cap. delta><phi>=1/2C(<cap. delta><phi>){2} (3)
for the electrostatic potential energy of the charges Q and -Q
(themselves and of their effect). We should emphasize again that in
the recent relation, <cap. delta><phi> is the potential difference
arising from the free charges Q and -Q (and not also from eg
polarization charges), and C depends only on the configuration of the
conductors (and not also on eg existence or nonexistence of linear
dielectrics).
At the end of this section let's obtain the capacitance of a
capacitor consisting of two parallel plates in which the plates
separation d is very small compared with the dimensions of the
plates:
Q Q Q
C=----------------------=------=---------------------
(<cap. delta><phi>)[Q] E[Q]d <sigma>d/<epsilon>[0]
Q <epsilon>[0]A
=-------------------=-------------- ,
(Q/A)d/<epsilon>[0] d
in which (<cap. delta><phi>)[Q] and E[Q] are the potential difference
and the electrostatic field arising from Q and -Q (and not also from
the polarization charges) respectively. Therefore, the capacitance of
this capacitor is <epsilon>[0]A/d regardless of whether there exist
any linear dielectrics between the parallel plates or not.
And now see the present books of Electricity and Magnetism in which
without attention to this fact that <cap. delta><phi> must be arising
only from the capacitor charge, the relation <cap. delta><phi>=Ed, in
which E is arising from not only the capacitor charge but also the
linear dielectrics polarization charges, is used and consequently
wrong expression <epsilon>A/d is obtained for the capacitance.
III.C. Dielectric as source of potential
----------------------------------------
We saw that the mathematical discussions presented so far proved
independence of the capacitance of a capacitor from its dielectric.
But this is doubtlessly surprising for the physicists and engineers,
because they know well that dielectric has a substantial part in
accumulation of charge in the capacitor. This section is intended for
obviating this surprise.
It is made use often of electroscope to show the effect of
dielectrics in capacitors. If the two conductors of a charged
capacitor are connected to an electroscope, leaves of the
electroscope will get away from each other. Now, if, without any
change in the configuration of the capacitor's conductors, a
dielectric is inserted between the two conductors of the capacitor,
the leaves of the electroscope will come close to each other. Current
justification of this phenomenon is as follows (eg see University
Physics by Sears, Zemansky and Young, Addison-Wesley 1987):
"The equation C=Q/<cap. delta><phi> shows the relation among the
capacitor's capacitance, capacitor's charge, and the potential
difference between the two conductors of the capacitor. When a
dielectric is inserted into the capacitor, due to the orientation of
the electric dipoles of the dielectric in the field inside the
capacitor some polarization charge opposite to the charge of each
conductor of the capacitor is induced on that surface of the
dielectric which is adjacent to this conductor, and then the
electrostatic field in the dielectric, and thereby the potential
difference (between the two conductors), arising from both the
capacitor's charge and this induced polarization charge is decreased.
Then, the denominator of C=Q/<cap. delta><phi> decreases which
results in increasing of the capacitance (C) considering that Q
remains uncharged, ie the capacitor's capacitance increases by
inserting a dielectric between the capacitor's conductors. That the
leaves of the electroscope come closer to each other by inserting the
dielectric is because of this same decreasing of the potential
difference, <cap. delta><phi>."
It is clear that considering the discussion presented in this
article, the above justification is quite wrong, because
<cap. delta><phi> is the potential difference arising only from the
capacitor's charge not also from the polarization charge formed in
the dielectric. But why do the leaves of the electroscope come closer
to each other when a dielectric is inserted into the capacitor? Its
reason is quite obvious. Metal housing and the leaves connected to
the metal knob of the electroscope, themselves, are in fact a
capacitor, which when are connected separately to the two conductors
of the capacitor under measurement, a new (equivalent) capacitor will
be formed consisting of two conductors: the first being one of the
conductors of the capacitor under measurement and the electroscope's
metal housing which is connected to it, and the second being the
other conductor of the capacitor under measurement and the set of the
knob and the leaves of the electroscope which is connected to this
conductor. It is obvious that if the capacitor under measurement is
charged at first, its charge now, after its connecting to the
electroscope, will be distributed throughout the new formed
capacitor and then a part of the charge of the primary capacitor now
will go to the electroscope because of which the leaves of the
electroscope will get away from each other (because the opposite
charges induced in the electroscope will attract each other causing
drawing of the leaves toward the electroscope's housing which itself
means more separation of the leaves from each other).
By inserting the dielectric into the capacitor we cause creation of
polarization charges in the dielectric which this, in turn, causes
more charges of the new formed capacitor to be drawn towards the
dielectric. Thus, the distribution of the charge will be changed in
such a manner that a part of the charge distribution in the
electroscope will go to the primary capacitor (or the one under
measurement) to be placed as close as possible to the dielectric;
this means decrease of the electroscope's charge which will cause its
leaves to come closer to each other. Therefore, the act of the
dielectric is change of the charge distribution in the new capacitor
formed from the primary capacitor and the electroscope, not change of
the capacitance of the primary capacitor.
Now, let's connect the two plates of a parallel-plate capacitor by a
wire in the space exterior to the space between the plates. What will
happen if a slice of a dielectric having a permanent electric
polarization is inserted between the two plates of the capacitor? The
polarized dielectric will cause induction of charge on the two
plates; the positive surface of the slice will induce negative charge
on the plate adjacent to it, and the negative surface will induce
positive charge on the (other) plate adjacent to it. Induction of
charge on the two plates, while they had no charge beforehand, means
that while inserting the dielectric between the plates an electric
current has been flowing in the wire from one plate to the other. In
other words the dielectric acts like a power supply producing
electric current or charging the capacitor. Then, we can attribute
electric potential difference to it (like the potential difference
between the two poles of a battery).
Now, how will the situation be if the inserted dielectric is not to
have previous polarization but it is to be polarized because of the
charge (or in fact the electric field produced by the charge) of the
capacitor? Answer is that the situation will be similar to the same
state of permanent polarization, and again the dielectric acts as a
source of potential. Its physical and direct reason can be seen
easily in the discussion we presented about the electroscope. There,
we saw that inserting the dielectric, charge distribution was changed
in such a manner that some more charges were accumulated on the
conductors of the (primary) capacitor. It is clear that more
accumulation of charge on the capacitor necessitates flowing of
electric current in the circuit. Cause of this current and of the
more accumulation of charge on the capacitor is the source of
potential difference which we must attribute to the dielectric.
In this manner, the purpose of this section has been fulfilled
practically; in electric circuits wherever a dielectric is to exist
between the conductors of a capacitor, a proper source of voltage
must be considered in the circuit in the same place of the
dielectric. Such a voltage source causes accumulation of charges on
the conductors of the capacitor more than when there exists no
dielectric in the capacitor. One can say whether this act is not
equivalent to defining, in principle, the capacitance of a capacitor
equal to the charge accumulated on the capacitor (due to both the
configuration of the capacitor's conductors and the electric
induction in the conductors caused by the polarization of the
dielectric) divided by the potential difference between the two
conductors of the capacitor (which is the method that current
instruments measuring capacitor's capacitances work based on it) and
no longer considering the dielectric as a source of potential.
Following example shows that consequences of such a definition in
practice are not equivalent to the practical consequences of the main
definition of capacitance of capacitor (although can be close to it
under suitable conditions). We then shall investigate another example
which will show, well, considerable differences that can come into
existence if role of the dielectric as a power supply in the circuit
is not taken into consideration, according to which a quite practical
criterion for testing the theory presented in this section in
comparison with the current theory will be presented.
III.D. Some examples as test
----------------------------
Let's connect the two plates of a dielectricless parallel-plate
capacitor to the two poles of a battery. At the end of the section
III.B. we saw that the capacitance of such a capacitor is
<epsilon>[0]A/d in which A is the capacitor's area and d is the
distance between its plates. Then, according to the relation
C=Q/<cap. delta><phi> for the capacitor's capacitance, we have
<epsilon>[0]A/d=<sigma>A/V in which <sigma> is the surface density
of the charge accumulated on the capacitor and V is the potential
difference given to the two plates of the capacitor by the battery.
In this manner we have:
<sigma>d=<epsilon>[0]V (4)
Now we fill the space between the two plates with a linear dielectric
with the permitivity <epsilon>. We indicate the magnitude of the
formed electric polarization in the dielectric by P. P is in fact
equal to the surface density of the polarization charge in the
dielectric. Suppose that a charge exactly equal to the polarization
charge is induced on the plates of the capacitor. (Indeed, in the
state of induction of charge in the capacitor due to the polarized
dielectric between the capacitor's plates we should suppose that the
two plates of the capacitor are connected to each other by a wire in
the space exterior to the space between the plates; in other words in
this state the battery existent in the circuit does not play any role
except as a short circuit.) Then the charge induced on the capacitor
due to the polarization of the dielectric is equal to PA. This
charge, as we said, has been stored in the capacitor because of a
source of potential difference, equal to V', which we must attribute
to the dielectric; ie because of the potential difference V' exerted
to the two plates of the capacitor the charge PA has been accumulated
in the capacitor, and then the ratio PA/V' is equal to the
capacitor's capacitance <epsilon>[0]A/d=PA/V'. Considering that
P=(<epsilon>-<epsilon>[0])E=(<epsilon>-<epsilon>[0])<sigma>/<epsilon>
in which E is the electrostatic field arising from both the external
and polarization charges we infer from this relation that
V'=(<epsilon>-<epsilon>[0])<sigma>d/(<epsilon><epsilon>[0])
which considering Eq.(4) results in
V'=(1-<epsilon>[0]/<epsilon>)V=(1-1/K)V (5)
Let's calculate sum of the charges (Q) accumulated on this capacitor
(due to both the configuration of the capacitor's conductors and the
induction arising from the (polarization of the) dielectric). For
this act we must add the potential difference arising from the
dielectric to the potential difference given by the battery and after
that multiply the sum by the (real) capacitance of the capacitor
C=<epsilon>[0]A/d:
Q=(V+(1-<epsilon>[0]/<epsilon>)V)<epsilon>[0]A/d=
(2-<epsilon>[0]/<epsilon>)(<epsilon>[0]A/d)V=(2-1/K)CV (6)
Can we present another definition of capacitance of capacitor, for
convenience in practice, equal to sum of the charges accumulated on
the capacitor (consisting of the charges arising from both the
configuration of the capacitor's conductors and the induction due to
the dielectric) divided by the potential difference between the two
capacitor's conductors, given to the capacitor only by the battery
(or the circuit)? Considering Eq.(6) such a definition gives the
following (newly defined) capacitance of our capacitor equal to
Q/V=(2-<epsilon>[0]/<epsilon>)<epsilon>[0]A/d. (7)
Is this definition useful in practice, and does it yield real
consequences? The answer is negative. It is sufficient only instead
of a single capacitor to consider n capacitors connected in series
such that the space between the plates of only one of them is filled
with dielectric and to try to calculate the accumulated charges on
the equivalent capacitor.
If all of these n capacitors were dielectricless, because of the
identity between the capacitors the (shared) potential difference
between the two plates of each of these capacitors would be V/n.
When only one of these capacitors is filled with a linear dielectric
with the permittivity <epsilon>, the potential difference related to
this dielectric (as a source of potential), similar to Eq.(5) will be
(1-<epsilon>[0]/<epsilon>)V/n. Since these n capacitors are identical
and the capacitance of each of them is <epsilon>[0]A/d, the
equivalent capacitance of these n capacitors which are connected in
series will be obtained by solving the equation
1/C[1]=n/(<epsilon>[0]A/d) for C[1] equal to <epsilon>[0]A/(nd).
Therefore, the charge accumulated on each capacitor is equal to
<epsilon>[0] V <epsilon>[0]A
( V + ( 1 - ------------ ) --- ) -------------
<epsilon> n nd
<epsilon>-<epsilon>[0] <epsilon>[0]A
= ( 1 + ---------------------- ) ------------- V . (8)
n<epsilon> nd
But now let's see if the capacitance of the capacitor having
dielectric is to be equal to (7) while the capacitance of each of the
other capacitors is equal to <epsilon>[0]A/d, whether or not the
charge accumulated on each capacitor will be obtained still equal to
(8) when no longer the source of potential difference related to the
dielectric is considered in lieu of considering (7) for the
capacitance of the capacitor having dielectric. Equivalent
capacitance of the capacitors which are in series will be obtained by
solving the equation
1 n-1 1
---- = --------------- + -----------------------------------------
C[2] <epsilon>[0]A/d (2-<epsilon>[0]/<epsilon>)<epsilon>[0]A/d
for C[2], and charge of each capacitor should be considered equal to
C[2]V:
1 <epsilon>[0]A
C[2]V= --------------------------------------- . ------------- V (9)
n-1+<epsilon>/(2<epsilon>-<epsilon>[0]) d
Obviously the coefficient of <epsilon>[0]AV/d in Eq.(8) is not equal
to the coefficient of <epsilon>[0]AV/d in Eq.(9) except when
<epsilon>=<epsilon>[0] or n=1. Thus, we see that the new definition
we tried to present for capacitance of capacitor is not so useful in
practice (at least in this example does not give the real charge
accumulated on the capacitors). But the ratio of these two
coefficients is not so far from one. To see this fact let's indicate
<epsilon>/<epsilon>[0] by K and obtain the ratio of the coefficient
of <epsilon>[0]AV/d in Eq.(9) to the coefficient of <epsilon>[0]AV/d
in Eq.(8):
(n-1+<epsilon>/(2<epsilon>-<epsilon>[0])){-1} (K-1){2}(n-1)
---------------------------------------------- = 1/(1+ -------------)
1/n + (<epsilon>-<epsilon>[0])/(n{2}<epsilon>) (2K-1)Kn{2}
It is seen that the degree of the term (K-1){2}(n-1)/((2K-1)Kn{2})
with respect to K is zero and with respect to n is -1; thus this term
is close to zero practically, or in other words the ratio of the
above-mentioned coefficients is close to one practically. This matter
is itself a good reason that why the definition of capacitance in the
form of capacitor's charge divided by the potential difference
exerted on the capacitor's conductors (Eq.(7)) has been able to
endure practically and the difficulties due to such a definition has
remained hidden in practice. But, important for a physicist should be
mathematical much exactness and discovery of what actually occurs or
exists. In order to find out that such an exactness can be important
even in practice (and then won't be negligible even for engineers)
notice the following example.
Consider a series circuit of RLC, which its capacitor is
parallel-plate and dielectricless, connected to a constant voltage V.
After connection of the switch in the time t=0, the equation of the
circuit will be
V=RI+LdI/dt+1/(2C)<integral from t=0 to t>I(t)dt. (10)
(We should notice that as it will be proven in the last section of
this article, in this circuit we must consider the circuital
potential difference of the capacitor, ie the third term of the
right-hand side of (10), not as it is usual wrongly its electrostatic
potential difference ie 1/C<integral from t=0 to t>I(t)dt. There,
also we shall see that what the current instruments measure as
capacitance is in fact two times more than the capacitance. Another
noticeable point being that as it has been explained in the section 5
of the 13th article of the book, L in (10) is in fact equal to
<mu><epsilon>'a'L{*} not equal to only
d<cap. phi>{*}/dI(=L{*}) according to its usual definition. But
since the current instruments for measuring L work based on the
formula <cursive E>=-LdI/dt, they are in fact measuring
<mu><epsilon>'a'L{*} as L because as we can see in that article
the correct relation is in fact
<cursive E> = -<mu><epsilon>'a'L{*}dI/dt.)
With one time differentiation of this equation with respect to time,
the following equation will be obtained considering that V is
constant: Ld{2}I/dt{2}+RdI/dt+I/(2C)=0. If R/(2L)<(2LC){-1/2}, this
equation will be solved as I= a.exp(-Rt/(2L))cos(<omega>[n]t-<theta>)
in which
1 R{2}
<omega>[n] = (--- - -----){1/2} (11)
2LC 4L{2}
and a and <theta> are two arbitrary constants. Since in t=0 we have
I=0 and then also from Eq.(10) we have dI/dt=V/L, we conclude that
a=V/(<omega>[n]L) and <theta>=<pi>/2, and then
V
I= ---------- exp(-Rt/(2L))sin(<omega>[n]t). (12)
<omega>[n]L
For calculating the voltage drop in the capacitor we should calculate
the third term of the right-hand side of Eq.(10):
1
--<integral from t =
2C
V
0 to to t>-----------exp(-Rt/(2L))sin(<omega>[n]t)dt =
<omega>[n]L
R
V(1-exp(-Rt/(2L))(cos(<omega>[n]t)+------------sin(<omega>[n]t)))
2<omega>[n]L (13)
Now, if the space between the two plates of the capacitor (without
any change in the configuration of the plates) is to be filled by a
linear dielectric with the permittivity <epsilon>, we must multiply
the negative of the voltage drop in the capacitor ((13)) by
(1-<epsilon>[0]/<epsilon>) till according to Eq.(5) the potential
difference which we must attribute to the dielectric as source of
potential is otained. We then should add this source to the previous
constant source and equate the sum to the right-hand side of Eq.(10):
V+V(exp(-Rt/(2L))(cos(<omega>[n]t)+R/(2<omega>[n]L)sin(<omega>[n]t))-
1)(1-<epsilon>[0]/<epsilon>)=RI+LdI/dt+1/(2C)<integral from t=0 to
t>I(t)dt (14)
With one time differentiation of this equation with respect to time
the following equation will be obtained:
d{2}I dI 1
L----- +R-- + --I =
dt{2} dt 2C
<epsilon>[0] 2<omega>[n]L
V(1- ------------)------------exp(-Rt/(2L))sin(<omega>[n]t)
<epsilon> R{2}C-2L
Particular solution of this equation is
V <epsilon>[0]
--------(1- ------------)t.exp(-Rt/(2L))cos(<omega>[n]t),
2L-R{2}C <epsilon>
and general solution of its corresponding homogeneous equation is
a.exp(-Rt/(2L))cos(<omega>[n]t-<theta>) with the two arbitrary
constants a and <theta>. Then general solution of this equation is
I=a.exp(-Rt/(2L))cos(<omega>[n]t-<theta>) +
V <epsilon>[0]
--------(1- ------------)t.exp(-Rt/(2L))cos(<omega>[n]t)
2L-R{2}C <epsilon>
with the two arbitrary constants a and <theta>. For obtaining a and
<theta> by means of the initial conditions, we should be careful that
initial conditions must be fit, ie t=0 should be the same moment
that, without dielectric, the current in the circuit was zero and we
had dI/dt=V/L; and now, when the dielectric has been inserted, we
should see how the conditions change, and in this moment (t=0) what
the current and its time derivative are as initial conditions. The
physics of the problem says that we have in this state I=0 in this
moment too, and then also it is clear from Eq.(14) that in this
moment we have dI/dt=V/L too. Then
V V <epsilon>[0]
a = ----------- + --------------------(1- ------------) =
<omega>[n]L <omega>[n](R{2}C-2L) <epsilon>
L(1+<epsilon>[0]/<epsilon>)-R{2}C V
---------------------------------.----------- and <theta>=<pi>/2.
2L-R{2}C <omega>[n]L
Thus
L(1+<epsilon>[0]/<epsilon>)-R{2}C V
I=---------------------------------.-----------exp(-Rt/(2L))sin(<
2L-R{2}C <omega>[n]L
V <epsilon>[0]
omega>[n]t)+--------(1- ------------)t.exp(-Rt/(2L))cos(<omega>[n]t).
2L-R{2}C <epsilon> (15)
(It is noticeable that when <omega>=<omega>[0] the same Eq.(12) will
be obtained from this equation.) We obtained Eq.(15) for the current
of the circuit, while what is current at present is that inserting
the linear dielectric (with the permittivity <epsilon>) between the
plates of the capacitor only the capacitor's capacitance changes from
C to KC where K=<epsilon>/<epsilon>[0] (without any addition of new
source of potential to the circuit), and then the circuit's current
has the same form of Eq.(12) with this only difference that in the
equation related to <omega>[n] (Eq.(11)) we must write KC instead
of C.
Now suppose that instead of the constant voltage V we have an
alternating voltage in the form of V(t)=V[0]sin(<omega>t-<theta>')(in
which <theta>' is a constant value) as the main source of potential
in the series circuit of RLC which its parallel-plate capacitor is
dielectricless. In such a case we have
dI 1
V[0]sin(<omega>t-<theta>')=RI+L-- + --<integral from t =
dt 2C
0 to t>I(t)dt, (16)
and then
Ld{2}I/dt{2}+RdI/dt+I/(2C)=V[0]<omega>cos(<omega>t-<theta>').
Particularl solution of this equation is
a[1]cos(<omega>-<theta>'-<thata>[1]) in which
a[1]=V[0]/((1/(2C<omega>)-L<omega>){2}+R{2}){1/2} (17)
and
<theta>[1]=cot{-1}((1/(2<omega>C)-L)/R). (18)
Since solution of its corresponding homogeneous equation is
a.exp(-Rt/(2L))cos(<omega>[n]t-<theta>), the general siolution of
this equation is
I=a.exp(-Rt/(2L))cos(<omega>[n]t-<theta>)+
a[1]cos(<omega>t-<theta>'-<theta>[1]) (19)
with the two arbitrary constants a and <theta> (of course assuming
that R/(2L)<(2LC){-1/2}).
We suppose that we have I=0 in t=0 and from Eq.(16) we have
dI/dt=-V[0]sin<theta>'/L in this moment. Having these initial values
we can obtain a and <theta>, but since the first term of the right-
hand side of Eq.(19) is transient, this act is of no importance for
us. (Nevertheless, they should be obtained by solving the system of
( acos<theta>=-a[1]cos(<theta>'+<theta>[1])
<
( asin<theta>=-(1/<omega>[n])(V[0]sin<theta>'/L+(Ra[1]/(2L))cos(<
theta>'+<theta>[1])+a[1]<omega>sin(<theta>'+<theta>[1]))
for a and <theta>.)
Now, as before, having the form of current (Eq.(19)) we obtain
voltage drop in the capacitor:
1/(2C)<integral from t=0 to t>(a.exp(-Rt/(2L))cos(<omega>[n]t-<
theta>)+a[1]cos(<omega>t-<theta>'-<theta>[1]))dt=a(exp(-Rt/(2L))(<
omega>[n]Lsin(<omega>[n]t-<theta>)-R/2cos(<omega>[n]t-<theta>))+<
omega>[n]Lsin<theta>+R/2cos<theta>)+a[1]/(2<omega>C)(sin(<omega>t-
<theta>'-<theta>[1])+sin(<theta>'+<theta>[1])) (20)
And now, as before, if the space between the two plates of the
capacitor is to be filled by a linear dielectric with the
permittivity <epsilon> (without any change in the plates'
configuration), in order to obtain the potential difference that we
must attribute to the dielectric as a source of potential in the
circuit, according to Eq.(5) we should multiply the negative of the
potential drop in the capacitor (20) by (1-<epsilon>[0]/<epsilon>).
We then must add this source to the initial alternating source and
equate the sum to the right-hand side of Eq.(16):
V[0]sin(<omega>t-<theta>')+a(1-<epsilon>[0]/<epsilon>)(exp(-Rt/(2L))(
R/2cos(<omega>[n]t-<theta>)-<omega>[n]Lsin(<omega>[n]t-<theta>))-<
omega>[n]Lsin<theta>-R/2cos<theta>)-a[1]/(2<omega>C)(1-<epsilon>[0]/<
epsilon>)(sin(<omega>t-<theta>'-<theta>[1])+sin(<theta>'+<theta>[1]))
=RI+LdI/dt+1/(2C)<integral from t=0 to t>I(t)dt
With one time differentiation of this equation with respect to time
the following equation will be obtained:
Ld{2}I/dt{2}+RdI/dt+1/(2C)I=V[0]<omega>cos(<omega>t-<theta>')-a(1-<
epsilon>[0]/<epsilon>)(R{2}/(4L)+<omega>[n]{2}L)exp(-Rt/(2L))cos(<
omega>[n]t-<theta>)-a[1]/(2C)(1-<epsilon>[0]/<epsilon>)cos(<omega>t-
<theta>'-<theta>[1]) (21)
For obtaining the particular solution of this equation we must add up
particular solutions of the following equations (for reason see
Differential Equations with Application and Historical Notes by
Simmons, McGraw-Hill Inc., 1972):
Ld{2}I/dt{2}+RdI/dt+1/(2C)I=V[0]<omega>cos(<omega>t-<theta>') (22)
Ld{2}I/dt{2}+RdI/dt+1/(2C)I=-a(1-<epsilon>[0]/<epsilon>)(R{2}/(4L)+
<omega>[n]{2}L)exp(-Rt/(2L))cos(<omega>[n]t-<theta>) (23)
Ld{2}I/dt{2}+RdI/dt+1/(2C)I=-a[1]/(2C)(1-<epsilon>[0]/<epsilon>)cos(
<omega>t-<theta>'-<theta>[1]) (24)
We then must add the obtained particular solution to the general
solution of the corresponding homogeneous equation to obtain the
general solution of Eq.(21).
Both the general solution of the homogeneous equation and particular
solution of Eq.(23) are (trigonometric) multiples of exp(-Rt/(2L)),
thus these two terms in the general solution of Eq.(21) are transient
and then unimportant for us. Thus, for obtaining the nontransient
part of the general solution of Eq.(21) we should obtain the
particular solution of the equations (22) and (24) and then add
them up.
Particular solution of Eq.(22) is
(2V[0]C<omega>/(4L{2}C{2}<omega>{4}+4(R{2}C-L)C<omega>{2}+1))((1-2LC<
omega>{2})cos(<0mega>t-<theta>')+2RC<omega>sin(<omega>t-<theta>'))
and particular solution of Eq.(24) is
( -a[1]/(4L{2}C{2}<omega>{4}+4(R{2}C-L)C<omega>{2}+1) )(1-<epsilon>[0
]/<epsilon>)((1-2LC<omega>{2})cos(<omega>t-<theta>'-<theta>[1])+2RC<
omega>sin(<omega>t-<theta>'-<theta>[1])).
If we write the trigonometric terms in the recent solution in terms
of the sine and cosine of the arguments (<omega>t-<theta>') and
<theta>[1], and add up the particular solutions obtained for the
equations (22) and (24), and equate the sum to the expression
a[2]cos(<omega>t-<theta>'-<theta>[2]), and use the equations (17) and
(18), we shall finally obtain:
and
a[2]sin<theta>[2]=4V[0]RC{2}<omega>{2}.(4R{2}C{2}<omega>{2}+(1-2LC<
omega>{2}){2}-2(1-<epsilon>[0]/<epsilon>)(1-2LC<omega>{2}))/(4R{2}C{
2}<omega>{2}+(1-2LC<omega>{2}){2}){2} (26)
We can solve these equations to obtain a[2] and <theta>[2] in order
that the nontransient solution a[2]cos(<omega>t-<theta>'-<theta>[2])
for the circuit current will be obtained unambiguously. The value
which is obtained for the amplitude a[2] from these equations is
2V[0]C<omega>(4R{2}C{2}<omega>{2}+(1/K-2LC<omega>{2}){2}){1/2}
a[2]=--------------------------------------------------------------
4R{2}C{2}<omega>{2}+(1-2LC<omega>{2}){2} (27)
in which K=<epsilon>/<epsilon>[0]. (It is easily seen that for K=1
the same amplitude a[1] presented in Eq.(17) will be obtained from
a[2].)
Now if, as it is thought at present, after inserting the dielectric
between the capacitor's plates its capacitance is to increase to KC
and no more, then we must conclude that the amplitude of the
(nontransient) current is in the same form shown in Eq.(17) except
that KC must be substituted for C in this equation. Namely, the
magnitude of such an amplitude will be:
(V[0]/((1/(2KC<omega>)-L<omega>){2}+R{2}){1/2}=) 2V[0]C<omega>/(4R{2
}C{2}<omega>{2}+(1/K-2LC<omega>{2}){2}){1/2} (=2V[0]C<omega>(4R{2}C{2
}<omega>{2}+(1/K-2LC<omega>{2}){2}){1/2}/(4R{2}C{2}<omega>{2}+(1/K-
2LC<omega>{2}){2})) (28)
A comparison between (27) and (28) shows that their variations with K
is opposite to each other, ie if (27) increases with increase of K,
(28) will decrease with increase of K, and if (27) decreases with
increase of K, (28) will increase with increase of K, and vice versa.
For example on condition that <omega>{2} being greater than or equal
to 1/(2LC) the expression (27) indicates that the current's amplitude
increases by inserting the dielectric, while the expression (28) says
that this amplitude must decrease under the same condition.
Investigating that whether or not experiment shows that provided that
<omega>{2} being greater than or equal to 1/(2LC) current intensity
of the circuit increases by inserting dielectric between the
capacitor's plates is a good test for accepting the theory presented
here and rejecting the current one or vice versa.
To find the resonance frequency of the circuit it is sufficient to
differentiate from the right-hand side of Eq.(27) with respect to
<omega> and then to equate the obtained result to zero and to solve
the obtained equation for <omega>. By doing this act we obtain the
following result for the square of the resonance frequency
<omega>[r]:
2(K-1)+(4(K-1){2}+1){1/2}
<omega>[r]{2}= ------------------------- (29)
2LCK
(It is seen that for K=1, square of the resonance frequency is
1/(2LC) which is just the same result which Eq.(17) predicts for the
square of the resonance frequency. (Reminding of this point is
necessary that as we said we have L=<mu><epsilon>'a'L{*} here.))
Now, let's see what the prediction of the present current belief is
for the resonance frequency of the circuit. It says that since
inserting the dielectric (according to its belief) the amplitude of
the current is V[0]/((1/(2KC<omega>)-L<omega>){2}+R{2}){1/2} (see
Eq.(28)), the square of the resonance frequency will be:
1
<omega>[r]{2}= ---- (30)
2LCK
A simple mathematical try shows that the coefficient of 1/(2LC) in
(29) (ie (2(K-1)+(4(K-1){2}+1){1/2})/K) is an ascending function of
K, while the coefficient of 1/(2LC) in (30) (ie 1/K) is a descending
function of K. Namely, the analysis presented here shows that by
inserting dielectric between the capacitor's plates the resonance
frequency increases, while according to the current belief this
frequency must decrease.
NOTE:
----
That actually whether or not the resonance frequency of the circuit
increases with inserting dielectric between the plates of the
capacitor (without any change in the plates' configuration) is a
quite practical test for establishing the validity of the theory
presented in this article and invalidity of the current belief in
this respect, or vice versa. Recently this experiment has been
performed with a briliant success for the theory presented in this
article showing specifically increase of the resonance frequency when
inserting the dielectric. Here is the report of an electronics
engineer who could not believe the result of his experiments in
this respect:
| Oh, yes, indeed the resonant frequencies do change as
| drastically as you suggest if you put a dielectric with high
| dielectric constant between the parallel plates of a capacitor.
| I've put an example at the end of this posting.
|
| Example of capacitor with high-K dielectric...
| You can buy "disc ceramic" capacitors with about 1.0nF capacitance.
| These are nominally 1cm diameter, with nominally 0.5mm plate
| separation, with dielectric only between the conductive plates.
| The dielectric has a very high dielectric constant. If you resonant
| such a capacitor with, say, a 5mH inductor, you will find its
| resonant frequency will be about 70kHz. You can replace that
| capacitor with one with the same plate size and spacing but air
| dielectric, resulting in roughly 0.5pF capacitance. Then you will
| find that the measured resonant frequency depends on the self-
| resonance of the inductor, because you will be very hard-pressed to
| make a 5mH inductor with self-capacitance as low as 0.5pF. If you
| choose an inductor of, say, 1uH, properly constructed, then you
| might reasonably see the effects of 0.5pF, but now you will be
| dealing with much more awkward (especially if you have limited
| access to good test equipment) resonant frequencies in the hundreds
| of MHz. You will indeed find that the resonant frequency of that
| inductor with the nominal 1.0nF ceramic-dielectric capacitor will
| be on the order of 5MHz. The Q in each case should be high enough
| (with a well-constructed inductor) to give an easily measured
| resonant frequency. I _could_ do the experiment to specifically
| demonstrate the _dramatic_ shift in resonance, and even use other
| dielectrics less extreme, but I feel no need to: as I told you
| before, I _routinely_ design resonant circuits and filters, even
| taking into account the effects of stray capacitance and inductance
| and the resistances of things like circuit board traces where
| appropriate, and within my understanding of the tolerances of the
| parts and the effects of the strays, I'm never surprised. I am
| CERTAINLY never surprised by a resonance shifting higher as I
| increase capacitance, so long as I'm within the practical range of
| the parts I'm using.
|
| Note on 1uH coil: If you make a coil with #18AWG wire, which is
| about 1.0mm diameter, and make that coil with uniformly spaced
| turns, about 2.6cm diameter turns, spaced out 2.5cm total coil
| length, it will have an inductance about 1.0uH, and its first
| parallel self-resonance at about 190MHz. That implies about 0.7pF
| effective self-capacitance. Adding an external 0.5pF capacitance
| would drop the resonant frequency to about 145MHz.
(It is probable that the instrument by which one measures resonance
frequency needs to obtain the capacitance of the capacitor before
calculating the resonance frequency based on the formula
<omega>[r]{2}=1/(2LC). If so, there are two errors in such a
measurement:
1. The process in which the current instruments measure capacitance
of a capacitor is not accurate, because as we explained at the end of
Section III.C (in these instruments) this capacitance is defined
(wrongly) as the charge accumulated on the capacitor divided by the
potential difference between the two conductors of the capacitor.
2. As it has been proven (Eq. (29)), the above formula is not
correct.)
At present dielectric constant is determined in one of the two
following manners:
1. A parallel-plate capacitor is connected to a constant voltage two
times:
(first) when its dielectric is vacuum (or air), and (second) when
its dielectric is the substance under measurement. The geometry of
the conductors remains unchanged. Since it is supposed that the
capacitance of the capacitor is increased with the dielectric, the
ratio of the gathered charge in the second state to the gathered
charge in the first state is the dielectric constant of the
substance.
2. Instead of retaining the voltage unchanged, we put a unique charge
on the capacitor two times: (first) when its dielectric is vacuum,
and (second) when its dielectric is the substance under
measurement. The geometry of the conductors remains unchanged.
Since it is supposed that the capacitance of the dielectric is
increased with the dielectric, the ratio of the potential
difference between the plates in the first state to the potential
difference between them in the second state is the dielectric
constant of the substance.
Certainly the above methods don't give the dielectric constant
according to the contents of this article in which it has been proven
that the capacitance of a capacitor depends only on the geometry of
the conductors and not also on its dielectric. Thus, what are in fact
those measured as dielectric constant by these methods?
At the beginning of Section III.D it has been proven that the charge
accumulated on a parallel-plate capacitor with area A and plates'
separation d and a linear dielectric with the permittivity <epsilon>,
which the potential difference between its plates is V, is: (Eq. (6))
Q = (2-<epsilon>[0]/<epsilon>)<epsilon>[0]A/d V
What has been done in the first method above is in fact calculating
(2-<epsilon>[0]/<epsilon>)<epsilon>[0]A/d V
------------------------------------------- =
<epsilon>[0]A/d V
2-(<epsilon>[0]/<epsilon>) = 2-(1/K)
as the dielectric constant (K). And what has been done in the second
method is in fact calculating
Q
---------------
<epsilon>[0] A/d
--------------------------------------------- =
Q
-----------------------------------------
(2-<epsilon>[0]/<epsilon>)<epsilon>[0]A/d
2-(<epsilon>[0]/<epsilon>) = 2-(1/K)
as the dielectric constant (K). Anyhow, what is at present considered
as K is indeed 2-(1/K). Since ideally K at least is 1 (for vacuum)
and at most is infinity, what is measured as K at present (ie in fact
2-1/K) can be at least 1 and at most, for the best linear
dielectrics, 2.
Reviewing different tables of the dielectric constants in different
texts shows that these constants scarcely exceed 2 or 3 for the best
linear dielectrics, although for some materials this constant even
exceeds 100. (For example it is 2 for paper, paraffin, mineral oil,
indian rubber, ebonite, benzene, teflon, mica, wood, polyethylene,
liquid CCl4 and CS2 (while for liquid O2 and A is 1.5), ...., but is
suddenly near 100 for water.) In addition, there is notable
difference between the constants registered in different texts. It
seems that there is a drastic uncertainty in the results obtained by
the above-mentioned methods (esp when there is a huge difference
from 2). The cause of this uncertainty should be searched (maybe in
the nonlinearity of the dielectrics), but anyway it can be said that
for almost all of the best linear dielectrics (the permittivity of
which can be taken infinity) the constant registered as (wrong)
dielectric constant, as the above reasoning predicts, is about 2
(indeed the (true) dielectric constant of these good dielectrics is
infinity).
Separate from the theory, now let's prove physically that the
above-mentioned ratio of the gathered charges in the method 1 can not
exceed 2: Suppose that a parallel-plate capacitor, connected to a
constant voltage, when is dielectricless, gathers a charge Q. In this
state suppose we insert an ideal linear dielectric, with an infinite
permittivity, between its plates. When this linear dielectric is set
in the field between the plates it begins to become polarized, ie by
ordering the molecular electric dipoles of the dielectric the charges
of the capacitor begin to be canceled, but the potential source to
which the capacitor is connected compensates for the canceled charges
of the capacitor in such a manner that the dielectric is always in a
constsant electric field which its presence is essential for the
linear dielectric to maintain the polarization. (Notice the relation
~P=(<epsilon>-<epsilon>[0])~E for a linear dielectric in which when
<epsilon>=<infinity> we shall have ~E=~0 where ~E is arising from
both polarized charge and that part of the conductors' charges which
are gathered by these polarized charges. (If we wish to consider ~E
as the field arising from the polarized charges and the whole charge
of the capacitor, then the <epsilon> won't be infinity (because
indeed in such a case it is not related to only the dielectric but
the role of the conductors (or capacitor) has been added to it).))
Thus, the dielectric can attract, onto the capacitor, some additional
charge at most equal to the original charge of the capacitor (related
to when there is no dielectric). Then, the ratio of the charge of the
capacitor with dielectric to one without dielectric is at most 2Q/Q=2
(and at least is Q/Q=1 when there is no order for the molecular
electric dipoles even in the electric field between the plates).
Surely there are some persons reckoning these reasonings as fantasy.
The following material may help them not to think so:
The current usual prediction for the resonance frequency of a series
RLC circuit which its dielectricless capacitor is parallel-plate,
when its capacitor is filled with a linear dielectric having
dielectric constant K, is that square of the resonance frequency
drops by 1/K. If, in addition, the limitation of 2 is also a fantasy
for K in the above-mentioned 1/K, and K, depending on the used
dielectric, can take amounts like 20, 30, 40, 80, 100, 200, 300, ...,
then we should conclude that the resonance frequency becomes almost
zero when these dielectrics are used (since eg square root of 1/300
is about zero). A question: Is this the case or not? And in
principle, is this reasonable? But as we saw in this article the
coefficient by which the square of the resonace frequency, when the
dielectric is inserted, increases is:
2(K-1)+(4(K-1){2}+1){1/2}
------------------------- = 2(1-(1/K))+(4-(8/K)+(5/K{2})){1/2}
K
It is seen when K=1, square of the resonance frequency is 1, and when
K is infinity, square of the resonance frequency is 4. This means
that by inserting a linear dielectric we expect that the resonance
frequency will become double at most (when we have an ideal linear
dielectric with infinite permittivity). That ratio of the resonance
frequency with dielectric to the one without dielectric is a number
between 1 and 2 is analogous to that the ratio of the charge gathered
in the capacitor with dielectric to the one without dielectric is a
number between 1 and 2.
III.E. Again parallel-plate capacitor as another test
-----------------------------------------------------
Now we obtain the electrostatic potential energy of the
parallel-plate capacitor mentioned at the end of the section III.B.
by two methods. First, using the relation
U=1/2C(<cap. delta><phi>)[Q]{2} we obtain
U=1/2(<epsilon>[0]A/d)(<cap. delta><phi>)[Q]{2}.
In the second method we use the relation (2), ie U=1/2<integral over
V[h]>~D.~E[Q]dv in which ~E[Q] is the field arising from Q and -Q
(and not also from the polarization charges). We have the following
relation:
~D = <epsilon>~E = <epsilon>(~E[Q]+~E[P]) =
<epsilon>~E[Q]+<epsilon>~E[P] (31)
in which ~E[P] is the field arising only from the polarization
charges of the dielectric set between the two plates. Let's obtain
~E[P] in terms of ~D. Suppose that ~P is the polarization of the
dielectric and ^n is the unit vector in the direction of ~E.
~P.(-^n) is the polarization charge surface density formed adjacent
to the plate bearing the (positive) charge Q, and ~P.^n is the
polarization charge surface density formed adjacent to the plate
bearing the charge -Q. Since ~P=(<epsilon>-<epsilon>[0])~E, we have
~P.(-^n)=(<epsilon>[0]-<epsilon>)E and
~P.^n=(<epsilon>-<epsilon>[0])E which the first is negative and the
second is positive obviously. Then, the electrostatic field arising
from these (polarization) charges in the dielectric is
~P.^n <epsilon>[0]-<epsilon>
~E[P] = ------------(-^n) = ----------------------~E (32)
<epsilon>[0] <epsilon>[0]
and since ~D=<epsilon>~E we have
~E[P]=(<epsilon>[0]-<epsilon>)/(<epsilon>[0]<epsilon>)~D. Combining
this result with the relation (31) yields
<epsilon>[0]-<epsilon>
~D=<epsilon>~E[Q]+ ----------------------~D ==>
<epsilon>[0]
~D=<epsilon>[0]~E[Q]. (33)
Therefore, we have U=1/2<integral over V[h]>~D.~E[Q]dv=1/2<integral
over V=Ad><epsilon>[0]E[Q]{2}dv=1/2<epsilon>[0]AdE[Q]{2}=1/2<
epsilon>[0]Ad((<cap. delta><phi>)[Q]/d){2}=1/2(<epsilon>[0]A/d)(<cap.
delta><phi>)[Q]{2}, which is the same result obtained in the first
method.
Now we proceed to another case. Consider the following figure.
x
<----------->
______________________________
---------------------- /^\
......................| |
......................| | d
______________________| \|/
-----------------------------"
<---------------------------->
l
Figure. A linear dielectric block is pulled into the space between
the plates of a parallel-plate capacitor having the constant
electrostatic potential difference (<cap. delta><phi>)[Q].
The (unshown) width of the plates is w. A linear dielectric block is
along the l-dimension and only the length x is between the plates.
Potential difference between the two plates is constant (equal to
(<cap. delta><phi>)[Q]; we proved this fact beforehand). It is clear
that the charges on that part of a plate of the capacitor which is in
the empty part of the capacitor exert an attractive force on the
polarization charges adjacent to that plate and a repulsive force on
the polarization charges adjacent to the other plate, while the
charges on the empty part of the other plate act a similar work, and
the resultant force of all of these forces is an inward force along
the l-dimension magnitude of which must approach zero when d
approaches zero. Now let's try to obtain this force from the energy
method. First of all, according to what said so far, it is obvious
that with the dielectric displacement the electrostatic potential
energy of the capacitor being only of the capacitor charge (Q and -Q)
does not alter. Thus, only the electrostatic potential energy of the
dielectric and its alteration must be considered.
We know that the surface density of polarization charge of the
dielectric in the capacitor is +P or -P and then the electrostatic
field arising from it is ~E[P]=-~P/<epsilon>[0]. On the other hand,
by using each of the relations (1) and (2) we obtain a unique
expression for the electrostatic potential energy of only the
polarization charges of the dielectric:
(1) ==> U[P]=1/2<integral over V[h]><rho><phi>dv=1/2((-Pd/(2<
epsilon>[0])+0)(Q[P])+(-(-Pd)/(2<epsilon>[0])+0)(-Q[P]))=Pd/(2<
epsilon>[0])Q[P]=Pd/(2<epsilon>[0])P(wx)=P{2}d/(2<epsilon>[0])wx
considering that the potential arising from an infinite charged plate
with the surface charge density <sigma> is -<sigma>/(2<epsilon>[0])d
at the (nonnegative) distance d from the plate, and
(2) ==> U[P]=1/2<integral over V[h]><epsilon>[0]~E[P].~E[P]dv=<
epsilon>[0]/2<integral over V[h]>~E[P]{2}dv=<epsilon>[0]/2<integral
over V[h]>(P/<epsilon>[0]){2}dv=<epsilon>[0]/2 P{2}/<epsilon>[
0]{2} wxd=P{2}d/(2<epsilon>[0])wx.
We have also ~P=-<epsilon>[0]~E[P] from ~E[P]=-~P/<epsilon>[0]. If in
addition we apply the relations (32), (33) and (31), we shall obtain
~P=-<epsilon>[0]~E[P]=(<epsilon>-<epsilon>[0])~E=(<epsilon>[0](<
epsilon>-<epsilon>[0])/<epsilon>)~E[Q] and consequently
U[P] = P{2}d/(2<epsilon>[0]) wx = <epsilon>[0]E[P]{2}d/2 wx =
(<epsilon>-<epsilon>[0]){2}E{2}d/(2<epsilon>[0]) wx =
<epsilon>[0](<epsilon>-<epsilon>[0]){2}E[Q]{2}d/(2<epsilon>{2}) wx.
Since with displacement of the dielectric only x is changed,
dU[P] = <epsilon>[0]E[P]{2}d/2 wdx = (<epsilon>-<epsilon>[0]){2}E{2
}d/(2<epsilon>[0]) wdx = <epsilon>[0](<epsilon>-<epsilon>[0]){2}E[Q
]{2}d/(2<epsilon>{2}) wdx. (34)
We know that the above mentioned force pulling the dielectric into
the capacitor performs some work on the dielectric which, according
to the conservation law of energy, this work must be conserved in
some manner. By pulling inward, this force not only causes forming
more polarization charges, but also alters (and in fact increases)
the kinetic energy of the dielectric block. Thus, the above mentioned
work is conserved both as the electrostatic potential energy of the
formed polarization charges and as the alteration of the kinetic
energy. We show this work as dW and the alteration of the
electrostatic potential energy as dU[P] and the alteration of the
kinetic energy as dT. Therefore, we have:
dW=dU[P]+dT & dW=F[x]dx ==> F[x]dx=dU[P]+dT )
F[x]dx = <epsilon>[0](<epsilon>-<epsilon>[0]){2}wd/(2<epsilon>{2}) E[
Q]{2}dx + dT = <epsilon>[0](<epsilon>-<epsilon>[0]){2}/(2<epsilon>{2
}) w (<cap. delta><phi>)[Q]{2}/d dx + dT (35)
It is obvious that if in an especial case we have dT=0 then we shall
have
F[x] = <epsilon>[0](<epsilon>-<epsilon>[0]){2}/(2<epsilon>{2}) E[Q]{
2}wd = <epsilon>[0](<epsilon>-<epsilon>[0]){2}/(2<epsilon>{2}) w (<
cap. delta><phi>)[Q]{2}/d = 1/2 ((K-1){2}/K{2})<epsilon>[0]E[Q]{2}wd
(36)
(It is seen that as was predicted beforehand, this force will
approach zero if d approaches zero.)
Observing the present current mistakes (including what we saw about
the capacitance and electrostatic potential energy of a capacitor) we
see the following relation instead of Eq.(35) in the present books of
Electricity and Magnetism or Electromagnetism:
F[x]dx = 1/2 (<epsilon>-<epsilon>[0]) w (<cap. delta><phi>)[Q]{2
}/d dx = 1/2 (K-1)<epsilon>[0]E[Q]{2}(wd)dx (37)
where it is supposed that (<cap. delta><phi>)[Q] remains constant.
(How? I don't know (!) because even by connecting the two plates to a
battery the voltage of the battery is equal to the sum of
<cap. delta><phi>)[Q] and the potential difference caused by the
dielectric (also see the beginning part of Section III.D).)
And also by mistake the following general result (instead of the
especial result (36)) is inferred from the relation (37):
F[x] = 1/2 (<epsilon>-<epsilon>[0]) w (<cap. delta><phi>)[Q]{2}/d =
1/2 (K-1)<epsilon>[0]E[Q]{2}wd
Practical comparison of the above relations for experimental testing
of the truth of Eq.(35) should be possible preparing ideal conditions
and regarding fringing effects at the edges of the capacitor and
considering the real value of K (see Note section in the previous
section).
IV. Two kinds of potential difference for a capacitor
-----------------------------------------------------
At present in all the textbooks of Electricity and Magnetism wherever
electrostatic potential difference between the two conductors of a
capacitor is concerned if to its producer source, ie the battery, has
been pointed implicitly or explicitly, it is shown or stated
implicitly or explicitly that this electrostatic potential difference
is equal to the potential difference between the two poles of the
battery that has charged the capacitor. But we now shall prove easily
that the electrostatic potential difference between the two
conductors of a capacitor is twofold compared with the potential
difference between the two poles of the battery which has charged it.
Suppose that the potential difference between the two poles of the
battery is <cap. delta><phi> and the electrostatic potential
difference between the two conductors of the capacitor is
<cap. delta><phi>'. It is obvious that if the charge collected on the
capacitor is Q, the battery has transmitted it through itself under
the potential difference <cap. delta><phi> and then has given it an
energy equal to Q<cap. delta><phi>. But Eq.(3) states that the
electrostatic potential energy of the capacitor is
1/2 Q<cap. delta><phi>'. According to the conservation law of energy
then we must have Q<cap. delta><phi> = 1/2 Q<cap. delta><phi>' or
<cap. delta><phi>' = 2<cap. delta><phi>.
A simple physical reasoning shows this fact too: When stating that
the electrostatic potential energy between the two conductors of the
capacitor is <cap. delta><phi>' we mean that supposing that all the
capacitor charges are fixed, if supposedly a one-coulomb external
point charge starts to move from one of the two conductors under the
influence of the electrostatic force of the capacitor until it
reaches the other conductor, the work performed on it by this force
will be <cap. delta><phi>', without any change in the charges on the
conductors. But if we suppose that the magnitude of the charge on
each conductor of the above capacitor is one coulomb and it is
possible that charges separate from a conductor and moving in the
space between the two conductors reach the other conductor, then the
total work performed on this one-coulomb charge by the electrostatic
force of the capacitor will not be certainly equal to
<cap. delta><phi>', because with each transmission of some part of
the charge, magnitude of the charge on each conductor (and
consequently the electrostatic field between the two conductors) is
decreased and does not remain unchanged as before. The above argument
shows that this work will be 1/2 <cap. delta><phi>', because this is
in fact the same work done by the battery for charging the capacitor
being conserved in the capacitor in the form of potential energy
which is being released now. We show this matter in an analytical
manner too: Suppose that our capacitor is a parallel-plate one and
its charge is Q. If a separate Q-coulomb charge travels from a plate
to the other one, the work performed on it will be
Q d
QEd = Q ---------- d = ---------- Q{2}, (38)
<epsilon>A <epsilon>A
while for calculating the work performed on the charge of the
capacitor itself being plucked bit by bit traveling from a plate to
the other one, we should say that the work performed on a
differential charge -dQ (note that dQ is negative), similar to (38),
is
Q+dQ d
(-dQ)Ed = -dQ ---------- d = - ---------- (Q+dQ)dQ.
<epsilon>A <epsilon>A
Sum of these differential works is
<integral from Q=Q to 0>-d/(<epsilon>A) (Q+dQ)dQ =
1/2 d/(<epsilon>A) Q{2}
which is half of the previous work (shown in Eq.(38)).
Thus we should expect to have 2<cap. delta><phi>=d/(<epsilon>[0]A)Q
when a battery with the potential difference <cap. delta><phi> has
charged a parallel-plate capacitor, while hitherto it is thought that
<cap. delta><phi>=d/(<epsilon[0]>A)Q. Since all the parameters of
both the recent relations are measurable (<cap. delta><phi> by
voltmeter), the truth or untruth of each can be tested practically.
We should notice a point. When connecting a voltmeter to the two
conductors of a charged capacitor, it measures <cap. delta><phi> not
<cap. delta><phi>', because its operation is based on passing a weak
electric current through a circuit in the instrument and measuiring
the potential difference between the two ends of the circuit; and
passing of a current means in fact the same being plucked of the
capacitor charge bit by bit from the conductors, and then the
voltmeter measures <cap. delta><phi>.
We should also say that there is no need that in the existent
calculations of electrical cicuits the potential difference of each
capacitor to be made double, because in these calculations the same
<cap. delta><phi> has been in fact intended not <cap. delta><phi>',
because the electric current passing through the circuit including
the capacitor is the same process of gradual loading and unloading of
the capacitor, not passing of charge through the space between the
two conductors of the capacitor retaining the capacitor charge
unchanged. Therefore, it is proper to give <cap. delta><phi> a name
other than the electrostatic potential difference which is the name
of <cap. delta><phi>'. Let's call it (ie <cap.delta><phi>) as
circuital potential difference of the capacitor. In this manner when
it is necessary to apply closed circuit law we must consider just
this circuital potential difference when passing the capacitor not
its electrostatic potential difference.
Now, again, consider a closed circuit of a battery, with the
potential difference <cap. delta><phi>, and a capacitor, with the
capacitance C. Let's investigate the usual method of analysis of RC
(or generally RLC) circuits and see what the difficulty is in it.
Without missing anything we suppose that the circuit has no
resistance (ie R=0). When a differential electric charge dQ passes
through the battery causes a differential change in the electrostatic
energy of the capacitor. In the first instance it seems that when the
differential charge dQ passes through the battery it gains the
differential energy <cap. delta><phi>dQ which, as a rule according to
the conservation law of energy, this same energy must be conserved in
the capacitor in the form of d(Q{2}/(2C)), and then
<cap. delta><phi>dQ=d(Q{2}/(2C)) ==> <cap. delta><phi>dQ=(Q/C)dQ ==>
<cap. delta><phi>=Q/C ==> <cap. delta><phi>-Q/C=0
which is just the same result which we could obtain from the
closed circuit law by traveling one time round the circuit if the
potential difference between the two conductors of the capacitor was
taken electrostatic potential difference, ie <cap. delta><phi>'=Q/C,
not circuital potential difference, ie Q/(2C)! The difficulty is that
the relation <cap. delta><phi>dQ=d(Q{2}/(2C)) is not necessarily
true, for this reason: If we had a mathematical relation, in the form
of an equality, between the energy given by the battery and the
electrostatic energy stored in the capacitor (ie Q{2}/(2C)), we
could differentiate from each side of the equality relation and
understand that the change of energy in the capacitor in the form of
d(Q{2}/(2C))(=Q/CdQ) is exactly arising from what the differential
change in the battery. But since there is no such a relation, we
cannot necessarily infer that change of energy in the capacitor in
the form of Q/CdQ is arising from passing of the charge dQ through
the battery and consequently from differential change of
<cap. delta><phi>dQ in the energy given by the battery, because eg by
writing Q/(2C)(2dQ) instead of Q/CdQ we can claim that this change of
energy in the capacitor is arising from passing of the charge 2dQ
through the battery and consequently from differential change of
<cap. delta><phi>(2dQ) in the energy given by the battery (ie
<cap. delta><phi>(2dQ)=Q/(2C)(2dQ)), and the previous reasonings
showed that incidentally this is the case.
Thus, we should bear in mind that in the analysis of RLC circuits we
must attribute only the circuital potential difference, ie Q/(2C),
not the electrostatic potential difference, ie Q/C, to the capacitor
of the circuit. (Refer to the discussion of RLC circuit in this
article.) Also it is notable that since current instruments indeed
measure capacitance of a capacitor using the formula
C=Q/<cap. delta><phi>' while taking <cap. delta><phi> instead of
<cap. delta><phi>', they give us in fact Q/<cap. delta><phi> =
Q/(<cap. delta><phi>'/2) = 2(Q/<cap. delta><phi>') = 2C as the
capacitance; in other words what they measure as capacitance is in
fact double the capacitance. In this manner what we see as 2C in the
equations (29) and (30), for example, is the same amount our current
instruments give as the capacitance.
It is necessary to note the influence that inattention to the
above-mentioned problem (ie difference between <cap. delta><phi> and
<cap. delta><phi>') has on the results of the experiments of Millikan
and Thomson for determining charge and mass of the electron (and
similarly positive ions).
In the experiment of Millikan the electric charge of each charged oil
droplet is proportional to k/E in which k is the coefficient of
proportion of Stokes and E is the electrostatic field between the two
plates of the parallel-plate capacitor used in the experiment. As we
know E between the two plates of a parallel-plate capacitor is equal
to the electrostatic potential difference <cap. delta><phi>' divided
by the distance d between the two plates. So the charge of each
droplet is proportional to k/<cap. delta><phi>'. But for practical
determination of <cap. delta><phi>' the potential difference read by
the voltmeter connected to the plates of the capacitor is considered
erroneously, while as we said this potential difference,
<cap. delta><phi>, which we called it as circuital potential
difference, is half of <cap. delta><phi>'. In other words as a rule
the quantity so far recognized as the charge of a droplet should be
two times larger than the real charge of the droplet and then the
electron's charge obtained from the numerous repetitions of the
experiment of Millikan should be really half of what is at present
accepted as the charge of electron.
But this is not the case because the experiment of Millikan plainly
lacks sufficient accuracy (and a tolerance up to half of the real
amount seems natural for it because certainly it is unlikely that
the electrons are added or deducted only one by one). In fact it
seems that the results of this experiment have been adapted in some
manner for being in conformity with the results of the exact
experiment of determination of electric charge of electron by X-ray.
(As we know in this experiment the wavelength of X-ray can be
determined by its diffraction via a diffraction grating with quite
known specifications, and then having this wavelength and Bragg's
equation and analyzing the diffraction of the ray via a crystal
lattice the lattice spacing, d, of the crystal can be determined;
thereupon considering the molecular mass and crystal density
Avogadro's number N[0] can be calculated with sufficient accuracy and
using it in the formula F=N[0]e, in which F is the Faraday constant
and e is the charge of electron, e can be obtained which is the same
that has been accepted at present as the charge of the electron.)
In the experiment of Thomson too, for evaluation of q/m related to
the charge and mass of the electron in the cathodic ray, this
quantity, ie q/m, is obtained proportional to the electrostatic field
E between the two plates of the parallel-plate capacitor through
which the cathodic ray passes. But again for practical determination
of E the above-mentioned error is repeated and while E is really
equal to <cap. delta><phi>'/d the amount read on the voltmeter,
<cap. delta><phi>, (which is in fact equal to 1/2<cap. delta><phi>')
is set instead of <cap. delta><phi>'. In other words as a rule the
quantity hitherto considered as q/m of the electron (in the
experiment of Thomson) should be half of its real amount. Then, to
obtain the real value of q/m we must multiply the value accepted
presently as q/m by 2.
But here we should say that it seems that this experiment (or any
other similar one) is not accurate in determining q/m of electron or
positive ions since in it a shooting motion has been assumed for the
electron in the cathodic ray (or for the positive ion in the positive
ray), while as explained in detail in the 12th article of this book
we must consider for it a longitudinal wave motion in the gas medium
existent in the tube without any charge transferring, and it seems
that such a wave motion, although has many similarities with the
shooting motion, is not exactly the same shooting motion and has
difference with it. Thus, it is necessary to doubt what has been
accepted as the mass of electron.
Hamid V. Ansari
My email address: ansari18109<at>yahoo<dot>com
The contents of the book "Great mistakes of the physicists":
0 Physics without Modern Physics
1 Geomagnetic field reason
2 Compton effect is a Doppler effect
3 Deviation of light by Sun is optical
4 Stellar aberration with ether drag
5 Stern-Gerlach experiment is not quantized
6 Electrostatics mistakes; Capacitance independence from dielectric
7 Surface tension theory; Glaring mistakes
8 Logical justification of the Hall effect
9 Actuality of the electric current
10 Photoelectric effect is not quantized
11 Wrong construing of the Boltzmann factor; E=h<nu> is wrong
12 Wavy behavior of electron beams is classical
13 Electromagnetic theory without relativity
14 Cylindrical wave, wave equation, and mistakes
15 Definitions of mass and force; A critique
16 Franck-Hertz experiment is not quantized
17 A wave-based polishing theory
18 What the electric conductor is
19 Why torque on stationary bodies is zero
A1 Solution to four-color problem
A2 A proof for Goldbach's conjecture
----
1. See NOTE section at the end of Section III.D.
2. I recommend you not to forget to study the last section (IV) of
this article even as an independent part separate from the other
sections. There you can see interesting material about the
experiments for determination of charge and mass of the electron.
We use the following special terminology in this article:
{} indicates superscript (including the power).
[] indicates subscript.
~A means the vector A.
^a means the unit vector a.
<four> means 4.
We show integral around a closed space as <circulation>.
In a capital Greek letter, the word "cap." is written.
6 Mistakes in Electrostatics;
Ed 01.12.31 ---------------------------
Dreadful consequences in Modern Physics
---------------------------------------
Abstract
--------
It is shown that there exists a uniqueness theorem, stating that the
charges given to a constant configuration of conductors take a unique
distribution, which contrary to what is believed does not have any
relation to the uniqueness theorem of electrostatic potential. Using
this thorem we obtain coefficients of potential analytically. We show
that a simple carelessness has caused the famous formula for the
electrostatic potential to be written as U=1/2<integral>~D.~Edv while
its correct form is U=1/2<integral>~D.~E[<rho>]dv in which ~E[<rho>]
is the electrostatic field arising only from the external charges not
also from the polarization charges.
Considering the above-mentioned material it is shown that, contrary
to the current belief, capacitance of a capacitor does not at all
depend on the dielectric used in it and depends only on the
configuration of its conductors. We proceed to correct some current
mistakes resulted from the above-mentioned mistakes, eg electrostatic
potential energy of and the inward force exerted on a dielectric
block entering into a parallel-plate capacitor are obtained and
compared with the wrong current ones.
It is shown that existence of dielectric in the capacitor of a
circuit causes attraction of more charges onto the capacitor because
of the polarization of the dielectric. Then, in electric circuits we
should consider the capacitor's dielectric as a source of potential
not think wrongly that existence of dielectric changes the
capacitor's capacitance. Difference between these two understandings
are verified completely during some examples, and some experiments
are proposed for testing the theory. For example it is shown that
contrary to what the current theory predicts, resonance frequency of
a circuit of RLC will increase by inserting dielectric into the
capacitor (without any change of the geometry of its conductors).
It is also shown that what is calculated as K (dielectric constsant)
is in fact 2-(1/K).
It is also shown that contrary to this current belief that the
electrostatic potential difference between the two conductors of a
capacitor is the same potential difference between the two poles of
the battery which has charged it, the first is twofold compared with
the second. We see the influence of this in the experiments performed
for determination of charge and mass of the electron.
I. Introduction
---------------
In the current electrostatic discussions it is stated that a solution
of Laplace's equation which fits a set of boundary conditions is
unique, and while this matter has not been proven in the case that
these boundary conditions are the charges on the boundaries, the
known charges on the boundaries are taken as boundary conditions.
First section of this article solves this problem after which obtains
the coefficient of potential, while in the current electromagnetic
books these coefficients are obtained by using the above mentioned
unproven generalization of the boundary conditions which
incorrectness of this way is also shown.
The relation U=1/2<integral over V>~D.~Edv for the electrostatic
potential energy of a system is a quite familiar equation to every
physicist, but a careful scrutiny shows an existent undoubted mistake
in this equation. This mistake is easily arising from this fact that
in the process of obtaining this equation, while accepting that
<del>.~D=<rho> where <rho> is the external electric charge density,
it is forgotten that in the primary equation of the electrostatic
potential energy of the system the potential arising only from this
<rho>, <phi>[<rho>], not also from the polarization charges be taken
into account resulting in considering ~E (obtained from -<del><phi>)
instead of ~E[<rho>] (obtained from -<del><phi>[<rho>]) which is the
electrostatic field arising only from <rho> not also from the
polarization. This careful scrutiny is presented in the third section
of this article. A great part of this section proceeds to some
consequences of this same mistake including this current belief that
the capacitance of a capacitor depends on its dielectric, while we
shall prove that this is not at all the case and it depends only on
the form of the configuration of the conductors of the capacitor.
To another much simple and obvious current mistake is paid in the
last section: We connect a battery, which the potential difference
between its poles is <cap. delta><phi>, to the two plates of an
uncharged capacitor until it will be charged. Then, what is the
electrostatic potential difference between the plates of the charged
capacitor? All the current literature on the subject answer that this
electrostatic potential difference is the same potential difference
between the poles of the battery, <cap. delta><phi>, while this
is not the case and is equal to 2<cap. delta><phi>.
As it is seen, the above current mistakes some of which being
fundamental are totally in bases of the subject of Electromagnetism,
and cannot be ignored, because not only are much widespread and
taught in all the universities but also some of them are basis for
some subsequent deductions in other branches of physics. This matter
shows that in the progress of physics the attention should not be
only to its rapidity but also to its profundity, otherwise, as in the
case of this article, sometimes some of the obvious mistakes remain
hidden from the physicists' view yielding probably very other wrong
consequences.
II. Another uniqueness theorem in Electrostatics
------------------------------------------------
II.A. Uniqueness theorem of charge distribution in conductors
-------------------------------------------------------------
In solving electrostatic problems there is a uniqueness theorem that
distinctly states that when the electrostatic potential or the normal
component of its gradient is given in each point of the bounding
surfaces then if the potential is given in at least one point, the
solution of Laplace's equation is uique, and otherwise we may add any
constant to a solution of this equation. Unfortunately, sometimes
negligence is seen in careful applying of the quite clear stated
above boundary conditions. For instance without any reason the
charges of bounding surfaces are taken as boundary conditions in
terms of which the above theorem is applied in obtaining coefficients
of potential of a system of conductors. The reasoning being used is
this (see Foundations of Electromagnetic Theory by Reitz, Milford and
Christy, Addison-Wesley, 1979): "Suppose there are N conductors in
fixed geometry. Let all the conductors be uncharged except conductor
j, which bears the charge Q[j0]. The appropriate solution to
Laplace's equation in the space exterior to the conductors will be
given the symbol <phi>{(j)}(x,y,z) and the potential of each of the
conductors will be indicated by <phi>{(j)}[1], <phi>{(j)}[2], ....,
<phi>{(j)}[j], ....,<phi>{(j)}[N]. Now let us change the charge of
the jth conductor to <lambda>Q[j0]. The function
<lambda><phi>{(j)}(x,y,z) satisfies Laplace's equation, since
<lambda> is a constant; that the new boundary conditions are
satisfied by this function may be seen from the following argument.
The potential at all points in space is multiplied by <lambda>; thus
all derivatives (and in particular the gradient) of the potential are
multiplied by <lambda>. Because <sigma>=<epsilon>[0]E[n], it follows
that all charge densities are multiplied by <lambda>. Thus the charge
of the jth conductor is <lambda>Q[j0] and all other conductors remain
uncharged. A solution of Laplace's equation which fits a particular
set of boundary conditions is unique; therefore we have found the
correct solution, <lambda><phi>{(j)}(x,y,z) to our modified problem.
The conclusion we draw from this discussion is that the potential of
each conductor is proportional to the charge Q[j] of conductor j,
that is <phi>{(j)}=p[ij]Q[j], (i=1,2,...,N) where p[ij] is a
constant which depends only on the geometry."
The fault may be found in this reasoning is arising from the same
incorrect distinction of boundary conditions. This fault is that a
solution to Laplace's equation other than <lambda><phi>{(j)} can be
found such that it can make the charge of the jth conductor
<lambda>fold retaining all other conductors uncharged. This solution
can be <lambda><phi>{(j)}(x,y,z)+c for a non-zero constant c. It is
obvious that its gradient and therefore <sigma>=<epsilon>[0]E[n]
arising from it compared with before are <lambda>fold and then the
charge of the jth conductor will be <lambda>fold while all other
conductors remain uncharged. But this solution is no longer
proportional to the charge of the jth conductor, Q[j], ie we won't
have <phi>{(j)}(x,y,z)=p[ij]Q[j].
In order to clear obviously that the uniqueness theorem of potential
does not include boundary conditions on charges, suppose that there
is an initially uncharged conductor. We then give it some charge. We
want to see when the given charge is definite whether potential
function outside the conductor will or won't be determined uniquely
by this theorem. We say that the given charge distributes itself onto
the surface of the conductor and remain fixed causing that the
potential of the equipotential surface of the conductor to become
specified. With specifying of the conductor potential, potential
function outside the conductor is determined uniquely according to
the theorem. But important for us is knowing that whether form of the
charge distribution onto the conductor surface is uniquely determined
or not. One can say that maybe the charge can take another form of
distribution on the surface causing another potential for the
equipotential surface of the conductor and according to the theorem
we shall have another unique function for the potential outside the
conductor. In a geometric illustration there is not anything to
prevent the above problem for a sharp conductor being solved with
equipotential surfaces concentrated near either the sharp end or the
other end; the charge is concentrated at the sharp end in the first
and at the other end in the second case. Which occurs really is a
matter that must be determined by another uniqueness theorem,
uniqueness theorem of charge distribution, which has no relation to
the uniqueness theorem of potential.
Analytical proof of this theorem is a problem that must be solved.
That this theorem is valid can be understood by some thinking and
visualizing. Separate from inner parts of the conductors consider
external surfaces of the conductors as some conducting thin shells.
Obviously if some charge is to distribute itself in these shells, the
components of the charge, as a result of the repulsive forces, will
take the most distant possible distances from one another, and even
when for instance uncharged conducting shells are set in the vicinity
of charged conducting shells, their conducting (or valence) charges
will be separated in order that like charges take the most distant
and unlike charges take the most neighboring possible distances from
one another. What is clear is that these "most"s indicate to some
unique situation. Therefore we can say that form of the surface
charge distribution is a function of geometrical form of the
conductors and then will be specified uniquely for a definite
configuration of conductors.
II.B. Proportion of charge density to net charge
-----------------------------------------------
Now suppose that for a particular configuration of and definite
amount of charge given to some conductors we can find two
distributions of charge in the conductors in each of which the
resultant electrostatic force on each infinitesimal partial charge
due to other infinitesimal partial charges is outward normal to the
conductor surface and there exists no tangential component for this
force. (Of course these outward normal forces are balanced by surface
stress in the material of the conductors.) Because there is not any
tangential component for the mentioned forces, existence of these two
charge distributions is possible. But because of the same
configuration for the both, the uniqueness theorem of charge
distribution necessitates that the both distribution be the same.
We shall benefit form this matter soon.
We prove that in a constant configuration of some conductors from
which only one has net charge, Q, change of this net charge form Q to
<lambda>Q causes that the surface density in each point of the
conductors' surfaces becomes <lambda>fold: Visualize the constant
situation existent before that Q becomes <lambda>Q. The charges in
the conductors have a unique distribution according to the uniqueness
theorem of charge distribution. In this distribution there exists a
resultant electrostatic force exerted on each infinitesimal partial
surface charge <sigma>da due to other partial charges which is
outward normal to the conductor surface. Suppose that this
distribution becomes nailed up in some manner, ie each partial charge
becomes fixed in its position and no longer has the state of a
conducting free charge (in order that won't probably change its
position as a result of change of the charge). Now suppose each
partial charge becomes <lambda>fold in its position, ie we have for
the new partial charge <sigma>'da=<lambda><sigma>da. Since the
partial charges are nailed up, they are not free to redistribute
themselves on the conductors' surfaces probably. It is obvious that
resultant electrostatic force exerted on a partial charge <sigma>'da
will be still outward normal to the conductor surface, since firstly
this partial charge is <lambda>fold of previous <sigma>da and
secondly each of other partial charges is <lambda>fold of previous
partial charges and then the only change in the resultant force on
<sigma>da will be in its magnitude which becomes <lambda>{2}fold,
while its direction will remain unchanged. Therefore, by changing
each <sigma>da to <lambda><sigma>da we have found a nailed up
distribution for the charges which exerts a resultant force on each
partial surface charge outward normal to the conductor surface, and
furthermore, the only change in the net charges of the conductors is
in the conductor bearing net charge Q previously which now bears
<lambda>Q, and then it is obvious that if the partial charges get
free from the nailed state will retain this distribution. Therefore,
this distribution is a possible one, and according to what said
previously based on the uniqueness theorem, is the same distribution
that really occurs on the conductors' surfaces when the net charge of
the mentioned conductor changes from Q to <lambda>Q.
II.C. Generalization of the uniqueness theorem and of the charge
----------------------------------------------------------------
density proportion to net charge
--------------------------------
In fact, the uniqueness theorem of charge distribution on the
conductors is true in case of a particular configuration of
conductors and a constant (nailed up) charge distribution and a
constant set of linear dielectrics in the space exterior to the
conductors, ie in such a case a charge given to the conductors causes
a unique charge distribution on their surfaces. The truth of this
theorem can be found out with some indications similar to previous
ones.
Now consider a constant configuration of conductors and a constant
set of linear dielectrics outside the conductors. There is no charge
outside the conductors. We give a net charge to only one of the
conductors. Certainly, according to the above theorem we shall have a
unique charge distribution in the conductors. Suppose that the given
charge of that conductor becomes <lambda>fold. We want to prove that
the surface free charge densties on all of the conductors and also
the dielectrics' polarizations will become <lambda>fold consequently.
Visualize the situation existent before that the given charge becomes
<lambda>fold. An outward resultant force normal to the conductor
surface is exerted on each partial surface charge <sigma>da due to
other nonpolarization and polarization partial charges. Now suppose
that all the nonpolarization (or free) partial charges be nailed up
in their positions and then all the nonpolarization and polarization
partial charges (ie the previous free charges and dielectrics'
polarizations) become <lambda>fold. Obviously, in this case the
resultant electrostatic force on each partial surface charge is
outward normal to the conductor surface too (and only its magnitude
has become <lambda>{2}fold). Furthermore, it is obvious that in each
point of each dielectric the electrostatic field has only become
<lambda>fold (without any change in its direction) and then we see
that this field is propotional to the polarization at that point as
must be so expectedly. Thus, if the charges get free from the nailed
state, they will remain on their positions, and furthermore, the only
change in net charges is in the above mentioned conductor, net charge
of which has now become <lambda>fold. Therefore, this is a possible
distribution and according to the above mentioned uniqueness theorem
of charge distribution is unique and then is the same distribution
that really occurs.
II.D. Superposition principle for the charge densities
------------------------------------------------------
We must also notice another point. We understood that in a
configuration of some conductors that only one of them has net
charge, charge distribution is unique. Suppose that we have N
conductors and only conductor i has net charge (Q). The unique
distribution that charges get, prescribes charge surface density
<sigma>(~r) (and polarization ~P(~r)) for each point of each
conductor (and each point outside the conductors).
Now consider this same configuration of these conductors from which
only conductor j (such that j is not equal to i) has net charge
(Q[j]). The unique distribution that charges get, prescribes charge
surface density <sigma>[j](~r) (and polarization ~P[j](~r)) for each
point of each conductor (and each point outside the conductors).
It is clear intuitively that if we have this same configuration of
the conductors from which only two conductors have net charges, the
ith conductor has the same relevant net charge (Q) and the jth
conductor has the same relevant net charge (Q[j]), then the unique
distribution that charges get, prescribes charge surface density
<sigma>(~r)+<sigma>[j](~r) (and polarization ~P(~r)+~P[j](~r))
for each point of each conductor (and each point outside the
conductors). This fact has generality for when each conductor has a
specified net charge or when there is a fixed distribution of
external charge outside the conductors (ie we can add contribution of
this distribution towards forming charge surface density on the
conductors (and forming polarization) to other contributions). We can
even, when there are linear dielectrics, obtain surface charge
distribution on the conductors by adding the charge surface density
in each point on the conductors related to charge distribution in the
absence of dielectrics to the charge surface density in the same
point produced only by the polarizations of the dielectrics assuming
that there exists no net charge in any conductor but only the
polarizations exist.
Therefore, considering the theorems we have proven so far, we can
conceive that in a system of some charged conductors and some fixed
external charge distribution and some linear dielectrics if the net
charge of a conductor becomes <lambda>fold, free partial charge
surface density arising from that conductor, assuming that other
conductors are uncharged and there are not any dielectrics or other
external charges, will become <lambda>fold in each point on the
conductors. It is evident that, considering the integral definition
of electrostatic potential and assuming that the potential is zero at
infinity, the partial potential arising from that conductor (ie in
fact from its effect on forming the free charges) will become
<lambda>fold in each point, too, and then the partial potential
arising from that conductor will become <lambda>fold in each
conductor which is an equipotential region for this partial
potential. In other words, the free net charge of one of the
conductors is proportional to the partial potential arising from the
(effect of the free net) charge of that conductor (assuming that
there are not any dielectrics or other external charges and that
other conductors are uncharged) in each of the conductors:
(i=1,2,3,...,N) <phi>{(j)}=p[ij]Q[j]. Furthermore, this fact that
each conductor is an equipotential region for this partial potential
proves that p[ij] depends only on the geometry of the configuration
of the conductors and even does not depend on the dielectrics and
their positions (or other external charge distributions outside the
conductors), because, as we mentioned, this constant coefficient of
the proportion, p[ij], is related to when we suppose that there
are not at all any dielectrics (or other external charges) and infer
that the charge surface densities will become <lambda>fold if the net
charge of a conductor (the jth one) becomes <lambda>fold (assuming
that other conductors are uncharged).
Now since the potential of each conductor is the sum of its partial
potentials plus a constant, we have
<phi>=<summation from j=1 to N>p[ij]Q[j]+c. (Adding of c removes
the worry arising from generalization of the necessity of the above
reasoning that the partial potentials must be zero at infinity.)
III. Static potential energy and current mistakes
-------------------------------------------------
III.A. Static potential energy
------------------------------
We know that if a closed surface S contains external electric charge
Q and polarization electric charge Q[P], then we shall have
<circulation over S>~E.^nda=(Q+Q[P])/<epsilon>[0]. In this relation
~E is the partial electrostatic field arising from both an elective
distribuition of external charge, the part of which inside the closed
surface being equal to Q, and an elective distribution of
polarization charge, the part of which inside the closed surface
being equal to Q[P]. (The word "elective" implies that the entire
existent charge distribution is not necessarily taken into
consideration, and similarly the word "partial" implies that maybe
only a part of the existent field is intended. Notice the
superposition principle of field and the linearity of potential.)
On the other hand we have
Q[P]=<integral over S'>~P.^nda+<integral over V>(-<del>.~P)dv in
which V is the volume of the dielectric enclosed by S, and S' is the
surface of the conductors inside the closed surface S. In this
relation ~P.^n and -<del>.~P are the the polarization charge
densities of the elective distribution of polarization charge, and
then we can say that in this relation ~P is an elective (ie not
necessarily entire) distribution of electrostatic polarization. If
using the divergence theorem we change the volume integral into the
surface integral, we finally shall obtain
Q[P]=-<circulation over S>~P.^nda. The comparison of this relation
with the first relation of this section shows that
<circulation over S>(<epsilon>[0]~E+~P).^nda=Q in which ~P is an
elective distribution of polarization, and Q is the total charge of
that part of the elective distribution of external charge which is
inside the closed surface S, and ~E is the partial field arising from
both the totality of the elective distribution of external charge and
the totality of the elective distribution of polarization. On
definition, the electric displacement vector is ~D=<epsilon>[0]~E+~P.
Then <circulation over S>~D.^nda=<integral over V><rho>dv. This
relation says that if ~E is arising from both <rho>, which is an
elective distribution of external electric charge, and ~P, which is
an elective distribution of electrostatic polarization, then the
surface integration of ~D=<epsilon>[0]~E+~P on the closed surface S
is equal to the totality of only that part of our elective external
charge which is inside the closed surface. If we use the divergence
theorem in the recent relation, we shall conclude <del>.~D=<rho>.
Considering the above discussions the following deduction may be
interesting. (In this deduction the expression "the ~E arising from
both <rho> and ~P" is shown as "~E(<rho>,P)".)
~D[2]=<epsilon>[0]~E(<rho>,P[2])+~P[2] ==> <del>.~D[2]=<rho> )~D[1]= said:
<del>.~D[1]=<del>.~D[2] or
<del>.(<epsilon>[0]~E(<rho>,P[1])+~P[1]) =
<del>.(<epsilon>[0]~E(<rho>,P[2])+~P[2])
The electrostatic potential energy of a bounded system of electric
charges (which can exist in various forms of external charge,
polarization charge, etc, eg in the form of canceled charges, from
the macroscopic viewpoint, in a molecule) having the density <rho>,
which is in fact the spent energy for assembling all the fractions of
the charge differentially from infinity, is
U=1/2<integral over V[h]><rho>(~r)<phi>(~r)dv (1)
in which V[h] is the whole space and <phi> is the partial
electrostatic potential due to the distribution of <rho>. The way of
obtaining the relation (1) can be seen in many of the electromagnetic
texts.
As it is so actually in the tridimentional world of matter, we
disburden ourselves from the dualizing the charge density as the
surface and volume ones and say we have only the volume density of
the electrostatic charge that, for instance, can have an excessive
absolute amount on the surface of a charged electric conductor. Now
we take into consideration an elective distribution of the volume
density of the external (ie nonpolarization) electric charge, <rho>.
We want to obtain the electrostatic potential energy of this
distribution. We know that <del>.~D=<rho> so that
~D=<epsilon>[0]~E+~P in which ~P is the elective distribution of the
electrostatic polarization and ~E is the resultant field arising from
both the elective distribution of the external electric charge
density (<rho>) and the polarization charge densities due to the
elective distribution of the electrostatic polarization (~P). Since
the electrostatic potential energy of this elective distribution of
the external electric charge is U=1/2<integral over
V[h]><rho><phi>dv, in which (V[h] is the whole space and) <phi> is
only arising from <rho> (not from both <rho> and ~P), we shall have
U=1/2<integral over V[h]><phi><del>.~Ddv, and since
<integral over V[h]><phi><del>.~Ddv=<integral over V[h]><del>.(<
phi>~D)dv-<integral over V[h]>~D.<del><phi>dv=<integral over S[h]><
phi>~D.^n'da-<integral over V[h]>~D.<del><phi>dv=0-<integral over
V[h]>~D.(-~E[<rho>])dv=<integral over V[h]>~D.~E[<rho>]dv
(V[h] and S[h] being in turn the whole space and the total surfaces
of the problem (which of course there is not any surface)), we shall
have
U=1/2<integral over V[h]>~D.~E[<rho>]dv (2)
in which as we said " U is the electrostatic potential energy of an
elective distribution of the external electric charge with the
density <rho>, and we have <del>.~D=<rho> in which
~D=<epsilon>[0]~E+~P in which ~P is an elective distribution of
electrostatic polarization and ~E is arising from both ~P and <rho>,
while ~E[<rho>] is the field arising only from <rho>." It is obvious
that this electrostatic potential energy has been distributed in the
space with the volume density u=1/2~D.~E[<rho>].
(We saw previously that <del>.~D[1]=<del>.~D[2]. Uniqueness of the
electrostatic potential energy of a definite distribution of external
electric charges with the density <rho> necessitates having
1/2<integral over V[h]>~D[1].~E[<rho>]dv =
1/2<integral over V[h]>~D[2].~E[<rho>]dv ;
but although these total energies are equal to each other this won't
necessarily mean that the energy densities are also the same, ie we
cannot infer ~D[1].~E[<rho>]=~D[2].~E[<rho>] or ~D[1]=~D[2] (although
their divergences are equal).)
It is very opportune to compare the above accurate definition of the
electrostatic potential energy with what is set forth for discussion
under this very title in the present electromagnetic books, and to
pay attention to the existent inaccuracy in the definitions of the
involved terms caused by the omission of the subscript <rho> from the
term ~E[<rho>]. This is a sample of the existent inaccuracies in the
present current electromagnetic theory specially in not correct
distinguishing between different electric fields. This mistake has
caused that, considering relation ~D=<epsilon>~E for linear
dielectrics, wrong relations like
u=1/2<epsilon>E{2}=1/2D{2}/<epsilon> to be current in present
electromagnetic textbooks. We shall pay to some other mistakes soon.
III.B. Independence of capacitance from dielectric
--------------------------------------------------
Consider a system consisting of some fixed perfect conductors and
some linear dielectrics in the space exterior to the conductors and
some fixed distribution of external charge density in this space. We
want to obtain electrostatic potential energy arising from all the
free net charges on these conductors, ie the electrostatic potential
energy of that part of the charge distribution in all of the
conductors which comes into existence as a result of these free net
charges (which of course does not include electrostatic potential
energy of the polarization and distribution of external charges and
that (other) part of the charge distribution in all of the conductors
which comes into existence as a result of these polarization and
external charges). Since each conductor is an equipotential region
for the potential arising from these free net charges, for this
electrostatic potential energy we have U=1/2<summation from j=1 to
N>Q[j]<phi>[j] from the relation (1), in which Q[j] is the net charge
of the conductor j and <phi>[j] is the electrostatic potential on the
conductor j arising from all free net charges of the conductors of
the system (ie one related to free net charges themselves and their
effect on the conductors, not also related to dielectric polarization
and other external charges and their effect on the conductors). What
is necessary to be emphasized again (and is important in the coming
discussion) is that the <phi>[j]'s are arising only from net charges
of the conductors not also from the polarization charges. Using the
coefficients of potential for this system we can also write
<phi>=<summation from j=1 to N>p[ij]Q[j] in which Q[j] is the net
charge of the conductor j, and <phi> is the electrostatic
potential on the conductor i arising from all (Q[j]'s ie all) net
charges of the conductors of the system (ie one related to free net
charges themselves and their effect in the conductors, not also
related to dielectric polarization and other external charges and
their effect on the conductors). Combining the two recent relations
yields
U=1/2<summation from i=1 to N><summation from j=1 to N>p[ij]QQ[j]
for the electrostatic potential energy arising from free net charges
of the conductors of a system consisting of some perfect conductors
and probably some linear dielectrics and external charge distribution
outside the conductors.
A capacitor is defined as two conductors (denoted by 1 and 2), from
among a definite configuration of some conductors, that one of them
bears net charge Q (Q being greater than or equal to zero) and the
other one bears -Q. (Existence of net charges on other conductors in
the configuration or of linear dielectrics or external charges
outside the conductors and the effect which each has on these two
conductors (ie 1 and 2) are not important at all. We shall find out
this soon.)
By using the relation <phi>=<summation from j=1 to N>p[ij]Q[j] for
the above capacitor we have:
<phi>[2]=p[21]Q+p[22](-Q)+0 )
<cap. delta><phi>=<phi>[1]-<phi>[2]=(p[11]+p[22]-2p[12])Q=Q/C
(We know that p[12]=p[21] proof of which can be seen in many of the
electromagnetic books.) We have attention that in the relation
<cap. delta><phi>=Q/C, <cap. delta><phi> is the potential difference
between the potential arising from net charges of the conductors 1
and 2 (related to themselves and their effect in other conductors) on
the conductor 1 and the potential arising from these charges (related
to themselves and their effect in other conductors) on the conductor
2. Therefore, since the potential of other charges is not considered
and considering linearity of potentials and that C, which is called
as the capacitance of the capacitor, depends only on the form of the
configuration of all (and not only two) of the conductors, it is
obvious that existence of net charges on the conductors other than
the conductors 1 and 2 and existence of any linear dielectrics or
external charges in the space exterior to the conductors, so far as
the configuration of the conductors is constant, are unimportant (and
there is no need that one of the conductors 1 and 2 be shielded by
the other, the way presented in some electromagnetic books for the
potential difference independence of whether other conductors are
charged). We specially emphasize again that so, we have proven that
the capacitance (C) of a capacitor does not depend on whether there
exist any dielectrics at all and only depends on the configuration of
the conductors introduced in the definition of the capacitor.
Using the relation
U=1/2<summation from i=1 to N><summation from j=1 to N>p[ij]QQ[j]
we obtain
U=1/2Q{2}/C=1/2Q<cap. delta><phi>=1/2C(<cap. delta><phi>){2} (3)
for the electrostatic potential energy of the charges Q and -Q
(themselves and of their effect). We should emphasize again that in
the recent relation, <cap. delta><phi> is the potential difference
arising from the free charges Q and -Q (and not also from eg
polarization charges), and C depends only on the configuration of the
conductors (and not also on eg existence or nonexistence of linear
dielectrics).
At the end of this section let's obtain the capacitance of a
capacitor consisting of two parallel plates in which the plates
separation d is very small compared with the dimensions of the
plates:
Q Q Q
C=----------------------=------=---------------------
(<cap. delta><phi>)[Q] E[Q]d <sigma>d/<epsilon>[0]
Q <epsilon>[0]A
=-------------------=-------------- ,
(Q/A)d/<epsilon>[0] d
in which (<cap. delta><phi>)[Q] and E[Q] are the potential difference
and the electrostatic field arising from Q and -Q (and not also from
the polarization charges) respectively. Therefore, the capacitance of
this capacitor is <epsilon>[0]A/d regardless of whether there exist
any linear dielectrics between the parallel plates or not.
And now see the present books of Electricity and Magnetism in which
without attention to this fact that <cap. delta><phi> must be arising
only from the capacitor charge, the relation <cap. delta><phi>=Ed, in
which E is arising from not only the capacitor charge but also the
linear dielectrics polarization charges, is used and consequently
wrong expression <epsilon>A/d is obtained for the capacitance.
III.C. Dielectric as source of potential
----------------------------------------
We saw that the mathematical discussions presented so far proved
independence of the capacitance of a capacitor from its dielectric.
But this is doubtlessly surprising for the physicists and engineers,
because they know well that dielectric has a substantial part in
accumulation of charge in the capacitor. This section is intended for
obviating this surprise.
It is made use often of electroscope to show the effect of
dielectrics in capacitors. If the two conductors of a charged
capacitor are connected to an electroscope, leaves of the
electroscope will get away from each other. Now, if, without any
change in the configuration of the capacitor's conductors, a
dielectric is inserted between the two conductors of the capacitor,
the leaves of the electroscope will come close to each other. Current
justification of this phenomenon is as follows (eg see University
Physics by Sears, Zemansky and Young, Addison-Wesley 1987):
"The equation C=Q/<cap. delta><phi> shows the relation among the
capacitor's capacitance, capacitor's charge, and the potential
difference between the two conductors of the capacitor. When a
dielectric is inserted into the capacitor, due to the orientation of
the electric dipoles of the dielectric in the field inside the
capacitor some polarization charge opposite to the charge of each
conductor of the capacitor is induced on that surface of the
dielectric which is adjacent to this conductor, and then the
electrostatic field in the dielectric, and thereby the potential
difference (between the two conductors), arising from both the
capacitor's charge and this induced polarization charge is decreased.
Then, the denominator of C=Q/<cap. delta><phi> decreases which
results in increasing of the capacitance (C) considering that Q
remains uncharged, ie the capacitor's capacitance increases by
inserting a dielectric between the capacitor's conductors. That the
leaves of the electroscope come closer to each other by inserting the
dielectric is because of this same decreasing of the potential
difference, <cap. delta><phi>."
It is clear that considering the discussion presented in this
article, the above justification is quite wrong, because
<cap. delta><phi> is the potential difference arising only from the
capacitor's charge not also from the polarization charge formed in
the dielectric. But why do the leaves of the electroscope come closer
to each other when a dielectric is inserted into the capacitor? Its
reason is quite obvious. Metal housing and the leaves connected to
the metal knob of the electroscope, themselves, are in fact a
capacitor, which when are connected separately to the two conductors
of the capacitor under measurement, a new (equivalent) capacitor will
be formed consisting of two conductors: the first being one of the
conductors of the capacitor under measurement and the electroscope's
metal housing which is connected to it, and the second being the
other conductor of the capacitor under measurement and the set of the
knob and the leaves of the electroscope which is connected to this
conductor. It is obvious that if the capacitor under measurement is
charged at first, its charge now, after its connecting to the
electroscope, will be distributed throughout the new formed
capacitor and then a part of the charge of the primary capacitor now
will go to the electroscope because of which the leaves of the
electroscope will get away from each other (because the opposite
charges induced in the electroscope will attract each other causing
drawing of the leaves toward the electroscope's housing which itself
means more separation of the leaves from each other).
By inserting the dielectric into the capacitor we cause creation of
polarization charges in the dielectric which this, in turn, causes
more charges of the new formed capacitor to be drawn towards the
dielectric. Thus, the distribution of the charge will be changed in
such a manner that a part of the charge distribution in the
electroscope will go to the primary capacitor (or the one under
measurement) to be placed as close as possible to the dielectric;
this means decrease of the electroscope's charge which will cause its
leaves to come closer to each other. Therefore, the act of the
dielectric is change of the charge distribution in the new capacitor
formed from the primary capacitor and the electroscope, not change of
the capacitance of the primary capacitor.
Now, let's connect the two plates of a parallel-plate capacitor by a
wire in the space exterior to the space between the plates. What will
happen if a slice of a dielectric having a permanent electric
polarization is inserted between the two plates of the capacitor? The
polarized dielectric will cause induction of charge on the two
plates; the positive surface of the slice will induce negative charge
on the plate adjacent to it, and the negative surface will induce
positive charge on the (other) plate adjacent to it. Induction of
charge on the two plates, while they had no charge beforehand, means
that while inserting the dielectric between the plates an electric
current has been flowing in the wire from one plate to the other. In
other words the dielectric acts like a power supply producing
electric current or charging the capacitor. Then, we can attribute
electric potential difference to it (like the potential difference
between the two poles of a battery).
Now, how will the situation be if the inserted dielectric is not to
have previous polarization but it is to be polarized because of the
charge (or in fact the electric field produced by the charge) of the
capacitor? Answer is that the situation will be similar to the same
state of permanent polarization, and again the dielectric acts as a
source of potential. Its physical and direct reason can be seen
easily in the discussion we presented about the electroscope. There,
we saw that inserting the dielectric, charge distribution was changed
in such a manner that some more charges were accumulated on the
conductors of the (primary) capacitor. It is clear that more
accumulation of charge on the capacitor necessitates flowing of
electric current in the circuit. Cause of this current and of the
more accumulation of charge on the capacitor is the source of
potential difference which we must attribute to the dielectric.
In this manner, the purpose of this section has been fulfilled
practically; in electric circuits wherever a dielectric is to exist
between the conductors of a capacitor, a proper source of voltage
must be considered in the circuit in the same place of the
dielectric. Such a voltage source causes accumulation of charges on
the conductors of the capacitor more than when there exists no
dielectric in the capacitor. One can say whether this act is not
equivalent to defining, in principle, the capacitance of a capacitor
equal to the charge accumulated on the capacitor (due to both the
configuration of the capacitor's conductors and the electric
induction in the conductors caused by the polarization of the
dielectric) divided by the potential difference between the two
conductors of the capacitor (which is the method that current
instruments measuring capacitor's capacitances work based on it) and
no longer considering the dielectric as a source of potential.
Following example shows that consequences of such a definition in
practice are not equivalent to the practical consequences of the main
definition of capacitance of capacitor (although can be close to it
under suitable conditions). We then shall investigate another example
which will show, well, considerable differences that can come into
existence if role of the dielectric as a power supply in the circuit
is not taken into consideration, according to which a quite practical
criterion for testing the theory presented in this section in
comparison with the current theory will be presented.
III.D. Some examples as test
----------------------------
Let's connect the two plates of a dielectricless parallel-plate
capacitor to the two poles of a battery. At the end of the section
III.B. we saw that the capacitance of such a capacitor is
<epsilon>[0]A/d in which A is the capacitor's area and d is the
distance between its plates. Then, according to the relation
C=Q/<cap. delta><phi> for the capacitor's capacitance, we have
<epsilon>[0]A/d=<sigma>A/V in which <sigma> is the surface density
of the charge accumulated on the capacitor and V is the potential
difference given to the two plates of the capacitor by the battery.
In this manner we have:
<sigma>d=<epsilon>[0]V (4)
Now we fill the space between the two plates with a linear dielectric
with the permitivity <epsilon>. We indicate the magnitude of the
formed electric polarization in the dielectric by P. P is in fact
equal to the surface density of the polarization charge in the
dielectric. Suppose that a charge exactly equal to the polarization
charge is induced on the plates of the capacitor. (Indeed, in the
state of induction of charge in the capacitor due to the polarized
dielectric between the capacitor's plates we should suppose that the
two plates of the capacitor are connected to each other by a wire in
the space exterior to the space between the plates; in other words in
this state the battery existent in the circuit does not play any role
except as a short circuit.) Then the charge induced on the capacitor
due to the polarization of the dielectric is equal to PA. This
charge, as we said, has been stored in the capacitor because of a
source of potential difference, equal to V', which we must attribute
to the dielectric; ie because of the potential difference V' exerted
to the two plates of the capacitor the charge PA has been accumulated
in the capacitor, and then the ratio PA/V' is equal to the
capacitor's capacitance <epsilon>[0]A/d=PA/V'. Considering that
P=(<epsilon>-<epsilon>[0])E=(<epsilon>-<epsilon>[0])<sigma>/<epsilon>
in which E is the electrostatic field arising from both the external
and polarization charges we infer from this relation that
V'=(<epsilon>-<epsilon>[0])<sigma>d/(<epsilon><epsilon>[0])
which considering Eq.(4) results in
V'=(1-<epsilon>[0]/<epsilon>)V=(1-1/K)V (5)
Let's calculate sum of the charges (Q) accumulated on this capacitor
(due to both the configuration of the capacitor's conductors and the
induction arising from the (polarization of the) dielectric). For
this act we must add the potential difference arising from the
dielectric to the potential difference given by the battery and after
that multiply the sum by the (real) capacitance of the capacitor
C=<epsilon>[0]A/d:
Q=(V+(1-<epsilon>[0]/<epsilon>)V)<epsilon>[0]A/d=
(2-<epsilon>[0]/<epsilon>)(<epsilon>[0]A/d)V=(2-1/K)CV (6)
Can we present another definition of capacitance of capacitor, for
convenience in practice, equal to sum of the charges accumulated on
the capacitor (consisting of the charges arising from both the
configuration of the capacitor's conductors and the induction due to
the dielectric) divided by the potential difference between the two
capacitor's conductors, given to the capacitor only by the battery
(or the circuit)? Considering Eq.(6) such a definition gives the
following (newly defined) capacitance of our capacitor equal to
Q/V=(2-<epsilon>[0]/<epsilon>)<epsilon>[0]A/d. (7)
Is this definition useful in practice, and does it yield real
consequences? The answer is negative. It is sufficient only instead
of a single capacitor to consider n capacitors connected in series
such that the space between the plates of only one of them is filled
with dielectric and to try to calculate the accumulated charges on
the equivalent capacitor.
If all of these n capacitors were dielectricless, because of the
identity between the capacitors the (shared) potential difference
between the two plates of each of these capacitors would be V/n.
When only one of these capacitors is filled with a linear dielectric
with the permittivity <epsilon>, the potential difference related to
this dielectric (as a source of potential), similar to Eq.(5) will be
(1-<epsilon>[0]/<epsilon>)V/n. Since these n capacitors are identical
and the capacitance of each of them is <epsilon>[0]A/d, the
equivalent capacitance of these n capacitors which are connected in
series will be obtained by solving the equation
1/C[1]=n/(<epsilon>[0]A/d) for C[1] equal to <epsilon>[0]A/(nd).
Therefore, the charge accumulated on each capacitor is equal to
<epsilon>[0] V <epsilon>[0]A
( V + ( 1 - ------------ ) --- ) -------------
<epsilon> n nd
<epsilon>-<epsilon>[0] <epsilon>[0]A
= ( 1 + ---------------------- ) ------------- V . (8)
n<epsilon> nd
But now let's see if the capacitance of the capacitor having
dielectric is to be equal to (7) while the capacitance of each of the
other capacitors is equal to <epsilon>[0]A/d, whether or not the
charge accumulated on each capacitor will be obtained still equal to
(8) when no longer the source of potential difference related to the
dielectric is considered in lieu of considering (7) for the
capacitance of the capacitor having dielectric. Equivalent
capacitance of the capacitors which are in series will be obtained by
solving the equation
1 n-1 1
---- = --------------- + -----------------------------------------
C[2] <epsilon>[0]A/d (2-<epsilon>[0]/<epsilon>)<epsilon>[0]A/d
for C[2], and charge of each capacitor should be considered equal to
C[2]V:
1 <epsilon>[0]A
C[2]V= --------------------------------------- . ------------- V (9)
n-1+<epsilon>/(2<epsilon>-<epsilon>[0]) d
Obviously the coefficient of <epsilon>[0]AV/d in Eq.(8) is not equal
to the coefficient of <epsilon>[0]AV/d in Eq.(9) except when
<epsilon>=<epsilon>[0] or n=1. Thus, we see that the new definition
we tried to present for capacitance of capacitor is not so useful in
practice (at least in this example does not give the real charge
accumulated on the capacitors). But the ratio of these two
coefficients is not so far from one. To see this fact let's indicate
<epsilon>/<epsilon>[0] by K and obtain the ratio of the coefficient
of <epsilon>[0]AV/d in Eq.(9) to the coefficient of <epsilon>[0]AV/d
in Eq.(8):
(n-1+<epsilon>/(2<epsilon>-<epsilon>[0])){-1} (K-1){2}(n-1)
---------------------------------------------- = 1/(1+ -------------)
1/n + (<epsilon>-<epsilon>[0])/(n{2}<epsilon>) (2K-1)Kn{2}
It is seen that the degree of the term (K-1){2}(n-1)/((2K-1)Kn{2})
with respect to K is zero and with respect to n is -1; thus this term
is close to zero practically, or in other words the ratio of the
above-mentioned coefficients is close to one practically. This matter
is itself a good reason that why the definition of capacitance in the
form of capacitor's charge divided by the potential difference
exerted on the capacitor's conductors (Eq.(7)) has been able to
endure practically and the difficulties due to such a definition has
remained hidden in practice. But, important for a physicist should be
mathematical much exactness and discovery of what actually occurs or
exists. In order to find out that such an exactness can be important
even in practice (and then won't be negligible even for engineers)
notice the following example.
Consider a series circuit of RLC, which its capacitor is
parallel-plate and dielectricless, connected to a constant voltage V.
After connection of the switch in the time t=0, the equation of the
circuit will be
V=RI+LdI/dt+1/(2C)<integral from t=0 to t>I(t)dt. (10)
(We should notice that as it will be proven in the last section of
this article, in this circuit we must consider the circuital
potential difference of the capacitor, ie the third term of the
right-hand side of (10), not as it is usual wrongly its electrostatic
potential difference ie 1/C<integral from t=0 to t>I(t)dt. There,
also we shall see that what the current instruments measure as
capacitance is in fact two times more than the capacitance. Another
noticeable point being that as it has been explained in the section 5
of the 13th article of the book, L in (10) is in fact equal to
<mu><epsilon>'a'L{*} not equal to only
d<cap. phi>{*}/dI(=L{*}) according to its usual definition. But
since the current instruments for measuring L work based on the
formula <cursive E>=-LdI/dt, they are in fact measuring
<mu><epsilon>'a'L{*} as L because as we can see in that article
the correct relation is in fact
<cursive E> = -<mu><epsilon>'a'L{*}dI/dt.)
With one time differentiation of this equation with respect to time,
the following equation will be obtained considering that V is
constant: Ld{2}I/dt{2}+RdI/dt+I/(2C)=0. If R/(2L)<(2LC){-1/2}, this
equation will be solved as I= a.exp(-Rt/(2L))cos(<omega>[n]t-<theta>)
in which
1 R{2}
<omega>[n] = (--- - -----){1/2} (11)
2LC 4L{2}
and a and <theta> are two arbitrary constants. Since in t=0 we have
I=0 and then also from Eq.(10) we have dI/dt=V/L, we conclude that
a=V/(<omega>[n]L) and <theta>=<pi>/2, and then
V
I= ---------- exp(-Rt/(2L))sin(<omega>[n]t). (12)
<omega>[n]L
For calculating the voltage drop in the capacitor we should calculate
the third term of the right-hand side of Eq.(10):
1
--<integral from t =
2C
V
0 to to t>-----------exp(-Rt/(2L))sin(<omega>[n]t)dt =
<omega>[n]L
R
V(1-exp(-Rt/(2L))(cos(<omega>[n]t)+------------sin(<omega>[n]t)))
2<omega>[n]L (13)
Now, if the space between the two plates of the capacitor (without
any change in the configuration of the plates) is to be filled by a
linear dielectric with the permittivity <epsilon>, we must multiply
the negative of the voltage drop in the capacitor ((13)) by
(1-<epsilon>[0]/<epsilon>) till according to Eq.(5) the potential
difference which we must attribute to the dielectric as source of
potential is otained. We then should add this source to the previous
constant source and equate the sum to the right-hand side of Eq.(10):
V+V(exp(-Rt/(2L))(cos(<omega>[n]t)+R/(2<omega>[n]L)sin(<omega>[n]t))-
1)(1-<epsilon>[0]/<epsilon>)=RI+LdI/dt+1/(2C)<integral from t=0 to
t>I(t)dt (14)
With one time differentiation of this equation with respect to time
the following equation will be obtained:
d{2}I dI 1
L----- +R-- + --I =
dt{2} dt 2C
<epsilon>[0] 2<omega>[n]L
V(1- ------------)------------exp(-Rt/(2L))sin(<omega>[n]t)
<epsilon> R{2}C-2L
Particular solution of this equation is
V <epsilon>[0]
--------(1- ------------)t.exp(-Rt/(2L))cos(<omega>[n]t),
2L-R{2}C <epsilon>
and general solution of its corresponding homogeneous equation is
a.exp(-Rt/(2L))cos(<omega>[n]t-<theta>) with the two arbitrary
constants a and <theta>. Then general solution of this equation is
I=a.exp(-Rt/(2L))cos(<omega>[n]t-<theta>) +
V <epsilon>[0]
--------(1- ------------)t.exp(-Rt/(2L))cos(<omega>[n]t)
2L-R{2}C <epsilon>
with the two arbitrary constants a and <theta>. For obtaining a and
<theta> by means of the initial conditions, we should be careful that
initial conditions must be fit, ie t=0 should be the same moment
that, without dielectric, the current in the circuit was zero and we
had dI/dt=V/L; and now, when the dielectric has been inserted, we
should see how the conditions change, and in this moment (t=0) what
the current and its time derivative are as initial conditions. The
physics of the problem says that we have in this state I=0 in this
moment too, and then also it is clear from Eq.(14) that in this
moment we have dI/dt=V/L too. Then
V V <epsilon>[0]
a = ----------- + --------------------(1- ------------) =
<omega>[n]L <omega>[n](R{2}C-2L) <epsilon>
L(1+<epsilon>[0]/<epsilon>)-R{2}C V
---------------------------------.----------- and <theta>=<pi>/2.
2L-R{2}C <omega>[n]L
Thus
L(1+<epsilon>[0]/<epsilon>)-R{2}C V
I=---------------------------------.-----------exp(-Rt/(2L))sin(<
2L-R{2}C <omega>[n]L
V <epsilon>[0]
omega>[n]t)+--------(1- ------------)t.exp(-Rt/(2L))cos(<omega>[n]t).
2L-R{2}C <epsilon> (15)
(It is noticeable that when <omega>=<omega>[0] the same Eq.(12) will
be obtained from this equation.) We obtained Eq.(15) for the current
of the circuit, while what is current at present is that inserting
the linear dielectric (with the permittivity <epsilon>) between the
plates of the capacitor only the capacitor's capacitance changes from
C to KC where K=<epsilon>/<epsilon>[0] (without any addition of new
source of potential to the circuit), and then the circuit's current
has the same form of Eq.(12) with this only difference that in the
equation related to <omega>[n] (Eq.(11)) we must write KC instead
of C.
Now suppose that instead of the constant voltage V we have an
alternating voltage in the form of V(t)=V[0]sin(<omega>t-<theta>')(in
which <theta>' is a constant value) as the main source of potential
in the series circuit of RLC which its parallel-plate capacitor is
dielectricless. In such a case we have
dI 1
V[0]sin(<omega>t-<theta>')=RI+L-- + --<integral from t =
dt 2C
0 to t>I(t)dt, (16)
and then
Ld{2}I/dt{2}+RdI/dt+I/(2C)=V[0]<omega>cos(<omega>t-<theta>').
Particularl solution of this equation is
a[1]cos(<omega>-<theta>'-<thata>[1]) in which
a[1]=V[0]/((1/(2C<omega>)-L<omega>){2}+R{2}){1/2} (17)
and
<theta>[1]=cot{-1}((1/(2<omega>C)-L)/R). (18)
Since solution of its corresponding homogeneous equation is
a.exp(-Rt/(2L))cos(<omega>[n]t-<theta>), the general siolution of
this equation is
I=a.exp(-Rt/(2L))cos(<omega>[n]t-<theta>)+
a[1]cos(<omega>t-<theta>'-<theta>[1]) (19)
with the two arbitrary constants a and <theta> (of course assuming
that R/(2L)<(2LC){-1/2}).
We suppose that we have I=0 in t=0 and from Eq.(16) we have
dI/dt=-V[0]sin<theta>'/L in this moment. Having these initial values
we can obtain a and <theta>, but since the first term of the right-
hand side of Eq.(19) is transient, this act is of no importance for
us. (Nevertheless, they should be obtained by solving the system of
( acos<theta>=-a[1]cos(<theta>'+<theta>[1])
<
( asin<theta>=-(1/<omega>[n])(V[0]sin<theta>'/L+(Ra[1]/(2L))cos(<
theta>'+<theta>[1])+a[1]<omega>sin(<theta>'+<theta>[1]))
for a and <theta>.)
Now, as before, having the form of current (Eq.(19)) we obtain
voltage drop in the capacitor:
1/(2C)<integral from t=0 to t>(a.exp(-Rt/(2L))cos(<omega>[n]t-<
theta>)+a[1]cos(<omega>t-<theta>'-<theta>[1]))dt=a(exp(-Rt/(2L))(<
omega>[n]Lsin(<omega>[n]t-<theta>)-R/2cos(<omega>[n]t-<theta>))+<
omega>[n]Lsin<theta>+R/2cos<theta>)+a[1]/(2<omega>C)(sin(<omega>t-
<theta>'-<theta>[1])+sin(<theta>'+<theta>[1])) (20)
And now, as before, if the space between the two plates of the
capacitor is to be filled by a linear dielectric with the
permittivity <epsilon> (without any change in the plates'
configuration), in order to obtain the potential difference that we
must attribute to the dielectric as a source of potential in the
circuit, according to Eq.(5) we should multiply the negative of the
potential drop in the capacitor (20) by (1-<epsilon>[0]/<epsilon>).
We then must add this source to the initial alternating source and
equate the sum to the right-hand side of Eq.(16):
V[0]sin(<omega>t-<theta>')+a(1-<epsilon>[0]/<epsilon>)(exp(-Rt/(2L))(
R/2cos(<omega>[n]t-<theta>)-<omega>[n]Lsin(<omega>[n]t-<theta>))-<
omega>[n]Lsin<theta>-R/2cos<theta>)-a[1]/(2<omega>C)(1-<epsilon>[0]/<
epsilon>)(sin(<omega>t-<theta>'-<theta>[1])+sin(<theta>'+<theta>[1]))
=RI+LdI/dt+1/(2C)<integral from t=0 to t>I(t)dt
With one time differentiation of this equation with respect to time
the following equation will be obtained:
Ld{2}I/dt{2}+RdI/dt+1/(2C)I=V[0]<omega>cos(<omega>t-<theta>')-a(1-<
epsilon>[0]/<epsilon>)(R{2}/(4L)+<omega>[n]{2}L)exp(-Rt/(2L))cos(<
omega>[n]t-<theta>)-a[1]/(2C)(1-<epsilon>[0]/<epsilon>)cos(<omega>t-
<theta>'-<theta>[1]) (21)
For obtaining the particular solution of this equation we must add up
particular solutions of the following equations (for reason see
Differential Equations with Application and Historical Notes by
Simmons, McGraw-Hill Inc., 1972):
Ld{2}I/dt{2}+RdI/dt+1/(2C)I=V[0]<omega>cos(<omega>t-<theta>') (22)
Ld{2}I/dt{2}+RdI/dt+1/(2C)I=-a(1-<epsilon>[0]/<epsilon>)(R{2}/(4L)+
<omega>[n]{2}L)exp(-Rt/(2L))cos(<omega>[n]t-<theta>) (23)
Ld{2}I/dt{2}+RdI/dt+1/(2C)I=-a[1]/(2C)(1-<epsilon>[0]/<epsilon>)cos(
<omega>t-<theta>'-<theta>[1]) (24)
We then must add the obtained particular solution to the general
solution of the corresponding homogeneous equation to obtain the
general solution of Eq.(21).
Both the general solution of the homogeneous equation and particular
solution of Eq.(23) are (trigonometric) multiples of exp(-Rt/(2L)),
thus these two terms in the general solution of Eq.(21) are transient
and then unimportant for us. Thus, for obtaining the nontransient
part of the general solution of Eq.(21) we should obtain the
particular solution of the equations (22) and (24) and then add
them up.
Particular solution of Eq.(22) is
(2V[0]C<omega>/(4L{2}C{2}<omega>{4}+4(R{2}C-L)C<omega>{2}+1))((1-2LC<
omega>{2})cos(<0mega>t-<theta>')+2RC<omega>sin(<omega>t-<theta>'))
and particular solution of Eq.(24) is
( -a[1]/(4L{2}C{2}<omega>{4}+4(R{2}C-L)C<omega>{2}+1) )(1-<epsilon>[0
]/<epsilon>)((1-2LC<omega>{2})cos(<omega>t-<theta>'-<theta>[1])+2RC<
omega>sin(<omega>t-<theta>'-<theta>[1])).
If we write the trigonometric terms in the recent solution in terms
of the sine and cosine of the arguments (<omega>t-<theta>') and
<theta>[1], and add up the particular solutions obtained for the
equations (22) and (24), and equate the sum to the expression
a[2]cos(<omega>t-<theta>'-<theta>[2]), and use the equations (17) and
(18), we shall finally obtain:
2}){2} (25)a[2]cos said:
and
a[2]sin<theta>[2]=4V[0]RC{2}<omega>{2}.(4R{2}C{2}<omega>{2}+(1-2LC<
omega>{2}){2}-2(1-<epsilon>[0]/<epsilon>)(1-2LC<omega>{2}))/(4R{2}C{
2}<omega>{2}+(1-2LC<omega>{2}){2}){2} (26)
We can solve these equations to obtain a[2] and <theta>[2] in order
that the nontransient solution a[2]cos(<omega>t-<theta>'-<theta>[2])
for the circuit current will be obtained unambiguously. The value
which is obtained for the amplitude a[2] from these equations is
2V[0]C<omega>(4R{2}C{2}<omega>{2}+(1/K-2LC<omega>{2}){2}){1/2}
a[2]=--------------------------------------------------------------
4R{2}C{2}<omega>{2}+(1-2LC<omega>{2}){2} (27)
in which K=<epsilon>/<epsilon>[0]. (It is easily seen that for K=1
the same amplitude a[1] presented in Eq.(17) will be obtained from
a[2].)
Now if, as it is thought at present, after inserting the dielectric
between the capacitor's plates its capacitance is to increase to KC
and no more, then we must conclude that the amplitude of the
(nontransient) current is in the same form shown in Eq.(17) except
that KC must be substituted for C in this equation. Namely, the
magnitude of such an amplitude will be:
(V[0]/((1/(2KC<omega>)-L<omega>){2}+R{2}){1/2}=) 2V[0]C<omega>/(4R{2
}C{2}<omega>{2}+(1/K-2LC<omega>{2}){2}){1/2} (=2V[0]C<omega>(4R{2}C{2
}<omega>{2}+(1/K-2LC<omega>{2}){2}){1/2}/(4R{2}C{2}<omega>{2}+(1/K-
2LC<omega>{2}){2})) (28)
A comparison between (27) and (28) shows that their variations with K
is opposite to each other, ie if (27) increases with increase of K,
(28) will decrease with increase of K, and if (27) decreases with
increase of K, (28) will increase with increase of K, and vice versa.
For example on condition that <omega>{2} being greater than or equal
to 1/(2LC) the expression (27) indicates that the current's amplitude
increases by inserting the dielectric, while the expression (28) says
that this amplitude must decrease under the same condition.
Investigating that whether or not experiment shows that provided that
<omega>{2} being greater than or equal to 1/(2LC) current intensity
of the circuit increases by inserting dielectric between the
capacitor's plates is a good test for accepting the theory presented
here and rejecting the current one or vice versa.
To find the resonance frequency of the circuit it is sufficient to
differentiate from the right-hand side of Eq.(27) with respect to
<omega> and then to equate the obtained result to zero and to solve
the obtained equation for <omega>. By doing this act we obtain the
following result for the square of the resonance frequency
<omega>[r]:
2(K-1)+(4(K-1){2}+1){1/2}
<omega>[r]{2}= ------------------------- (29)
2LCK
(It is seen that for K=1, square of the resonance frequency is
1/(2LC) which is just the same result which Eq.(17) predicts for the
square of the resonance frequency. (Reminding of this point is
necessary that as we said we have L=<mu><epsilon>'a'L{*} here.))
Now, let's see what the prediction of the present current belief is
for the resonance frequency of the circuit. It says that since
inserting the dielectric (according to its belief) the amplitude of
the current is V[0]/((1/(2KC<omega>)-L<omega>){2}+R{2}){1/2} (see
Eq.(28)), the square of the resonance frequency will be:
1
<omega>[r]{2}= ---- (30)
2LCK
A simple mathematical try shows that the coefficient of 1/(2LC) in
(29) (ie (2(K-1)+(4(K-1){2}+1){1/2})/K) is an ascending function of
K, while the coefficient of 1/(2LC) in (30) (ie 1/K) is a descending
function of K. Namely, the analysis presented here shows that by
inserting dielectric between the capacitor's plates the resonance
frequency increases, while according to the current belief this
frequency must decrease.
NOTE:
----
That actually whether or not the resonance frequency of the circuit
increases with inserting dielectric between the plates of the
capacitor (without any change in the plates' configuration) is a
quite practical test for establishing the validity of the theory
presented in this article and invalidity of the current belief in
this respect, or vice versa. Recently this experiment has been
performed with a briliant success for the theory presented in this
article showing specifically increase of the resonance frequency when
inserting the dielectric. Here is the report of an electronics
engineer who could not believe the result of his experiments in
this respect:
| Oh, yes, indeed the resonant frequencies do change as
| drastically as you suggest if you put a dielectric with high
| dielectric constant between the parallel plates of a capacitor.
| I've put an example at the end of this posting.
|
| Example of capacitor with high-K dielectric...
| You can buy "disc ceramic" capacitors with about 1.0nF capacitance.
| These are nominally 1cm diameter, with nominally 0.5mm plate
| separation, with dielectric only between the conductive plates.
| The dielectric has a very high dielectric constant. If you resonant
| such a capacitor with, say, a 5mH inductor, you will find its
| resonant frequency will be about 70kHz. You can replace that
| capacitor with one with the same plate size and spacing but air
| dielectric, resulting in roughly 0.5pF capacitance. Then you will
| find that the measured resonant frequency depends on the self-
| resonance of the inductor, because you will be very hard-pressed to
| make a 5mH inductor with self-capacitance as low as 0.5pF. If you
| choose an inductor of, say, 1uH, properly constructed, then you
| might reasonably see the effects of 0.5pF, but now you will be
| dealing with much more awkward (especially if you have limited
| access to good test equipment) resonant frequencies in the hundreds
| of MHz. You will indeed find that the resonant frequency of that
| inductor with the nominal 1.0nF ceramic-dielectric capacitor will
| be on the order of 5MHz. The Q in each case should be high enough
| (with a well-constructed inductor) to give an easily measured
| resonant frequency. I _could_ do the experiment to specifically
| demonstrate the _dramatic_ shift in resonance, and even use other
| dielectrics less extreme, but I feel no need to: as I told you
| before, I _routinely_ design resonant circuits and filters, even
| taking into account the effects of stray capacitance and inductance
| and the resistances of things like circuit board traces where
| appropriate, and within my understanding of the tolerances of the
| parts and the effects of the strays, I'm never surprised. I am
| CERTAINLY never surprised by a resonance shifting higher as I
| increase capacitance, so long as I'm within the practical range of
| the parts I'm using.
|
| Note on 1uH coil: If you make a coil with #18AWG wire, which is
| about 1.0mm diameter, and make that coil with uniformly spaced
| turns, about 2.6cm diameter turns, spaced out 2.5cm total coil
| length, it will have an inductance about 1.0uH, and its first
| parallel self-resonance at about 190MHz. That implies about 0.7pF
| effective self-capacitance. Adding an external 0.5pF capacitance
| would drop the resonant frequency to about 145MHz.
(It is probable that the instrument by which one measures resonance
frequency needs to obtain the capacitance of the capacitor before
calculating the resonance frequency based on the formula
<omega>[r]{2}=1/(2LC). If so, there are two errors in such a
measurement:
1. The process in which the current instruments measure capacitance
of a capacitor is not accurate, because as we explained at the end of
Section III.C (in these instruments) this capacitance is defined
(wrongly) as the charge accumulated on the capacitor divided by the
potential difference between the two conductors of the capacitor.
2. As it has been proven (Eq. (29)), the above formula is not
correct.)
At present dielectric constant is determined in one of the two
following manners:
1. A parallel-plate capacitor is connected to a constant voltage two
times:
(first) when its dielectric is vacuum (or air), and (second) when
its dielectric is the substance under measurement. The geometry of
the conductors remains unchanged. Since it is supposed that the
capacitance of the capacitor is increased with the dielectric, the
ratio of the gathered charge in the second state to the gathered
charge in the first state is the dielectric constant of the
substance.
2. Instead of retaining the voltage unchanged, we put a unique charge
on the capacitor two times: (first) when its dielectric is vacuum,
and (second) when its dielectric is the substance under
measurement. The geometry of the conductors remains unchanged.
Since it is supposed that the capacitance of the dielectric is
increased with the dielectric, the ratio of the potential
difference between the plates in the first state to the potential
difference between them in the second state is the dielectric
constant of the substance.
Certainly the above methods don't give the dielectric constant
according to the contents of this article in which it has been proven
that the capacitance of a capacitor depends only on the geometry of
the conductors and not also on its dielectric. Thus, what are in fact
those measured as dielectric constant by these methods?
At the beginning of Section III.D it has been proven that the charge
accumulated on a parallel-plate capacitor with area A and plates'
separation d and a linear dielectric with the permittivity <epsilon>,
which the potential difference between its plates is V, is: (Eq. (6))
Q = (2-<epsilon>[0]/<epsilon>)<epsilon>[0]A/d V
What has been done in the first method above is in fact calculating
(2-<epsilon>[0]/<epsilon>)<epsilon>[0]A/d V
------------------------------------------- =
<epsilon>[0]A/d V
2-(<epsilon>[0]/<epsilon>) = 2-(1/K)
as the dielectric constant (K). And what has been done in the second
method is in fact calculating
Q
---------------
<epsilon>[0] A/d
--------------------------------------------- =
Q
-----------------------------------------
(2-<epsilon>[0]/<epsilon>)<epsilon>[0]A/d
2-(<epsilon>[0]/<epsilon>) = 2-(1/K)
as the dielectric constant (K). Anyhow, what is at present considered
as K is indeed 2-(1/K). Since ideally K at least is 1 (for vacuum)
and at most is infinity, what is measured as K at present (ie in fact
2-1/K) can be at least 1 and at most, for the best linear
dielectrics, 2.
Reviewing different tables of the dielectric constants in different
texts shows that these constants scarcely exceed 2 or 3 for the best
linear dielectrics, although for some materials this constant even
exceeds 100. (For example it is 2 for paper, paraffin, mineral oil,
indian rubber, ebonite, benzene, teflon, mica, wood, polyethylene,
liquid CCl4 and CS2 (while for liquid O2 and A is 1.5), ...., but is
suddenly near 100 for water.) In addition, there is notable
difference between the constants registered in different texts. It
seems that there is a drastic uncertainty in the results obtained by
the above-mentioned methods (esp when there is a huge difference
from 2). The cause of this uncertainty should be searched (maybe in
the nonlinearity of the dielectrics), but anyway it can be said that
for almost all of the best linear dielectrics (the permittivity of
which can be taken infinity) the constant registered as (wrong)
dielectric constant, as the above reasoning predicts, is about 2
(indeed the (true) dielectric constant of these good dielectrics is
infinity).
Separate from the theory, now let's prove physically that the
above-mentioned ratio of the gathered charges in the method 1 can not
exceed 2: Suppose that a parallel-plate capacitor, connected to a
constant voltage, when is dielectricless, gathers a charge Q. In this
state suppose we insert an ideal linear dielectric, with an infinite
permittivity, between its plates. When this linear dielectric is set
in the field between the plates it begins to become polarized, ie by
ordering the molecular electric dipoles of the dielectric the charges
of the capacitor begin to be canceled, but the potential source to
which the capacitor is connected compensates for the canceled charges
of the capacitor in such a manner that the dielectric is always in a
constsant electric field which its presence is essential for the
linear dielectric to maintain the polarization. (Notice the relation
~P=(<epsilon>-<epsilon>[0])~E for a linear dielectric in which when
<epsilon>=<infinity> we shall have ~E=~0 where ~E is arising from
both polarized charge and that part of the conductors' charges which
are gathered by these polarized charges. (If we wish to consider ~E
as the field arising from the polarized charges and the whole charge
of the capacitor, then the <epsilon> won't be infinity (because
indeed in such a case it is not related to only the dielectric but
the role of the conductors (or capacitor) has been added to it).))
Thus, the dielectric can attract, onto the capacitor, some additional
charge at most equal to the original charge of the capacitor (related
to when there is no dielectric). Then, the ratio of the charge of the
capacitor with dielectric to one without dielectric is at most 2Q/Q=2
(and at least is Q/Q=1 when there is no order for the molecular
electric dipoles even in the electric field between the plates).
Surely there are some persons reckoning these reasonings as fantasy.
The following material may help them not to think so:
The current usual prediction for the resonance frequency of a series
RLC circuit which its dielectricless capacitor is parallel-plate,
when its capacitor is filled with a linear dielectric having
dielectric constant K, is that square of the resonance frequency
drops by 1/K. If, in addition, the limitation of 2 is also a fantasy
for K in the above-mentioned 1/K, and K, depending on the used
dielectric, can take amounts like 20, 30, 40, 80, 100, 200, 300, ...,
then we should conclude that the resonance frequency becomes almost
zero when these dielectrics are used (since eg square root of 1/300
is about zero). A question: Is this the case or not? And in
principle, is this reasonable? But as we saw in this article the
coefficient by which the square of the resonace frequency, when the
dielectric is inserted, increases is:
2(K-1)+(4(K-1){2}+1){1/2}
------------------------- = 2(1-(1/K))+(4-(8/K)+(5/K{2})){1/2}
K
It is seen when K=1, square of the resonance frequency is 1, and when
K is infinity, square of the resonance frequency is 4. This means
that by inserting a linear dielectric we expect that the resonance
frequency will become double at most (when we have an ideal linear
dielectric with infinite permittivity). That ratio of the resonance
frequency with dielectric to the one without dielectric is a number
between 1 and 2 is analogous to that the ratio of the charge gathered
in the capacitor with dielectric to the one without dielectric is a
number between 1 and 2.
III.E. Again parallel-plate capacitor as another test
-----------------------------------------------------
Now we obtain the electrostatic potential energy of the
parallel-plate capacitor mentioned at the end of the section III.B.
by two methods. First, using the relation
U=1/2C(<cap. delta><phi>)[Q]{2} we obtain
U=1/2(<epsilon>[0]A/d)(<cap. delta><phi>)[Q]{2}.
In the second method we use the relation (2), ie U=1/2<integral over
V[h]>~D.~E[Q]dv in which ~E[Q] is the field arising from Q and -Q
(and not also from the polarization charges). We have the following
relation:
~D = <epsilon>~E = <epsilon>(~E[Q]+~E[P]) =
<epsilon>~E[Q]+<epsilon>~E[P] (31)
in which ~E[P] is the field arising only from the polarization
charges of the dielectric set between the two plates. Let's obtain
~E[P] in terms of ~D. Suppose that ~P is the polarization of the
dielectric and ^n is the unit vector in the direction of ~E.
~P.(-^n) is the polarization charge surface density formed adjacent
to the plate bearing the (positive) charge Q, and ~P.^n is the
polarization charge surface density formed adjacent to the plate
bearing the charge -Q. Since ~P=(<epsilon>-<epsilon>[0])~E, we have
~P.(-^n)=(<epsilon>[0]-<epsilon>)E and
~P.^n=(<epsilon>-<epsilon>[0])E which the first is negative and the
second is positive obviously. Then, the electrostatic field arising
from these (polarization) charges in the dielectric is
~P.^n <epsilon>[0]-<epsilon>
~E[P] = ------------(-^n) = ----------------------~E (32)
<epsilon>[0] <epsilon>[0]
and since ~D=<epsilon>~E we have
~E[P]=(<epsilon>[0]-<epsilon>)/(<epsilon>[0]<epsilon>)~D. Combining
this result with the relation (31) yields
<epsilon>[0]-<epsilon>
~D=<epsilon>~E[Q]+ ----------------------~D ==>
<epsilon>[0]
~D=<epsilon>[0]~E[Q]. (33)
Therefore, we have U=1/2<integral over V[h]>~D.~E[Q]dv=1/2<integral
over V=Ad><epsilon>[0]E[Q]{2}dv=1/2<epsilon>[0]AdE[Q]{2}=1/2<
epsilon>[0]Ad((<cap. delta><phi>)[Q]/d){2}=1/2(<epsilon>[0]A/d)(<cap.
delta><phi>)[Q]{2}, which is the same result obtained in the first
method.
Now we proceed to another case. Consider the following figure.
x
<----------->
______________________________
---------------------- /^\
......................| |
......................| | d
______________________| \|/
-----------------------------"
<---------------------------->
l
Figure. A linear dielectric block is pulled into the space between
the plates of a parallel-plate capacitor having the constant
electrostatic potential difference (<cap. delta><phi>)[Q].
The (unshown) width of the plates is w. A linear dielectric block is
along the l-dimension and only the length x is between the plates.
Potential difference between the two plates is constant (equal to
(<cap. delta><phi>)[Q]; we proved this fact beforehand). It is clear
that the charges on that part of a plate of the capacitor which is in
the empty part of the capacitor exert an attractive force on the
polarization charges adjacent to that plate and a repulsive force on
the polarization charges adjacent to the other plate, while the
charges on the empty part of the other plate act a similar work, and
the resultant force of all of these forces is an inward force along
the l-dimension magnitude of which must approach zero when d
approaches zero. Now let's try to obtain this force from the energy
method. First of all, according to what said so far, it is obvious
that with the dielectric displacement the electrostatic potential
energy of the capacitor being only of the capacitor charge (Q and -Q)
does not alter. Thus, only the electrostatic potential energy of the
dielectric and its alteration must be considered.
We know that the surface density of polarization charge of the
dielectric in the capacitor is +P or -P and then the electrostatic
field arising from it is ~E[P]=-~P/<epsilon>[0]. On the other hand,
by using each of the relations (1) and (2) we obtain a unique
expression for the electrostatic potential energy of only the
polarization charges of the dielectric:
(1) ==> U[P]=1/2<integral over V[h]><rho><phi>dv=1/2((-Pd/(2<
epsilon>[0])+0)(Q[P])+(-(-Pd)/(2<epsilon>[0])+0)(-Q[P]))=Pd/(2<
epsilon>[0])Q[P]=Pd/(2<epsilon>[0])P(wx)=P{2}d/(2<epsilon>[0])wx
considering that the potential arising from an infinite charged plate
with the surface charge density <sigma> is -<sigma>/(2<epsilon>[0])d
at the (nonnegative) distance d from the plate, and
(2) ==> U[P]=1/2<integral over V[h]><epsilon>[0]~E[P].~E[P]dv=<
epsilon>[0]/2<integral over V[h]>~E[P]{2}dv=<epsilon>[0]/2<integral
over V[h]>(P/<epsilon>[0]){2}dv=<epsilon>[0]/2 P{2}/<epsilon>[
0]{2} wxd=P{2}d/(2<epsilon>[0])wx.
We have also ~P=-<epsilon>[0]~E[P] from ~E[P]=-~P/<epsilon>[0]. If in
addition we apply the relations (32), (33) and (31), we shall obtain
~P=-<epsilon>[0]~E[P]=(<epsilon>-<epsilon>[0])~E=(<epsilon>[0](<
epsilon>-<epsilon>[0])/<epsilon>)~E[Q] and consequently
U[P] = P{2}d/(2<epsilon>[0]) wx = <epsilon>[0]E[P]{2}d/2 wx =
(<epsilon>-<epsilon>[0]){2}E{2}d/(2<epsilon>[0]) wx =
<epsilon>[0](<epsilon>-<epsilon>[0]){2}E[Q]{2}d/(2<epsilon>{2}) wx.
Since with displacement of the dielectric only x is changed,
dU[P] = <epsilon>[0]E[P]{2}d/2 wdx = (<epsilon>-<epsilon>[0]){2}E{2
}d/(2<epsilon>[0]) wdx = <epsilon>[0](<epsilon>-<epsilon>[0]){2}E[Q
]{2}d/(2<epsilon>{2}) wdx. (34)
We know that the above mentioned force pulling the dielectric into
the capacitor performs some work on the dielectric which, according
to the conservation law of energy, this work must be conserved in
some manner. By pulling inward, this force not only causes forming
more polarization charges, but also alters (and in fact increases)
the kinetic energy of the dielectric block. Thus, the above mentioned
work is conserved both as the electrostatic potential energy of the
formed polarization charges and as the alteration of the kinetic
energy. We show this work as dW and the alteration of the
electrostatic potential energy as dU[P] and the alteration of the
kinetic energy as dT. Therefore, we have:
dW=dU[P]+dT & dW=F[x]dx ==> F[x]dx=dU[P]+dT )
(34) )
F[x]dx = <epsilon>[0](<epsilon>-<epsilon>[0]){2}wd/(2<epsilon>{2}) E[
Q]{2}dx + dT = <epsilon>[0](<epsilon>-<epsilon>[0]){2}/(2<epsilon>{2
}) w (<cap. delta><phi>)[Q]{2}/d dx + dT (35)
It is obvious that if in an especial case we have dT=0 then we shall
have
F[x] = <epsilon>[0](<epsilon>-<epsilon>[0]){2}/(2<epsilon>{2}) E[Q]{
2}wd = <epsilon>[0](<epsilon>-<epsilon>[0]){2}/(2<epsilon>{2}) w (<
cap. delta><phi>)[Q]{2}/d = 1/2 ((K-1){2}/K{2})<epsilon>[0]E[Q]{2}wd
(36)
(It is seen that as was predicted beforehand, this force will
approach zero if d approaches zero.)
Observing the present current mistakes (including what we saw about
the capacitance and electrostatic potential energy of a capacitor) we
see the following relation instead of Eq.(35) in the present books of
Electricity and Magnetism or Electromagnetism:
F[x]dx = 1/2 (<epsilon>-<epsilon>[0]) w (<cap. delta><phi>)[Q]{2
}/d dx = 1/2 (K-1)<epsilon>[0]E[Q]{2}(wd)dx (37)
where it is supposed that (<cap. delta><phi>)[Q] remains constant.
(How? I don't know (!) because even by connecting the two plates to a
battery the voltage of the battery is equal to the sum of
<cap. delta><phi>)[Q] and the potential difference caused by the
dielectric (also see the beginning part of Section III.D).)
And also by mistake the following general result (instead of the
especial result (36)) is inferred from the relation (37):
F[x] = 1/2 (<epsilon>-<epsilon>[0]) w (<cap. delta><phi>)[Q]{2}/d =
1/2 (K-1)<epsilon>[0]E[Q]{2}wd
Practical comparison of the above relations for experimental testing
of the truth of Eq.(35) should be possible preparing ideal conditions
and regarding fringing effects at the edges of the capacitor and
considering the real value of K (see Note section in the previous
section).
IV. Two kinds of potential difference for a capacitor
-----------------------------------------------------
At present in all the textbooks of Electricity and Magnetism wherever
electrostatic potential difference between the two conductors of a
capacitor is concerned if to its producer source, ie the battery, has
been pointed implicitly or explicitly, it is shown or stated
implicitly or explicitly that this electrostatic potential difference
is equal to the potential difference between the two poles of the
battery that has charged the capacitor. But we now shall prove easily
that the electrostatic potential difference between the two
conductors of a capacitor is twofold compared with the potential
difference between the two poles of the battery which has charged it.
Suppose that the potential difference between the two poles of the
battery is <cap. delta><phi> and the electrostatic potential
difference between the two conductors of the capacitor is
<cap. delta><phi>'. It is obvious that if the charge collected on the
capacitor is Q, the battery has transmitted it through itself under
the potential difference <cap. delta><phi> and then has given it an
energy equal to Q<cap. delta><phi>. But Eq.(3) states that the
electrostatic potential energy of the capacitor is
1/2 Q<cap. delta><phi>'. According to the conservation law of energy
then we must have Q<cap. delta><phi> = 1/2 Q<cap. delta><phi>' or
<cap. delta><phi>' = 2<cap. delta><phi>.
A simple physical reasoning shows this fact too: When stating that
the electrostatic potential energy between the two conductors of the
capacitor is <cap. delta><phi>' we mean that supposing that all the
capacitor charges are fixed, if supposedly a one-coulomb external
point charge starts to move from one of the two conductors under the
influence of the electrostatic force of the capacitor until it
reaches the other conductor, the work performed on it by this force
will be <cap. delta><phi>', without any change in the charges on the
conductors. But if we suppose that the magnitude of the charge on
each conductor of the above capacitor is one coulomb and it is
possible that charges separate from a conductor and moving in the
space between the two conductors reach the other conductor, then the
total work performed on this one-coulomb charge by the electrostatic
force of the capacitor will not be certainly equal to
<cap. delta><phi>', because with each transmission of some part of
the charge, magnitude of the charge on each conductor (and
consequently the electrostatic field between the two conductors) is
decreased and does not remain unchanged as before. The above argument
shows that this work will be 1/2 <cap. delta><phi>', because this is
in fact the same work done by the battery for charging the capacitor
being conserved in the capacitor in the form of potential energy
which is being released now. We show this matter in an analytical
manner too: Suppose that our capacitor is a parallel-plate one and
its charge is Q. If a separate Q-coulomb charge travels from a plate
to the other one, the work performed on it will be
Q d
QEd = Q ---------- d = ---------- Q{2}, (38)
<epsilon>A <epsilon>A
while for calculating the work performed on the charge of the
capacitor itself being plucked bit by bit traveling from a plate to
the other one, we should say that the work performed on a
differential charge -dQ (note that dQ is negative), similar to (38),
is
Q+dQ d
(-dQ)Ed = -dQ ---------- d = - ---------- (Q+dQ)dQ.
<epsilon>A <epsilon>A
Sum of these differential works is
<integral from Q=Q to 0>-d/(<epsilon>A) (Q+dQ)dQ =
1/2 d/(<epsilon>A) Q{2}
which is half of the previous work (shown in Eq.(38)).
Thus we should expect to have 2<cap. delta><phi>=d/(<epsilon>[0]A)Q
when a battery with the potential difference <cap. delta><phi> has
charged a parallel-plate capacitor, while hitherto it is thought that
<cap. delta><phi>=d/(<epsilon[0]>A)Q. Since all the parameters of
both the recent relations are measurable (<cap. delta><phi> by
voltmeter), the truth or untruth of each can be tested practically.
We should notice a point. When connecting a voltmeter to the two
conductors of a charged capacitor, it measures <cap. delta><phi> not
<cap. delta><phi>', because its operation is based on passing a weak
electric current through a circuit in the instrument and measuiring
the potential difference between the two ends of the circuit; and
passing of a current means in fact the same being plucked of the
capacitor charge bit by bit from the conductors, and then the
voltmeter measures <cap. delta><phi>.
We should also say that there is no need that in the existent
calculations of electrical cicuits the potential difference of each
capacitor to be made double, because in these calculations the same
<cap. delta><phi> has been in fact intended not <cap. delta><phi>',
because the electric current passing through the circuit including
the capacitor is the same process of gradual loading and unloading of
the capacitor, not passing of charge through the space between the
two conductors of the capacitor retaining the capacitor charge
unchanged. Therefore, it is proper to give <cap. delta><phi> a name
other than the electrostatic potential difference which is the name
of <cap. delta><phi>'. Let's call it (ie <cap.delta><phi>) as
circuital potential difference of the capacitor. In this manner when
it is necessary to apply closed circuit law we must consider just
this circuital potential difference when passing the capacitor not
its electrostatic potential difference.
Now, again, consider a closed circuit of a battery, with the
potential difference <cap. delta><phi>, and a capacitor, with the
capacitance C. Let's investigate the usual method of analysis of RC
(or generally RLC) circuits and see what the difficulty is in it.
Without missing anything we suppose that the circuit has no
resistance (ie R=0). When a differential electric charge dQ passes
through the battery causes a differential change in the electrostatic
energy of the capacitor. In the first instance it seems that when the
differential charge dQ passes through the battery it gains the
differential energy <cap. delta><phi>dQ which, as a rule according to
the conservation law of energy, this same energy must be conserved in
the capacitor in the form of d(Q{2}/(2C)), and then
<cap. delta><phi>dQ=d(Q{2}/(2C)) ==> <cap. delta><phi>dQ=(Q/C)dQ ==>
<cap. delta><phi>=Q/C ==> <cap. delta><phi>-Q/C=0
which is just the same result which we could obtain from the
closed circuit law by traveling one time round the circuit if the
potential difference between the two conductors of the capacitor was
taken electrostatic potential difference, ie <cap. delta><phi>'=Q/C,
not circuital potential difference, ie Q/(2C)! The difficulty is that
the relation <cap. delta><phi>dQ=d(Q{2}/(2C)) is not necessarily
true, for this reason: If we had a mathematical relation, in the form
of an equality, between the energy given by the battery and the
electrostatic energy stored in the capacitor (ie Q{2}/(2C)), we
could differentiate from each side of the equality relation and
understand that the change of energy in the capacitor in the form of
d(Q{2}/(2C))(=Q/CdQ) is exactly arising from what the differential
change in the battery. But since there is no such a relation, we
cannot necessarily infer that change of energy in the capacitor in
the form of Q/CdQ is arising from passing of the charge dQ through
the battery and consequently from differential change of
<cap. delta><phi>dQ in the energy given by the battery, because eg by
writing Q/(2C)(2dQ) instead of Q/CdQ we can claim that this change of
energy in the capacitor is arising from passing of the charge 2dQ
through the battery and consequently from differential change of
<cap. delta><phi>(2dQ) in the energy given by the battery (ie
<cap. delta><phi>(2dQ)=Q/(2C)(2dQ)), and the previous reasonings
showed that incidentally this is the case.
Thus, we should bear in mind that in the analysis of RLC circuits we
must attribute only the circuital potential difference, ie Q/(2C),
not the electrostatic potential difference, ie Q/C, to the capacitor
of the circuit. (Refer to the discussion of RLC circuit in this
article.) Also it is notable that since current instruments indeed
measure capacitance of a capacitor using the formula
C=Q/<cap. delta><phi>' while taking <cap. delta><phi> instead of
<cap. delta><phi>', they give us in fact Q/<cap. delta><phi> =
Q/(<cap. delta><phi>'/2) = 2(Q/<cap. delta><phi>') = 2C as the
capacitance; in other words what they measure as capacitance is in
fact double the capacitance. In this manner what we see as 2C in the
equations (29) and (30), for example, is the same amount our current
instruments give as the capacitance.
It is necessary to note the influence that inattention to the
above-mentioned problem (ie difference between <cap. delta><phi> and
<cap. delta><phi>') has on the results of the experiments of Millikan
and Thomson for determining charge and mass of the electron (and
similarly positive ions).
In the experiment of Millikan the electric charge of each charged oil
droplet is proportional to k/E in which k is the coefficient of
proportion of Stokes and E is the electrostatic field between the two
plates of the parallel-plate capacitor used in the experiment. As we
know E between the two plates of a parallel-plate capacitor is equal
to the electrostatic potential difference <cap. delta><phi>' divided
by the distance d between the two plates. So the charge of each
droplet is proportional to k/<cap. delta><phi>'. But for practical
determination of <cap. delta><phi>' the potential difference read by
the voltmeter connected to the plates of the capacitor is considered
erroneously, while as we said this potential difference,
<cap. delta><phi>, which we called it as circuital potential
difference, is half of <cap. delta><phi>'. In other words as a rule
the quantity so far recognized as the charge of a droplet should be
two times larger than the real charge of the droplet and then the
electron's charge obtained from the numerous repetitions of the
experiment of Millikan should be really half of what is at present
accepted as the charge of electron.
But this is not the case because the experiment of Millikan plainly
lacks sufficient accuracy (and a tolerance up to half of the real
amount seems natural for it because certainly it is unlikely that
the electrons are added or deducted only one by one). In fact it
seems that the results of this experiment have been adapted in some
manner for being in conformity with the results of the exact
experiment of determination of electric charge of electron by X-ray.
(As we know in this experiment the wavelength of X-ray can be
determined by its diffraction via a diffraction grating with quite
known specifications, and then having this wavelength and Bragg's
equation and analyzing the diffraction of the ray via a crystal
lattice the lattice spacing, d, of the crystal can be determined;
thereupon considering the molecular mass and crystal density
Avogadro's number N[0] can be calculated with sufficient accuracy and
using it in the formula F=N[0]e, in which F is the Faraday constant
and e is the charge of electron, e can be obtained which is the same
that has been accepted at present as the charge of the electron.)
In the experiment of Thomson too, for evaluation of q/m related to
the charge and mass of the electron in the cathodic ray, this
quantity, ie q/m, is obtained proportional to the electrostatic field
E between the two plates of the parallel-plate capacitor through
which the cathodic ray passes. But again for practical determination
of E the above-mentioned error is repeated and while E is really
equal to <cap. delta><phi>'/d the amount read on the voltmeter,
<cap. delta><phi>, (which is in fact equal to 1/2<cap. delta><phi>')
is set instead of <cap. delta><phi>'. In other words as a rule the
quantity hitherto considered as q/m of the electron (in the
experiment of Thomson) should be half of its real amount. Then, to
obtain the real value of q/m we must multiply the value accepted
presently as q/m by 2.
But here we should say that it seems that this experiment (or any
other similar one) is not accurate in determining q/m of electron or
positive ions since in it a shooting motion has been assumed for the
electron in the cathodic ray (or for the positive ion in the positive
ray), while as explained in detail in the 12th article of this book
we must consider for it a longitudinal wave motion in the gas medium
existent in the tube without any charge transferring, and it seems
that such a wave motion, although has many similarities with the
shooting motion, is not exactly the same shooting motion and has
difference with it. Thus, it is necessary to doubt what has been
accepted as the mass of electron.
Hamid V. Ansari
My email address: ansari18109<at>yahoo<dot>com
The contents of the book "Great mistakes of the physicists":
0 Physics without Modern Physics
1 Geomagnetic field reason
2 Compton effect is a Doppler effect
3 Deviation of light by Sun is optical
4 Stellar aberration with ether drag
5 Stern-Gerlach experiment is not quantized
6 Electrostatics mistakes; Capacitance independence from dielectric
7 Surface tension theory; Glaring mistakes
8 Logical justification of the Hall effect
9 Actuality of the electric current
10 Photoelectric effect is not quantized
11 Wrong construing of the Boltzmann factor; E=h<nu> is wrong
12 Wavy behavior of electron beams is classical
13 Electromagnetic theory without relativity
14 Cylindrical wave, wave equation, and mistakes
15 Definitions of mass and force; A critique
16 Franck-Hertz experiment is not quantized
17 A wave-based polishing theory
18 What the electric conductor is
19 Why torque on stationary bodies is zero
A1 Solution to four-color problem
A2 A proof for Goldbach's conjecture
