See the original text.
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.)
See the continuation of this article in
http://www.geocities.com/hvansari/6.txt
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
My email addresses: hamidvansari<at>yahoo<dot>com or
hvansari<at>gmail<dot>com
To see all the articles send an email to one of my above-mentioned
email addresses.
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)".)
~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.)
See the continuation of this article in
http://www.geocities.com/hvansari/6.txt
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
My email addresses: hamidvansari<at>yahoo<dot>com or
hvansari<at>gmail<dot>com
To see all the articles send an email to one of my above-mentioned
email addresses.
