Catch-22 said:
Mathematically, any variable divided by zero
is considered 'undefined' or a non-number.
That much has nothing to do with electronics
or any other similar equation where division
by zero occurs.
DJ Delorie's answer was very good from the
practical point of view.
Laodao's answer is analytically recursive.
It points out that voltage is a function of resistance.
So your version of Ohm's law becomes i=v(r)/r.
But if you solve for v(0)=0 then i=0/0.
Which is it then, is the current infinite or zero?
If the voltages source is a capacitor that
could instantaneously discharge, then both
answers could be considered correct in the
sense that the discharge would be both infinite
and zero in an instantaneous amount of time.
You wouldn't be able to separate the two measurements.
Mike Warren's reply to Abstract Dissonance is
profound. Does an infinite precision measurment
take infinite time to measure? If so how can
we measure the current in an instaneous discharge?
If the denominator in your version of Ohms law
is finite but goes 'slowly' goes to zero then
it's more of a limits/calculus problem as Alan B
pointed out. In terms of measuring, you'd have to measure
or at least prove that you can always measure
the current as the resistance goes to zero.
At some point though you will not be able to
do this precisely. I cannot measure resistance
or current with infinite precision.
So using a value like zero for current in the
denominator can be considered impractical.
Bill Bowden brings up a good question about
alternating current. For instance what if there's
an impedance mismatch. Then there is current
flowing in both directions at the same time
and they can be of different values and they
can be frequency dependent as DJ Delorie points out.
The fact of the matter is that there are no zero resistance conductors or
infinite precision measure devices or anything like that. These are all
mathematical idealizations of a property that exists in the physical world.
Even super conductors do not have zero resistance(and many pople like to
think). Theres a very fine division between what happens in the physical
world and what happens in our minds. You can hypothesize all day long about
"what if's" and all it does it lead to mathematical contradictions like
this. I'm not saying its necessarily a bad thing but that one has to be
very very careful and really know the situation.
I think the sad fact is that "infinity" cannot exist but in our minds. What
does infinite current mean? If we expand the definition of current then it
must mean an infinite number of electrons(or protons or whatever you use as
your basis for current) flowing past a point in one second. Why? because we
already "know" they cannot have infinite speed. But theres a contradiction.
There are only a finite number of electrons in the universe.
Some people like to "what if" everything to death... "what if there was an
infinite number of electrons" or "what if there was zero resistance
conductors". These are very similar to questions like "What if there is a
god" or "What if I lived for ever". The only real thing that can come of
these types of questions is a waste of time(and it can be measured too).
I think people ask these types of questions because they want to feel smart
but are to stupid or to lazy to learn about much more important things that
actually have answers. They can sit around all day asking these questions
and when they don't arrive at an answer they won't feel stupid because no
one else has them either(or could prove them wrong).
It should be a crime to mention infinity outside the scientific community.
Far to many people don't have a clue about what it means(Actually no one
does but some more than others).
Anyways, thats my rant for today.
