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From: fransvo@maestro.htsa.aha.nl (Frans van Otten)
Newsgroups: sci.electronics,sci.physics
Subject: Re: HV Cap Fun!
Keywords: capacitor,energy,paradox
Message-ID: <955@maestro.htsa.aha.nl>
Date: 2 Jun 89 08:55:05 GMT
References: <4924@m2c.M2C.ORG> <3806@mit-amt> <20772@quacky.mips.COM> <8739@fluke.COM>
Reply-To: fransvo@htsa.UUCP (Frans van Otten)
Followup-To: sci.electronics
Organization: AHA-TMF (Technical Institute), Amsterdam The Netherlands
Lines: 52

Vaso Bovan writes:

>A Paradox of Capacitor Energy Storage
>
>I've heard several competing answers to this paradox. None is entirely
>satisfactory:
>
>Consider an ideal 2uF (for computational ease) capacitor charged by a 10 volt 
>source. Eventually, the energy stored is (1/2)*CV^2=100 joules.

>Consider the capacitor to be isolated from the voltage source, and then
>directly shorted across an identical (ideal) capacitor. Eventually, the
>voltage across each capacitor will be 5V. Now, there are two equally
>charged capacitors, each storing (1/2)*CV^2= 25 joules, for a total of
>of 50 joules.  What happened to the other 50 joules ?

If the circuit was really ideal, you're doing something "illegal".  The
voltage over the capacitor depends on the electrons in it.  They can't
move in zero time so the voltage over the capacitor can't be discontinu.
When you try to do that anyway, you get the same "illegal" situation as
when you short-circuit a voltage-source or disconnect a current-source.

So what happens:

  1. There is a non-zero voltage over a zero-Ohm connection;
  2. This results in a infinitely high current;
  3. Infinity current times zero Ohm can mathematically be anything;
  4. After zero time the capacitors are equally charged;
  5. Some energy can be dissipated.

Popular math:

i = infinitely many Amperes
R = 0 Ohm
t = 0 seconds

energy =                power          * time
       =      voltage     *   current  * time
       =       R * i      *      i     * t
       =   0 * infinity   *   infinity * 0
       = ( 0 * infinity ) * ( infinity * 0 )
       =       ?          *         ?
       =                  ?  (not neccisarily zero)

This for the impossible theoretical question.  In practice, this situation
is impossible because there will always be a non-zero resistance and other
non-ideal values.  A calculation then shows that the amount of dissipated
energy does not depend on the resistor value.
-- 
Frans van Otten                     |   fransvo@maestro.htsa.aha.nl    or
Algemene Hogeschool Amsterdam       |   fransvo@htsa.uucp              or
Technische en Maritieme Faculteit   |   [[...!]backbone!]htsa!fransvo