Xref: utzoo sci.electronics:6377 sci.physics:8372
Path: utzoo!utgpu!jarvis.csri.toronto.edu!rutgers!apple!arc!steve
From: steve@arc.UUCP (Steve Savitzky)
Newsgroups: sci.electronics,sci.physics
Subject: Re: HV Cap Fun!
Summary: what paradox?
Keywords: capacitor,energy,paradox
Message-ID: <349@arc.UUCP>
Date: 1 Jun 89 00:51:58 GMT
References: <4924@m2c.M2C.ORG> <3806@mit-amt> <20772@quacky.mips.COM>
Reply-To: apple.com!arc!steve (Steve Savitzky)
Followup-To: sci.electronics
Organization: Advansoft Research Corp, Santa Clara, CA
Lines: 71

In article <20772@quacky.mips.COM> vaso@mips.COM (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.

 call it C1

>Consider the capacitor to be isolated from the voltage source, and then
>directly shorted across an identical (ideal) capacitor.

 call it C2

>                                                         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 ?

... followed by several unsatisfactory responses from several people
who obviously didn't think about either the physics or the electronics
of the situation.

PHYSICS:

Consider the case where there is only a single electron in C1.  It has
potential energy.  When it moves over to the other capacitor, that
potential energy is converted to kinetic energy.  When it gets to C2,
it will have have just the same amount of potential energy again.  Now
it sloshes back to C1.  The system oscillates.  Think of it as a
marble between two hills.

If the wires between C1 and C2 have resistance, some of the kinetic
energy is converted to heat, and the oscillations eventually damp
down.  Think of friction on the marble.

ELECTRONICS:

We have the following circuit:

       ---------[ Z1 ]---------
       |                      |
     -----   C1             -----   C2
     -----                  -----
       |          \  S1       |
       ----------- \-----------

Look familiar?  Those "wires" have inductance and (unless they're made
of superconductors) resistance, represented by Z1.  (Ever try to draw
an inductor in ascii? :-)

When you close the switch, current flows from C1 to C2, and energy is
stored in the magnetic field of the inductors.  When the two
capacitors reach the same charge (voltage), the current keeps flowing,
powered by the collapsing magnetic field of the inductors.
Eventually, all the charge ends up in C2, and the current stops.  Then
the process starts again in the opposite direction.  The system
oscillates. 

If the wires are resistive, some of the stored energy is converted to
heat, and the oscillations damp down.

THE TWO EXPLANATIONS ABOVE ARE EQUIVALENT.

("kinetic energy" and "energy stored in a magnetic field" are
equivalent ways of looking at the energy of a moving charged
particle.) 

Satisfactory?