How many joules --> e.m. waves?

[LUDWIK KOWALSKI <kowalskil@alpha.montclair.edu>]

Two weeks ago Alex Burr wrote that he was able to "hear" electromagnetic 
waves of a discharging capacitor with a radio set. I also remember hearing 
them in that way. This posting ends with a related problem. But first let 
me remind you that the emission of e.m. waves is contradicted by the 
common "discharge current" formula:

             i=(V/R)*exp(-t/RC)  where V is the initial d.o.p.

To show the conflict calculate the amount of heat, H, produced during the
discharge (an intergral of i^2*R*dt from zero to infinity); the result is 
H=0.5*C*V^2. In other words, FOR ANY FINITE R, the amount of heat is equal 
to the initial electrostatic energy. No energy is left for the e.m. waves!
But sparking capacitors do emit waves. This is a reminder that the above 
formula is only an approximation; it is not valid for very small RC (fast 
discharging).

Suppose that V=10 volts, C=0.01 farads (50 joules of energy is available) 
and that R=0.001 ohms. To simplify assume that the wire through which the 
current is flowing is a circular loop whose radius is 2 cm. How many joules 
are taken away in the form of electromagnetic waves?
                                                        Ludwik Kowalski

P.S. The problem is related to the parallel capacitors "paradox" posted 
by Dan (partial discharging, of C1 into C=C1+C2=2*C1). Here is what 
Leigh wrote about this:

> The charges redistribute, so that each capacitor has charge Q/2.
> In a resistanceless circuit this redistribution will likely be of
> an oscillatory nature, the energy being radiated away. It is very
> difficult to make the circuit inductanceless! At any instant the
> energy which can be localized to the circuit is given by

                   1     2    1     2
              U = --- C V  + --- L I
                   2          2

> The localization of energy is, at best, imperfect, and depending
> on geometric considerations the energy will gradually leak into
> the radiation field.
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