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

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

1) The e.m. waves energy problem (repeated at the bottom of this message) 
   had an obvious typing error (V=100 volts, not 10, is needed to have 50 
   joules). It would probably be better if L was actaually given; what is 
   the purpose of combining two non-trivial tasks? 

2) I am puzzled by how small the resistance must be to efficiently convert 
   the electrostatic energy into the e.m. waves. This is illustrated by the
   results of calculations shown below. For the value of R=0.001, specified 
   in our problem, 99.9% of electrostatic energy is turned into heat. 

               R(ohms)  -->     0.001   0.0001   0.00001   

                % e.m.  -->      0.1     36.9      90.5

   All calculations were performed by assuming that L=20 nH. Does anybody 
   know what is a typical energy efficiency of a broadcasting station? 

3) The calculations were performed by using the formula for the current 
   i(t) in the serial LCR circuit. The reference used was "Principles of 
   Electricity" by Page and Adams (fourth edition, 1969). As usual, the
   formula for i(t) is parametrized in terms of Q(0)=initial charge, 
   alpha=0.5*R/L and omega_zero= sqrt[1/(L*C)-alph^2). In our problem: 
   q(0)=1, alpha=0.5*R/L=25000 and omega-zero=66144, all in SI units). 
   The formula, derived on pages 300-301), is:

               i(t)=i(0)*exp(-alpha*t)*sin((omega_zero*t)

   where i(0)=[(omega_zero^2+alph^2)*C*V(0)]/omega_zero. Note that our
   V(0)=100 volts and C*V(0) is the initial charge on the capacitor, Q(0).

4) Once the formula is written it can be used to find H (as an integral 
   of i^2*R*dt between zero and infinity). Analytical integration is
   likely to be easy but being out of practice with this craft I simply
   wrote a short program to do this numerically on my Mac. The energies of 
   the e.m. waves were then calculated as 0.5*C*V^2-H. All exotic forms of
   energy (neutrinos, gravitons, etc.) were ignored.

5) The textbbok I used reminded my that first radio waves were detected
   in 1888. "The oscillator used by Hertz consisted essentially of a
   capacitor discharging through an inductance and resistance". Does
   anybody know the values of C, R, L and V for his setup? I would be glad 
   to calculate the corresponding energy conversion efficiency.
............................................................................
6) Here is the original wording of our problem; it was part of an 
   observation that the common capacitor-discharge-current formula,
   i(t)=(V/R)*exp(-t/R*C), is in conflict with the emission of e.m. waves
   and that L must also be considered. This morning Brian reminded us that
   the exponential pulse can be decomposed (Fourier) into a broad spectrum 
   of sinusoidal components. This not enough to provide for the generation 
   of sinusoidal waves.

> Suppose that V=100 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