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



On 16 Apr 1997 10:33:05 -0700 (PDT) Mark Shapiro wrote:

The radiation resistance is not easy to calculate, because it depends
critically on how well coupled the oscillator is to the radiation fields,
i.e. on how good an antenna the circuit is. This will depend, in turn on,
the frequency of the oscillation. If some part of the circuit has a
dimension of order c/f, it will radiate reasonably well. Otherwise, most
of the energy will indeed end up as heat.

Please correct me if the interpretation below is wrong. The value of f,
in our problem (C=0.01 F, L=20 nH, R=0.001), is either 11254 or 10527 Hz,
depending on wheather or not R is neglected. Thus c/f is about 3e8/1e4
or 30,000 m. The size of our antenna (r=0.02 m) is too small to radiate.
Practically all electrostatic energy will be dissipated as heat in R.

But we know that capacitors discharged through short wires spark emitting
electromagnetic waves (which can be heard with radio sets). And if we accept
the conclusion that the dominant part of the initial energy becomes heat
for a single C why was it so hard to accept this for the partial discharging
of C1 into C=C1+C2? This thread evolved from the parallel capacitors "energy
paradox" posted by Donald.

To learn something (from those who know better) change the parameters of
the RLC cirquit so that c/f (the wavelength) is close to the radius of the
wire loop. How many joules are emitted? Here is the new version of the same
problem. I hope somebody will know how to solve it.
...........................................................................
We are discharging a capacitor in the LCR circuit. Suppose that V=10000 volts,
C=30 pF (1.5 mJ of energy is available) and that R=0.1 ohms. To simplify
assume that the wire through which the current is flowing is a circular loop
whose radius is 3 m. According to Alex's formula r=300 cm corresponds to
L=0.083 mH. How many mJ are taken away in the form of electromagnetic waves?
The setup is in the interplanetary vacuum.
...........................................................................
Ludwik Kowalski
P.S.
Bob Schiamanda wrote:

If you are looking for some universal radiation resistance that is fixed
once R, L and C are specified, there is none! The radiation resistance (and
the fraction of starting energy that is radiated) are determined by how
good an antenna your circuit is! This depends on the geometry; bending a
wire will change it!

What else should be specified to avoid an ambiguity about "how good" the
anthena is? It is a perfect flat circular wire loop. The wire thickness is
very small in comparison with the loop radius. The capacitor itself has
plates whose areas are small. Feel free to add additional specifications,
if needed to solve the problem.
...........................................................................