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If $\eta=\sqrt{\mu/\epsilon}$ is the wave impedance of a homogeneous infinite medium in which a free-space electromagnetic wave propagates (i.e. plane wave), then we know that the magnitude of the electric and magnetic fields are related by

$$ |E|=\eta |H| $$

Will this simple equation generally hold also for any field at resonance (e.g. say for an electromagnetic field resonating inside some cavity of arbitrary shape) or is it only known to be true for plane waves?

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For resonance, consider the simple case of bouncing off perpendicular to a mirror. This looks like two plane waves travelling in opposite directions. There will be points where the E fields interfere completely constructively (double) while the B fields interfere completely destructively (zero). So no, in general E is not proportional to B.

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  • $\begingroup$ This would be mainly because of the phase difference between the two waves, not because the amplitudes are different! $\endgroup$
    – user135626
    Jul 23, 2017 at 12:59

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