So, I have been having a hard time understanding why there is even a complex phase for EM waves:
$$\phi=\exp[i\omega t]=\cos(\omega t)+i\sin(\omega t)$$
Don't understand why it is there? Any one have an explanation?
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The field itself has to be real, of course. The imaginary part is added as a computational aid. Dealing with an exponential, whose derivative is itself, is much easier than dealing with a cosine. So we add the imaginary part, do our computation, then dump the imaginary part of the result. It's all simple when the computations are all linear. Sometimes, though, one is called upon to square a field, as in calculating intensity, and in those cases you have to be a little more careful. The resulting expressions then involve complex conjugation. Bottom line: it's a computational convenience.
garyp
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