I was reading the Jackson and suddenly in one line they wrote the electric field as:
$$E(x,t)=E(x)e^{-iwt}$$
why is this valid?
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$\begingroup$ Please type in the necessary information into the question. Pasting images is not encouraged as they don't allow the content to be searchable. Also please consider using MathJax for mathematical expressions $\endgroup$– ohneValOct 22, 2020 at 9:07
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1 Answer
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This is the complex wave equation,
$$ e^{\boldsymbol{i} z} = \cos z + \boldsymbol{i}\sin z $$
With $z=-\omega t$
Note, the imaginary unit $\boldsymbol{i}$ is missing in what you wrote. They are just saying that the solution being sought is a plane wave.
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1$\begingroup$ Hi, thank you for the help. I found another answer that says: As we are concerned with the harmonic modes of the system, we further decompose the fields by Fourier analysis into independent spatial- and time-dependent components. How is Fourier related in this case? If what sought is a plane wave $\endgroup$– WhoOct 21, 2020 at 19:05
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1$\begingroup$ Why not accept this question and ask a new one? Will be clearer. $\endgroup$ Oct 21, 2020 at 19:12