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Given $\tilde {E}$, there exist two formulas in my book (Cheng) to compute $\tilde {H}$:

  1. Maxwell's formula: $\nabla \times \tilde {E} = -j\omega\mu \tilde {H}$

  2. Plane wave formula: $\tilde {H} = \frac{\hat k\times \tilde {E}}{\eta} $, where $\hat k$ points to the direction of the traveling wave

The question is when to use the more complex version and when to use the plane wave formula.

Can someone please show me what factor decides which formula is used?

Thank you

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  • $\begingroup$ Use the plane wave formula for plane waves: it is then equivalent to the other, and easier, as detailed in your answer below. $\endgroup$ – WetSavannaAnimal Dec 6 '14 at 12:42
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The law of induction formulated in the most common way, that is regarding the fields as depending on space and time, reads

$$ \nabla\times E = -\frac{\partial B}{\partial t} $$

Fourier transforming in space ($\nabla\times E \rightarrow -j\vec{k}\times E$) or in time ($\partial_t B \rightarrow j\omega B$) yields the two formulas you cite.

One may also Fouriertransform in both space and time coordinates, which gives

$$ \vec{k}\times E(\vec{k},\omega) = \omega B(\vec{k},\omega)$$

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