# Given $\tilde {E}$, which formula should I use to find $\tilde {H}$

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

• Use the plane wave formula for plane waves: it is then equivalent to the other, and easier, as detailed in your answer below. – Selene Routley Dec 6 '14 at 12:42

$$\nabla\times E = -\frac{\partial B}{\partial t}$$
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.
$$\vec{k}\times E(\vec{k},\omega) = \omega B(\vec{k},\omega)$$