# Magnetic field of a circular wave

• kediss

#### kediss

Hi !

Does $\vec{H}=\frac{1}{\eta}\hat{k}\times \vec{E}$ always suit an electromagnetic wave ?

Because I'm getting some inconsistency for a circular wave

For instance:
$\vec{E}=E_0(\hat{e}_x+e^{j\frac{\pi}{2}}\hat{e}_y)e^{-jkz}$ (propagating $\hat{e}_z$)
$\Rightarrow\vec{H}=\frac{E_0}{\eta}(\hat{e}_y-e^{j\frac{\pi}{2}}\hat{e}_x)e^{-jkz}$

The poynting vector results:
$\vec{S}=\vec{E}\times\vec{H}=\frac{E^2_0}{\eta}e^{-2jkz}(1+e^{j\pi})\hat{e}_z=\vec{0}$ which is quite disconcerting :/

Thx a lot for your help !

Your expression is not circular wave. Actually, it is not a wave at all, it is not time-dependent.

However, neglecting prefactors and with fixed orthogonal E and H:
$\vec{E} \propto (e_x - e_y)$
$\vec{H} \propto (e_x + e_y)$

=> $S_z = E_x H_y - E_y H_x \propto 1-(-1) = 2 \neq 0$

I think that if you use complex representation, the Poynting vector is calculated as
S=1/2(E x H*) where the star means complex conjugate.
With this definition, the real part of S is the average power density.