Are $\mathbf{E}$ and $\mathbf{B}$ really always in phase in EM waves?

Question

I've been taught that in EM waves the electric and magnetic field are in phase. Nevertheless using Maxwell equation in absence of sources and solving the wave equation $$\square f=0$$ in cylindrical coordinates and under clindrical symmetry ($\frac{\partial f}{\partial \phi}=\frac{\partial f}{\partial z}=0$) one can get as solution the two fields with a only one nonzero component, and in the limit of $r>>\frac{\omega}{c}$ has:

$$\begin{cases} E_z\approx E_z^0\frac{1}{\sqrt{r}} \mathrm{sin}[\omega(\frac{r}{c}-t)-\frac{\pi}{4}] \\ B_{\phi}\approx-E_\phi^0\frac{1}{c\sqrt{r}} \mathrm{sin}[\omega(\frac{r}{c}-t)-\frac{\pi}{4}] \end{cases}$$

The two fields are out of phase! So is the in-phase relation between the two fields really a universal rule? Or is it valid only in some cases (such as the simplest case of a plane wave)

Answer 1

It's only for plane waves. It comes from the Maxwell's equations when assuming a plane wave solution in free space.

Secondly, a minus sign here isn't out of phase. A minus sign indicates a change in direction. You'll note that $E\times B$ is still pointing away from the axis, and the magnitude oscillates in phase. The interesting case is when the phase is not $0$, or $\pi$, which I believe occurs in metals under incident radiation.

Answer 2

$\vec E$ and $\vec B$ are in phase only in lossless media, for which the conductivity $\sigma=0$. In general, lossy media are modelled as having a complex permittivity $\epsilon$: the complex part is proportional to $\sigma$ and this leads not only to an exponential decay of the amplitude but also to a phase shift between $\vec E$ and $\vec B$.