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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)

Qmechanic
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Sørën
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2 Answers2

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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.

Enrique Mendez
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$\vec E$ 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 $\vec B$.

ZeroTheHero
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