Phase Difference between Electric and Magnetic Field in EM waves?

Question

Is it possible to get a phase difference between an Electric Field ($E$) and a Magnetic Field ($M$) in an electromagnetic wave and why are they in phase?

Answer 1

Yes. The relationship between $E_i$ and $H_j$ is given by the impedance $Z = \sqrt{\epsilon/\mu}.$ It can have an imaginary part, introducing a phase $\arg(Z)$. For example in the Debye model, you have $\epsilon \sim 1 + \alpha/(1 + i \omega \tau) $. More generally, the conductivity of a medium leads to losses, i.e. $\Im(k) < 0$. You effectively get an imaginary speed of light $c^* = \sqrt{\mu \epsilon}$ with $\Im{\epsilon} \sim$ -conductivity, leading to an imaginary impedance $Z = \omega \mu /k$. The imaginary part of $k$ represents the attenuation of the wave as it propagates through a medium. This attenuation is due to the absorption of energy by the medium. You don't have this in free space, where $c^* = c$ and $Z = \sqrt{\epsilon_0 / \mu_0}$.

Answer 2

Classical electrodynamics developed in order to fit the observations and data and be predictive of new setups.

Maxwell's equations gave the final mathematical model that fits all observations of light and is accurately predictive of any new interactions with matter. In vacuum, these equations lead to solutions where light is an electromagnetic wave with electric and magnetic fields in phase.

One cannot introduce a phase difference between electric and magnetic fields in the theory,and still fit the data in vacuum ( or where matter can be ignored in the solutions of the equations), so the answer is no,not in vacuum. It is possible mathematically but the theory would have no connection with the measurements and observations. Within and with matter this is a different story, and has to be examined for the given problem. For example see the answers here and here.