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Assume you have an isotropic dielectric. Now introduce an homogeneous and constant magnetic field and an electromagnetic wave (light) in the same direction of the magnetic field. I have to relate the polarization and the electric field, show that the field produces an anisotropy in the dielectric, find the dielectric permittivity tensor and then show that the dielectric es now optically active, meaning that the dielectric shifts the plane of polarization of the incoming wave.

Here's my attempt: Let $\epsilon$ be the dielectric permittivity tensor, $\epsilon_0$ the usual constant, $\vec{P}$ the polarization vector and $\vec{D}$ the electric displacement vector. Then I get:$$\vec{D}=\epsilon\vec{E}=\epsilon_0 \vec{E} + \vec{P} \Rightarrow \boxed{\vec{P}=\epsilon_0(\epsilon_r-I_3)\vec{E}}$$

But now I'm pretty much stuck. I think there $\textit{may}$ be a mistake in the formulation of the problem I was provided. I think that maybe the magnetic field is homogeneous but not constant, because if it's constant, it doesn't produce any additional electric field, and because of the vector $\vec{k}$ of the wave is in the same direction as $\vec{B}$, then when I use $\vec{H}\times\vec{k}=\omega \vec{D}$, the only field that affects the value of $\vec{D}$ is the magnetic field from the wave, not the "constant" one.

In conclusion, I'd like to know if it's possible (and how to do so) to solve the problem if $\vec{B}=B_0 \vec{u_k}$ and if it's not possible to do so without it being constant, then with an arbitrary magnetic field dependent on time.

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My recollections from Landafshitz on magneto-optical effects is that

$$D=\epsilon E + i f E\times H$$

where $\epsilon$ is the permeability without the external magnetic field, and $f$ is some constant, and $H$ is the magnetic field (Landashitz use $B$ for the magnetic induction but that would be the same here). If I recall correctly, they derive that purely based on some consideration of symmetry, not with a microscopic model of charges.

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