# Relation between the electric and magnetic fields for an arbitrary electromagnetic wave

When solving the wave equation for electromagnetic waves, $$\nabla^{2} \mathbf{E}=\mu_{0} \varepsilon_{0} \frac{\partial^{2} \mathbf{E}}{\partial t^{2}}, \quad \nabla^{2} \mathbf{B}=\mu_{0} \varepsilon_{0} \frac{\partial^{2} \mathbf{B}}{\partial t^{2}}$$

assuming solutions of the form $$\mathbf{E}=\mathbf{E}_{0} f\left(\hat{\mathbf{k}} \cdot \mathbf{x}-c t\right)$$, it arises that the relation between the electric and magnetic fields is $$\mathbf{B}=\frac{1}{c} \hat{\mathbf{k}} \times \mathbf{E} \tag{1}$$

However, $$\mathbf{E}=\mathbf{E}_{0} f\left(\hat{\mathbf{k}} \cdot \mathbf{x}-c t\right)$$ is actually a travelling plane wave, since it has the structure $$F(\vec{x}, t)=G(\vec{x} \cdot \vec{n}-c t)$$.

Is $$(1)$$ still a valid relation between electric and magnetic fields for non plane waves?

• As I recall, even non traveling plane waves can be expressed as traveling plane waves using Fourier series if f is square integrable , so it should hold under those conditions at least. Jun 9, 2021 at 21:36
• The statement is only true in an isotropic medium. In anisotropic medium, the electric field is not perpendicular to wavevector in general. Aug 2, 2021 at 14:00