Past exam of the physics B course of the University of Cambridge / 2011 / Paper 1 / Section B / B6 / d

In charge- and current-free vacuum, Maxwell equations reduce toTaking a curl and using $\nabla \times \nabla \times \mathbf{E}=\nabla (\nabla \cdot \mathbf{E})-{\nabla }^2\mathbf{E}$ givesand the same wave equation for $\mathbf{B}$. For a plane wave $\mathbf{E}=\mathrm{Re}[{\mathbf{E}}_0{e}^{i(\mathbf{k}\cdot \mathbf{r}-\omega t)}]$, $\omega =ck$, $\mathbf{k}\cdot {\mathbf{E}}_0=\mathbf{k}\cdot {\mathbf{B}}_0=0$ and ${\mathbf{B}}_0=\hat{\mathbf{k}}\times {\mathbf{E}}_0/c$. Thus the fields are transverse, mutually perpendicular and in phase; polarization describes the electric-field trajectory at a fixed point.

The energy density and Poynting vector areFor a linearly polarized sinusoid with peak electric amplitude $E_0$, the mean intensity is ${ϵ}_{0}c{E}_0^2/2$. Electromagnetic momentum density is $\mathbf{S}/c^2$; normal absorption produces radiation pressure $⟨S⟩/c$, while ideal retroreflection doubles it. Linear superposition permits interference and standing waves, whose local electric and magnetic fields need not have the traveling-plane-wave equality point by point.