Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / Section E / E15 / d / Solution

In vacuum, Maxwell's equations in SI units areThe displacement current term makes the last equation consistent with the charge continuity equation. In a source-free region, take the curl of Faraday's law and use $\nabla \times (\nabla \times \mathbf{E})=\nabla (\nabla \cdot \mathbf{E})-{\nabla }^2\mathbf{E}$. This givesA plane electromagnetic wave therefore propagates at $c=1/\sqrt{{\mu }_0{ϵ}_0}$. The divergence equations make both fields transverse; the curl equations giveThe electric fields and magnetic fields are in phase and perpendicular to one another and to the propagation direction. Their energy densities are equal:For electric-field peak amplitude $E_a$, the time-averaged intensity is $⟨S⟩={ϵ}_{0}c{E}_a^2/2$. Mutually coupled changing electric fields and magnetic fields transport energy through vacuum, even where no charges or conduction currents are present.