Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / J / J2 / Solution

In SI units Maxwell's equations areThe given vector identity yieldsDivide by ${\mu }_0$ and collect the two energy derivatives to obtain the local identity ${\partial }_t\mathcal{U}+\nabla \cdot P+E\cdot J=0$. Integration over the fixed region and the divergence theorem give the required integral conservation law. Its terms are, respectively, the rate of change of stored electromagnetic energy, the outward electromagnetic energy flux through the boundary, and work per unit time done by the field on charges. The last term transfers energy from the field to matter when positive.

In vacuum put ${\rho }_e=0$, $J=0$ and write $\varphi =kz-\omega t$. Faraday's law gives the plane-wave magnetic fieldBoth divergence equations are satisfied, and the remaining curl equation gives $k^2={\mu }_0{ϵ}_0{\omega }^2$. Hence $\boxed{\omega =ck,c=({\mu }_0{ϵ}_0{)}^{-1/2}}$, with $B_y=E_x/c$. Here the corresponding field is the pure wave, with no added static background.

Its electric and magnetic energy densities are equal. Explicitly,The energy flux points in the propagation direction, toward increasing $z$.