Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 1 / 15D (Electromagnetism)

In SI units the Maxwell equations in free space are

Take the divergence of the Ampère-Maxwell equation. Since the divergence of a curl is zero, Gauss's law givesThuswhich is charge conservation from Maxwell equations. Integrating over a fixed volume and using the divergence theorem yieldsCharge in $V$ is conserved provided no current crosses $\partial V$. For total charge in all space, the corresponding assumption is sufficient decay of $J$ so that the flux at infinity vanishes.

In empty space the fields are divergence-free. Taking the curl of Faraday's law and usinggivesThe same calculation, starting from the Ampère-Maxwell equation, gives the corresponding equation for $B$. Thus every component satisfies the electromagnetic wave equation with speedIts agreement with the measured speed of light identifies light as an electromagnetic wave.

Dot the Ampère-Maxwell equation with $E$, dot Faraday's law with $B/{\mu }_0$, and use the stated vector identity. The result is the local balance lawHere $S$ is the Poynting vector. Integration over $V$ gives the Poynting theoremThe field energy decreases through outward electromagnetic energy flux and through work done on charges.

The source-free equations imply $k\cdot E_0=0$, $\omega =c|k|$, andThereforeIt is parallel to $k$, and its period average isThe average electric and magnetic energy densities are equal:Hence $⟨u⟩={ϵ}_0|E_0{|}^2/2$ and $⟨S⟩=c⟨u⟩\hat{k}$, as summarized by energy density and flux of a plane electromagnetic wave.

For isotropic radiation, the average power is spread over a sphere. At $r=0.10\mathrm{m}$, the intensity of an isotropic radiator isUsing the result of part (d),