Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 36A (Electrodynamics)

The macroscopic Maxwell equations in matter areHere $\rho$ and $\mathbf{J}$ are the free charge and current densities.

Dot the Ampère-Maxwell equation with $\mathbf{E}$ and Faraday's law with $\mathbf{H}$:Adding and using the vector identitygivesThis is the local Poynting theorem before specializing the constitutive relations.

Because ${\epsilon }_{ij}$ and ${\mu }_{ij}$ are symmetric and time-independent, the linear anisotropic dielectric relations giveand similarlyIntegrate the identity from part (b) over $V$ and apply the divergence theorem to obtainWhen $\mathbf{J}=0$, this is conservation of electromagnetic energy. The Poynting theorem in a linear anisotropic medium identifiesas the energy density andas the energy flux, or Poynting vector.

Substitution of the sinusoidal plane waves into Faraday's law givesThe source-free Ampère-Maxwell equation, with $\mathbf{H}=\mathbf{B}/\mu$ and $D_i={\epsilon }_{ij}{E}_j$, givesEliminating $\mathbf{b}=(\mathbf{k}\times \mathbf{e})/\omega$ yields the wave equation for a plane wave in an anisotropic dielectric:

Suppose ${\epsilon }_{ij}{e}_j={\epsilon }_e{e}_i$. Taking the scalar product of the equation in part (d) with $\mathbf{k}$ gives $\mathbf{k}\cdot \mathbf{e}=0$. The vector triple product then reduces the equation toso $k^2={\omega }^2\mu {\epsilon }_e$. From part (d),ConsequentlyThus the instantaneous flux, except at its zeros, and its time average both point in the wave-propagation direction $\mathbf{k}$.