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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 4

Substitution of the sinusoidal <plane electromagnetic wave>[plane waves] into <Faraday's law> gives
$$
\mathbf k\times\mathbf e=\omega\mathbf b.
$$
The source-free <Ampère-Maxwell equation>, with $\mathbf H=\mathbf B/\mu$ and $D_i=\varepsilon_{ij}E_j$, gives
$$
\mathbf k\times\frac{\mathbf b}{\mu}
=-\omega\boldsymbol\varepsilon\mathbf e.
$$
Eliminating $\mathbf b=(\mathbf k\times\mathbf e)/\omega$ yields the wave equation for a <plane wave in an anisotropic dielectric>:
$$
\boxed{
[\mathbf k\times(\mathbf k\times\mathbf e)]_i
+\omega^2\mu\varepsilon_{ij}e_j=0}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 36A / d / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 36A / d / Solution

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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 36A / d / Solution

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