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

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- Solution d

Solution

Substitution of the sinusoidal plane waves into Faraday's law gives

k \times e = ω b .

The source-free Ampère-Maxwell equation, with

H = B / μ

and

D_{i} = ε_{ij} E_{j}

, gives

k \times \frac{b}{μ} = - ω ε e .

Eliminating

b = (k \times e) / ω

yields the wave equation for a plane wave in an anisotropic dielectric:

[k \times (k \times e)]_{i} + ω^{2} μ ε_{ij} e_{j} = 0 .

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