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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 36B / a / Solution

The source-free Maxwell equations in a linear medium are

\nabla \cdot D = 0, \nabla \cdot B = 0,

\nabla \times E = - \frac{\partial B}{\partial t}, \nabla \times H = \frac{\partial D}{\partial t},

with

D = ϵ E

and

B = μ H

.

Take complex plane waves

E = E_{0} e^{i (k \cdot x - ω t)}, B = B_{0} e^{i (k \cdot x - ω t)} .

The divergence equations give

k \cdot E_{0} = k \cdot B_{0} = 0,

and Faraday's law gives

B_{0} = \frac{1}{ω} k \times E_{0} .

Ampère's law gives

k \times \frac{B _{0}}{μ} = - ω ϵ E_{0} .

Substituting the expression for

B_{0}

and using

k \times (k \times E_{0}) = - ∣ k ∣^{2} E_{0}

yields

\frac{∣ k ∣ ^{2}}{μ ω} E_{0} = ω ϵ E_{0} .

Hence the plane electromagnetic wave in a linear medium has

ω^{2} = \frac{∣ k ∣ ^{2}}{ϵ μ}, v = \frac{ω}{∣ k ∣} = \frac{1}{ϵ μ},

and equivalently

B_{0} = ϵ μ k \times E_{0} .

Solved by gpt-5.6-sol high.

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