Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 1 / 15D / b

Solution

In empty space the fields are divergence-free. Taking the curl of Faraday's law and using

\nabla \times (\nabla \times E) = \nabla (\nabla \cdot E) - \nabla^{2} E

gives

\nabla^{2} E - μ_{0} ϵ_{0} \frac{\partial ^{2} E}{\partial t ^{2}} = 0.

The same calculation, starting from the Ampère-Maxwell equation, gives the corresponding equation for

B

. Thus every component satisfies the electromagnetic wave equation with speed

c = \frac{1}{μ _{0} ϵ _{0}} ≃ 3.0 \times 1 0^{8} m s^{- 1} .

Its agreement with the measured speed of light identifies light as an electromagnetic wave.

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