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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 36B / b / Solution

The outgoing Green function of the electromagnetic wave equation has support on the past light cone. Convolution with that retarded Green function yields the retarded electromagnetic potential

A^{μ} (x, t) = \frac{μ _{0}}{4 π} \int_{R^{3}} \frac{J ^{μ} ( x ^{'} , t ^{'} )}{∣ x - x ^{'} ∣} d^{3} x^{'},

where the source is evaluated at the retarded time

t^{'} = t - \frac{∣ x - x ^{'} ∣}{c} .

This is the causal solution with no incoming radiation.

Solved by gpt-5.6-sol high.

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