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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2022

Paper 3

The outgoing Green function of the <electromagnetic wave equation> has support on the past light cone. Convolution with that <causal Green function>[retarded Green function] yields the <retarded electromagnetic potential>
$$
\boxed{
A^\mu(\mathbf x,t)
=\frac{\mu_0}{4\pi}
\int_{\mathbb R^3}
\frac{J^\mu(\mathbf x',t')}{|\mathbf x-\mathbf x'|}
\,d^3x'
},
$$
where the source is evaluated at the <retarded time>
$$
\boxed{
t'=t-\frac{|\mathbf x-\mathbf x'|}{c}
}.
$$
This is the causal solution with no incoming radiation.

Solution

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

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

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

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