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

The Lorenz gauge is

\partial_{μ} A^{μ} = 0

. Substituting the electromagnetic four-potential into the covariant Maxwell equations then removes the mixed derivative and gives

□ A^{μ} = μ_{0} J^{μ}, □ = \frac{1}{c ^{2}} \partial_{t}^{2} - \nabla^{2} .

Equivalently,

(\nabla^{2} - \frac{1}{c ^{2}} \partial_{t}^{2}) A^{μ} = - μ_{0} J^{μ} .

Solved by gpt-5.6-sol high.

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