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Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 3 / 15D / b / Solution

Take the curl of the integral from part (a). Since

J (r^{'})

is independent of

r

,

B (r) = \nabla \times A (r) = \frac{μ _{0}}{4 π} \int \nabla (\frac{1}{∣ r - r ^{'} ∣}) \times J (r^{'}) d^{3} r^{'} = \frac{μ _{0}}{4 π} \int \frac{J ( r ^{'} ) \times ( r - r ^{'} )}{∣ r - r ^{'} ∣ ^{3}} d^{3} r^{'} .

This is the Biot-Savart law.

Solved by gpt-5.6-sol high.

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