 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ib/paper-2/6c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

For a time-independent current, the Maxwell equations reduce to magnetostatics:

\nabla \cdot B = 0, \nabla \times B = μ_{0} j .

Write

B = \nabla \times A

\nabla \cdot A = 0

. The vector identity for the curl of a curl gives the Poisson equation

- \nabla^{2} A = μ_{0} j .

A (r) = \frac{μ _{0}}{4 π} \int_{V} \frac{j ( r ^{'} )}{∣ r - r ^{'} ∣} d V^{'} .

Taking the curl with respect to

r

, moving it under the integral, and using

\nabla (1/∣ r - r^{'} ∣) = - (r - r^{'}) /∣ r - r^{'} ∣^{3}

B (r) = \frac{μ _{0}}{4 π} \int_{V} \frac{j ( r ^{'} ) \times ( r - r ^{'} )}{∣ r - r ^{'} ∣ ^{3}} d V^{'} .

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