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Past exam of the mathematical methods II course of the University of Oxford / 2019 resit / 8 / b / i / Solution 2026-10-25

Away from the origin, the electric field is

E = A r / r^{3}

, with

r

here denoting the position vector in the numerator and its Euclidean norm in the denominator. The spherical-coordinate formula for divergence gives

\nabla \cdot E = \frac{1}{r ^{2}} \frac{d}{d r} (r^{2} \frac{A}{r ^{2}}) = 0, r > 0.

At the origin the electric field is singular, so the ordinary divergence is undefined there. The equation above must not be extended to the whole space as an ordinary smooth-field identity. Its correct distributional divergence, derived in solution, includes the point source.

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