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

The electric dipole moment of a localized charge density is

p (t) = \int_{R^{3}} x ρ (x, t) d^{3} x .

Differentiate and use the continuity equation

\overset{ρ}{˙} + \nabla \cdot J = 0

:

\overset{p}{˙}_{i} = - \int x_{i} \partial_{j} J_{j} d^{3} x = \int J_{i} d^{3} x - \int_{\partial R^{3}} x_{i} J_{j} n_{j} d S .

Localization makes the boundary integral vanish. Component by component,

\dot{p} (t) = \int_{R^{3}} J (x, t) d^{3} x .

This is the time derivative of the electric dipole moment.

Solved by gpt-5.6-sol high.

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