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 isDifferentiate and use the continuity equation $\dot{\rho }+\nabla \cdot \mathbf{J}=0$:Localization makes the boundary integral vanish. Component by component,This is the time derivative of the electric dipole moment.