Classical Electrodynamics · Problem 4.2

Magnetostatics, Faraday's Law, Quasi-Static Fields — John David Jackson

Physics — E&M · point-dipole-effective-charge-density

Problem

A point dipole with dipole moment p\mathbf{p} is located at the point x0\mathbf{x}_0. From the properties of the derivative of a Dirac delta function, show that for calculation of the potential Φ\Phi or the energy of a dipole in an external field, the dipole can be described by an effective charge density ρeff(x)=−p⋅∇δ(x−x0)\rho_{\text{eff}}(\mathbf{x}) = -\mathbf{p} \cdot \boldsymbol{\nabla}\delta(\mathbf{x} - \mathbf{x}_0)

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