Classical Electrodynamics · Problem 6.21

Plane Electromagnetic Waves and Wave Propagation — John David Jackson

Physics — E&M · off-center-moving-electric-dipole-fields

Problem

An electric dipole of dipole moment p\mathbf{p}, fixed in direction, is located at a position r0(t)\mathbf{r}_0(t) with respect to the origin. Its velocity v=dr0/dt\mathbf{v} = d\mathbf{r}_0/dt is nonrelativistic. (a) Show that the dipole's charge and current densities can be expressed formally as ρ(x,t)=−(p⋅∇)δ(x−r0(t));J(x,t)=−v(p⋅∇)δ(x−r0(t))\rho(\mathbf{x}, t) = -(\mathbf{p} \cdot \nabla)\delta(\mathbf{x} - \mathbf{r}_0(t)); \quad \mathbf{J}(\mathbf{x}, t) = -\mathbf{v}(\mathbf{p} \cdot \nabla)\delta(\mathbf{x} - \mathbf{r}_0(t)) (b) Show that the off-center moving dipole gives rise to a magnetic dipole field and an electric quadrupole field in addition to an electric dipole field, with moments m=12p×v\mathbf{m} = \tfrac{1}{2}\mathbf{p} \times \mathbf{v} and Qij=3(x0ipj+x0jpi)−2r0⋅p δijQ_{ij} = 3(x_{0i}p_j + x_{0j}p_i) - 2\mathbf{r}_0 \cdot \mathbf{p}\,\delta_{ij} [There are, of course, still higher moments.] (c) Show that the quasi-static electric quadrupole field is E(x)=14πϵ01r4[15n(n⋅r0)(n⋅p)−3r0(n⋅p)−3p(n⋅r0)−3n(r0⋅p)]\mathbf{E}(\mathbf{x}) = \frac{1}{4\pi\epsilon_0}\frac{1}{r^4}\left[15\mathbf{n}(\mathbf{n} \cdot \mathbf{r}_0)(\mathbf{n} \cdot \mathbf{p}) - 3\mathbf{r}_0(\mathbf{n} \cdot \mathbf{p}) - 3\mathbf{p}(\mathbf{n} \cdot \mathbf{r}_0) - 3\mathbf{n}(\mathbf{r}_0 \cdot \mathbf{p})\right] where n\mathbf{n} is a unit vector in the radial direction.

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