Classical Electrodynamics · Problem 4.5

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

Physics — E&M · force-torque-multipole-expansion-external-field

Problem

A localized charge density ρ(x,y,z)\rho(x, y, z) is placed in an external electrostatic field described by a potential Φ(0)(x,y,z)\Phi^{(0)}(x, y, z). The external potential varies slowly in space over the region where the charge density is different from zero. ((a)) From first principles calculate the total force acting on the charge distribution as an expansion in multipole moments times derivatives of the electric field, up to and including the quadrupole moments. Show that the force is F=qE(0)(0)+{∇[p⋅E(0)(x)]}0+{∇[16∑j,kQjk∂Ej(0)∂xk(x)]}0+⋯\mathbf{F} = q\mathbf{E}^{(0)}(0) + \{\boldsymbol{\nabla}[\mathbf{p} \cdot \mathbf{E}^{(0)}(\mathbf{x})]\}_0 + \left\{\boldsymbol{\nabla}\left[\frac{1}{6}\sum_{j,k} Q_{jk} \frac{\partial E_j^{(0)}}{\partial x_k}(\mathbf{x})\right]\right\}_0 + \cdots Compare this to the expansion (4.24) of the energy WW. Note that (4.24) is a number---it is not a function of x\mathbf{x} that can be differentiated! What is its connection to F\mathbf{F}? ((b)) Repeat the calculation of part a for the total torque. For simplicity, evaluate only one Cartesian component of the torque, say N1N_1. Show that this component is N1=[p×E(0)(0)]1+13[∂∂x3(∑jQ2jEj(0))−∂∂x2(∑jQ3jEj(0))]0+⋯N_1 = [\mathbf{p} \times \mathbf{E}^{(0)}(0)]_1 + \frac{1}{3}\left[\frac{\partial}{\partial x_3}\left(\sum_j Q_{2j} E_j^{(0)}\right) - \frac{\partial}{\partial x_2}\left(\sum_j Q_{3j} E_j^{(0)}\right)\right]_0 + \cdots

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