Introduction to Electrodynamics · Problem 8.20

Electromagnetic Waves — David J. Griffiths

Physics · electromagnetic-momentum

Problem

Consider an ideal stationary magnetic dipole m\mathbf{m} in a static electric field E\mathbf{E}. Show that the fields carry momentum p=−ϵ0μ0(m×E).(8.45)\mathbf{p} = -\epsilon_0\mu_0(\mathbf{m} \times \mathbf{E}). \tag{8.45} [Hint: There are several ways to do this. The simplest method is to start with p=ϵ0∫(E×B) dτ\mathbf{p} = \epsilon_0 \int(\mathbf{E} \times \mathbf{B})\,d\tau, write E=−∇V\mathbf{E} = -\nabla V, and use integration by parts to show that p=ϵ0μ0∫VJ dτ.\mathbf{p} = \epsilon_0\mu_0 \int V\mathbf{J}\,d\tau. So far, this is valid for any localized static configuration. For a current confined to an infinitesimal neighborhood of the origin we can approximate V(r)≈V(0)−E(0)⋅rV(\mathbf{r}) \approx V(\mathbf{0}) - \mathbf{E}(\mathbf{0}) \cdot \mathbf{r}. Treat the dipole as a current loop, and use Eqs. 5.82 and 1.108.]

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