Classical Electrodynamics · Problem 14.19

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · radiation-magnetic-moment-nonrelativistic

Problem

Consider the angular and frequency spectrum of radiation produced by a magnetic moment in nonrelativistic motion, using (14.70) and the fact that a magnetization density M\mathcal{M} produces an effective current density Jeff=c∇×M\mathbf{J}_{\text{eff}} = c\nabla \times \mathcal{M}. (a) Show that a magnetic moment μ\boldsymbol{\mu} with magnetization, M=μ(t) δ[x−r(t)]\mathcal{M} = \boldsymbol{\mu}(t)\,\delta[\mathbf{x} - \mathbf{r}(t)], in nonrelativistic motion gives a radiation intensity (energy radiated per unit solid angle per unit frequency interval), d2Imagdω dΩ=ω44π2c3∣∫dt n×μ(t)eiω(t−n⋅r(t)/c)∣2\frac{d^2I_{\text{mag}}}{d\omega\, d\Omega} = \frac{\omega^4}{4\pi^2 c^3}\left|\int dt\, \mathbf{n}\times\boldsymbol{\mu}(t)e^{i\omega(t - \mathbf{n}\cdot\mathbf{r}(t)/c)}\right|^2 (b) The magnetic moment is located at the origin and is caused to precess by an external torque such that μx=μ0sin⁡ω0t\mu_x = \mu_0\sin\omega_0 t and μz=μ0cos⁡ω0t\mu_z = \mu_0\cos\omega_0 t for the time interval t=−T/2t = -T/2 to t=T/2t = T/2, where ω0T/2π≫1\omega_0 T/2\pi \gg 1. Show that the frequency distribution of the radiation is very strongly peaked at ω=ω0\omega = \omega_0, that the angular distribution of radiation is proportional to (1+sin⁡2θsin⁡2ϕ)(1 + \sin^2\theta\sin^2\phi), and that for T→∞T\to\infty, the total time-averaged power radiated is ⟨P⟩=2ω043c3μ02\langle P\rangle = \frac{2\omega_0^4}{3c^3}\mu_0^2 Compare the result with the power calculated by the method of Section 9.3.

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