Classical Electrodynamics · Problem 15.1

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · bremsstrahlung-radiation-amplitude-multipole-expansion

Problem

In radiative collision problems it is useful to have the radiation amplitude expressed explicitly as an integral involving the accelerations of the particles, as in (14.65), for example. In the nonrelativistic limit, particles do not move rapidly or far during the period of acceleration; only the lowest order velocity and retardation effects need be kept in an approximate description. (a) Show that the integral (times ee) in (14.65), which is, apart from an inessential phase, cRcR times the Fourier transform of the Liénard–Wiechert electric field at distance RR, can be expanded in inverse powers of cc (remembering that β=v/c\beta = v/c and k=ω/ck = \omega/c) as follows: J=e∫dt eiωt(β˙⊥+β∥β˙⊥+β˙∥β⊥−ikr∥β˙⊥+⋯ )\mathbf{J} = e\int dt\, e^{i\omega t}(\dot{\boldsymbol{\beta}}_\perp + \beta_\parallel \dot{\boldsymbol{\beta}}_\perp + \dot{\beta}_\parallel \boldsymbol{\beta}_\perp - ikr_\parallel \dot{\boldsymbol{\beta}}_\perp + \cdots) or J=e∫dt eiωt(β˙⊥+ddt(β∥β⊥)−ikr∥β˙⊥+⋯ )\mathbf{J} = e\int dt\, e^{i\omega t}\left(\dot{\boldsymbol{\beta}}_\perp + \frac{d}{dt}(\beta_\parallel \boldsymbol{\beta}_\perp) - ikr_\parallel \dot{\boldsymbol{\beta}}_\perp + \cdots\right) where β∥=β⋅n\beta_\parallel = \boldsymbol{\beta}\cdot\mathbf{n}, r∥=r⋅nr_\parallel = \mathbf{r}\cdot\mathbf{n}, and β⊥=(n×β)×n\boldsymbol{\beta}_\perp = (\mathbf{n}\times\boldsymbol{\beta})\times\mathbf{n}. Neglected terms are of order 1/c31/c^3 and higher. (b) Show that the first term in part a corresponds to the electric dipole approximation, while the next terms are the magnetic dipole and the electric quadrupole contributions. [Some integrations by parts are required, with a convergence factor e−ϵ∣t∣e^{-\epsilon|t|} to give meaning to the integrals, as discussed following (14.67).] For a group of charges, show that the generalization of part a can be written as J=1c∫dt eiωt(d2p⊥(t)dt2+d2m(t)dt2×n+16cd3Q⊥dt3(n,t)+⋯ )\mathbf{J} = \frac{1}{c}\int dt\, e^{i\omega t}\left(\frac{d^2\mathbf{p}_\perp(t)}{dt^2} + \frac{d^2\mathbf{m}(t)}{dt^2}\times\mathbf{n} + \frac{1}{6c}\frac{d^3\mathbf{Q}_\perp}{dt^3}(\mathbf{n},t) + \cdots\right) and the radiated intensity per unit solid angle and per unit frequency as d2Idω dΩ=14π2c3∣∫dt eiωt(d2p⊥(t)dt2+d2m(t)dt2×n+16cd3Q⊥dt3(n,t)+⋯ )∣2\frac{d^2I}{d\omega\, d\Omega} = \frac{1}{4\pi^2 c^3}\left|\int dt\, e^{i\omega t}\left(\frac{d^2\mathbf{p}_\perp(t)}{dt^2} + \frac{d^2\mathbf{m}(t)}{dt^2}\times\mathbf{n} + \frac{1}{6c}\frac{d^3\mathbf{Q}_\perp}{dt^3}(\mathbf{n},t) + \cdots\right)\right|^2 where p=∑jqjrj,m=∑jqjrj×βj2,Qαβ=∑jqj(3xjαxjβ−rj2δαβ)\mathbf{p} = \sum_j q_j \mathbf{r}_j, \qquad \mathbf{m} = \sum_j q_j \frac{\mathbf{r}_j\times\boldsymbol{\beta}_j}{2}, \qquad Q_{\alpha\beta} = \sum_j q_j(3x_{j\alpha}x_{j\beta} - r_j^2\delta_{\alpha\beta}) and the vector Q(n)\mathbf{Q}(\mathbf{n}) has components Qα=∑βQαβnβQ_\alpha = \sum_\beta Q_{\alpha\beta}n_\beta. Relate to the treatment of multipole radiation in Sections 9.2 and 9.3 and Problem 9.7.

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