Classical Electrodynamics · Problem 14.22

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · radiation-elliptic-orbit-hydrogen

Problem

Periodic motion of charges gives rise to a discrete frequency spectrum in multiples of the basic frequency of the motion. Appreciable radiation in multiples of the fundamental can occur because of relativistic effects (Problems 14.14 and 14.15) even though the components of velocity are truly sinusoidal, or it can occur if the components of velocity are not sinusoidal, even though periodic. An example of this latter motion is an electron undergoing nonrelativistic elliptic motion in a hydrogen atom. The orbit can be specified by the parametric equations x=a(cos⁡u−ϵ)x = a(\cos u - \epsilon) y=a1−ϵ2sin⁡uy = a\sqrt{1-\epsilon^2}\sin u where ω0t=u−ϵsin⁡u\omega_0 t = u - \epsilon\sin u aa is the semimajor axis, ϵ\epsilon is the eccentricity, ω0\omega_0 is the orbital frequency, and uu is an angle related to the polar angle θ\theta of the particle by tan⁡(u/2)=(1−ϵ)/(1+ϵ)tan⁡(θ/2)\tan(u/2) = \sqrt{(1-\epsilon)/(1+\epsilon)}\tan(\theta/2). In terms of the binding energy BB and the angular momentum LL, the various constants are a=e22B,ϵ=1−2BL2me4,ω02=8B3me4a = \frac{e^2}{2B}, \qquad \epsilon = \sqrt{1 - \frac{2BL^2}{me^4}}, \qquad \omega_0^2 = \frac{8B^3}{me^4} (a) Show that the power radiated in the kkth multiple of ω0\omega_0 is Pk=4e23c3(kω0)4a2{1k2[(Jk′(kϵ))2+(1−ϵ2ϵ2)Jk2(kϵ)]}P_k = \frac{4e^2}{3c^3}(k\omega_0)^4 a^2\left\{\frac{1}{k^2}\left[(J_k'(k\epsilon))^2 + \left(\frac{1-\epsilon^2}{\epsilon^2}\right)J_k^2(k\epsilon)\right]\right\} where Jk(x)J_k(x) is a Bessel function of order kk. (b) Verify that for circular orbits the general result above agrees with part a of Problem 14.21.

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