Apply the approach of Problem 9.1b to the current and magnetization densities of the particle of charge q rotating about the origin in the x-y plane in a circle of radius R at constant angular speed ω0. The motion is such that ω0R≪c.
(a) Find (Jx)n, (Jy)n, and (Jz)n in terms of cylindrical coordinates for all n. Also determine the components of the orbital ``magnetization,'' (x×Jn)/2, and its divergence [which plays the role of a magnetic charge density for magnetic multipoles, as in Mlm (9.172)].
(b) What long-wavelength magnetic multipoles (l,m) occur and at what frequencies? [Remember that the multipole order l does not necessarily equal the harmonic number n.]
(c) Use linear superposition to generalize your argument to the four charges rotating in Problem 9.2 at radius R=a/2. What harmonics occur, and what magnetic multipoles at each harmonic? Is there a magnetic multipole contribution at the E2 frequency of Problem 9.2? Is it significant relative to the E2 radiation?