A common textbook example of a radiating system (see Problem 9.2) is a configuration of charges fixed relative to each other but in rotation. The charge density is obviously a function of time, but it is not in the form of (9.1).
(a) Show that for rotating charges one alternative is to calculate real time-dependent multipole moments using ρ(x,t) directly and then compute the multipole moments for a given harmonic frequency with the convention of (9.1) by inspection or Fourier decomposition of the time-dependent moments. Note that care must be taken when calculating qlm(t) to form linear combinations that are real before making the connection.
(b) Consider a charge density ρ(x,t) that is periodic in time with period T=2π/ω0. By making a Fourier series expansion, show that it can be written as
ρ(x,t)=ρ0(x)+n=1∑∞Re[2ρn(x)e−inω0t]
where
ρn(x)=T1∫0Tρ(x,t)einω0tdt
This shows explicitly how to establish connection with (9.1).
(c) For a single charge q rotating about the origin in the x-y plane in a circle of radius R at constant angular speed ω0, calculate the l=0 and l=1 multipole moments by the methods of parts a and b and compare. In method b express the charge density ρn(x) in cylindrical coordinates. Are there higher multipoles, for example, quadrupole? At what frequencies?