Exercises

1. An almost spherical surface defined by
   $$R(\theta)=R_0\left[1+\beta\,P_2(\cos\theta)\right]$$
   has inside it a uniform volume distribution of charge totaling $Q$ . The small parameter $\beta$ varies harmonically in time at the angular frequency $\omega$ . This corresponds to a surface waves on a sphere. Keeping only the lowest-order terms in $\beta$ , and making the long-wavelength approximation, calculate the nonvanishing multipole moments, the angular distribution of radiation, and the total radiated power.

2. The uniform charge density of the previous exercise is replaced by a uniform magnetization parallel to the $z$ -axis and having a total magnetic moment $M$ . With the same approximations as in the previous exercise, calculate the nonvanishing multipole moments, the angular distribution of radiation, and the total radiated power.