Physics undergraduate upper level E&M homework help

Question

Spherical symmetric charge-current system: A sphere (surface) is uniformly charged with an

amount 𝑞. Its radius oscillates as a function of time as 𝑅(𝑡) = 𝑅0 (1 + cos𝜔 𝑡)/2. You may make the

usual assumption that the characteristic radiation wavelength is much longer than 𝑅0.

(a) Does the system have electric dipole radiation? Magnetic dipole radiation? Electric

quadrupole radiation? Answer these questions by calculating the oscillating components of the

corresponding multipole moments.

(b) Does the system have electromagnetic radiation at all? Explain your answer.

Answer 1

By Gauss's Law, because of symmetry, all non-induced electric field beyond R0 stays constant. By Faraday's law, and the symmetry of the charge (and therefore of the current), magnetic field is constant everywhere, because the curl of the electric field is zero everywhere. Since magnetic field is constant everywhere, there are no induced electric fields. Therefore, since E and B fields are both constant past R0, there is no radiation.