Other radiation problems

Problem:

An electric charge Q is distributed within a volume V, so that the distribution is spherically symmetric for all times.  The distribution undergoes radial oscillations.  Find the electric and magnetic fields outside the volume, assuming that no other sources of electric or magnetic fields are present anywhere in space.

Solution:

• Concepts:
  Symmetry, Gauss' law, the Poynting vector
• Reasoning:
  Gauss' law alone can be used in a situation with spherical symmetry to find E and to determine that B = 0.
• Details of the calculation:
  Symmetry requires that E = E(r) r/r,  B = B(r) r/r.
  Gauss' law:  E(r)4πr² = Q_inside/ε₀,  B(r)4πr² = 0.  (SI units)
  E(r) = Q_inside/(4πε₀r²),  B(r) = 0.
  S = (1/μ₀)E×B = 0  -->  no radiation.