Consider a set of positive charges moving in a confined region, like protons in a nucleus, and interacting with an external field of electromagnetic radiation. The charge density is ρ, so the current density is ∼ ρ v, where v is the characteristic velocity of the moving charges. Show that the energy of interaction between the magnetic dipole moment of the charges and the external magnetic field is smaller by a factor of ∼ v/c than the energy of interaction between the electric dipole moment and the external electric field. Since the values of the matrix elements for magnetic dipole and electric dipole radiation are proportional to these interaction energies, and since the transition rates are proportional to the "squares" of the matrix elements, the magnetic dipole transition rate is smaller than the electric dipole transition rate by a factor of ∼ (v/c)2. (Hint:

(i) Show that the ratio of the interaction energies equals the product of the ratio of magnetic to electric dipole moments times the ratio of the magnetic to electric field strengths.

(ii) Argue that the ratio of the magnetic to electric dipole moments equals the ratio of the current density to the charge density.

(iii) Evaluate the ratio of the magnetic to electric field strengths for electromagnetic radiation in a vacuum.)