(a) From the electric dipole fields with general time dependence of Problem 9.6, show that the total power and the total rate of radiation of angular momentum through a sphere at large radius r and time t are

where the dipole moment p is evaluated at the retarded time t = t – r/c.
(b) The dipole moment is caused by a particle of mass m and charge e moving nonrelativistically in a fixed central potential V(r). Show that the radiated power and angular momentum for such a particle can be written as

where I? = ??2/6I?N?0mN?3 (= 2e2/3mc3 in Gaussian units) is a characteristic time, L is the particle’s angular momentum, and the right-hand sides are evaluated at the retarded time. Relate these results to those from the Abraham-Lorentz equation for radiation damping [Section 16.2].
(c) Suppose the charged particle is an electron in a hydrogen atom. Show that the inverse time defined by the ratio of the rate of angular momentum radiated to the particle’s angular momentum is of the order of a4c/a0, where a = e2/4I?N?0hc ?^ 1/137 is the fine structure constant and a0 is the Bohr radius. How does this inverse time compare to the observed rate of radiation in hydrogen atoms?
(d) Relate the expressions in parts a and b to those for harmonic time dependence in Problem9.8.