Classical Electrodynamics · Problem 9.9

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · total-power-angular-momentum-radiation-general-time-dependence

Problem

(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

\begin{aligned} P(t) &= \frac{1}{6\pi\epsilon_0 c^3}\left(\frac{\partial^2 \mathbf{p}_{\text{ret}}}{\partial t^2}\right)^2 \\ \frac{d\mathbf{L}_{em}}{dt} &= \frac{1}{6\pi\epsilon_0 c^3}\left(\frac{\partial \mathbf{p}_{\text{ret}}}{\partial t} \times \frac{\partial^2 \mathbf{p}_{\text{ret}}}{\partial t^2}\right) \end{aligned}

where the dipole moment

\mathbf{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

\begin{aligned} P(t) &= \frac{\tau}{m}\left(\frac{dV}{dr}\right)^2 \\ \frac{d\mathbf{L}_{em}}{dt} &= \frac{\tau}{m}\left(\frac{dV}{rdr}\right)\mathbf{L} \end{aligned}

where

\tau = e^2/6\pi\epsilon_0 mc^3

(

= 2e^2/3mc^3

in Gaussian units) is a characteristic time,

\mathbf{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

\alpha^4 c/a_0

, where

\alpha = e^2/4\pi\epsilon_0 \hbar c \approx 1/137

is the fine structure constant and

a_0

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 Problem 9.8.

Intuitive explanation

The key is recognizing that the radiated power and angular momentum depend on the time derivatives of the dipole moment evaluated at the retarded time. For part (a), you'll integrate the Poynting vector over a large sphere; the far-field $\mathbf{E}$ and $\mathbf{B}$ from a time-varying dipole fall off as $1/r$ , making the radiated energy flux scale as $(\ddot{\mathbf{p}})^2$ . In part (b), connect the dipole moment $\mathbf{p} = e\mathbf{r}$ to the particle's position and use the equation of motion $m\ddot{\mathbf{r}} = -dV/dr$ to express the second derivative of $\mathbf{p}$ in terms of the force—this reveals how the potential gradient controls radiation losses. Part (c) asks you to estimate the ratio of radiated angular momentum loss to orbital angular momentum for the ground state, exposing why classical radiation damping fails catastrophically in atoms. Finally, part (d) shows that when the dipole oscillates harmonically, these general

Intuitive explanation

Related theory

Angular Momentum Of A Particle

Browse the problem library