Jackson Electrodynamics problem 9.8a

[andrew1982]

Homework Statement

9.8a) Show that a classical oscillating eletric dipole p with fields given by (9.18) radiates electromagnetic angular momentum to infinity at the rate
$$\frac{d\mathbf{L}}{dt}=\frac{k^3}{12\pi\epsilon_0}\textrm{Im}[\mathbf{p^*\times p}]$$
Hint: The electromagnetic angular momentum density comes from more than the transverse (radiation zone) components of the field.

Homework Equations

Dipole fields (9.18):
$$\mathbf{H}=\frac{ck^2}{4\pi}(\mathbf{n\times p}) \frac{e^{ikr}}{r}(1-\frac{1}{ikr})$$
$$\mathbf{E}=\frac{1}{4\pi \epsilon_0}(k^2\mathbf{(n\times p)\times n} \frac{e^{ikr}}{r}+(3\mathbf{n(n\cdot p)-p})(\frac{1}{r^3}-\frac{ik}{r^2})e^{ikr})$$
(n is the unit vector in direction x)

Electromagnetic momentum density (6.118)
$$\mathbf{g}=\frac{1}{c^2}(\mathbf{E\times H})$$

The Attempt at a Solution

So I guess the angular momentum density is
$$\mathbf{x\times g}$$
which with the fields in (9.18) simplifies to
$$\frac{ik^2}{8\pi^2 \epsilon_0}\mathbf{(n\cdot p)(n \times p^*)}(\frac{k}{r^2}+\frac{1}{ikr^4})$$
if I use the complex Poynting vector
$$\mathbf{E\times H^*}.$$
From here I'm not sure how to continue. This is the angular momentum density (per volume). If I integrate it over the whole space I get the total angular momentum, not only the part radiated to infinity. Since it is per volume I will not get the right dimension if I do as for the power radiated to infinity (integrate the Poynting vector over a spherical surface, radius R, and let R->infinity). What integration should I do?

I tried to integrate only the part prop. 1/r^2 (since I guess the other part will not "reach infinity") over a sphere with radius R and got
$$\mathbf{L}=\frac{ik^3}{6 c \pi \epsilon_0}(\mathbf{p^*\times p})R$$

This looks similar to the answer, but will diverge in the limit R->inf. I also don't see how to take the time derivative of this. If I use the complex Poynting vector (which I guess I should?) the complex exponentials containing the (harmonic) time dependence will cancel.

Any hint would be appreciated!

 

[Clau]

Look at this link:
http://www-personal.umich.edu/~pran/jackson/P506/hw04a.pdf

 

[andrew1982]

Hi
thanks a lot for the link! Very useful website :)

Ok, so my understanding after reading this solution is that the electromagnetic angular momentum that is radiated to infinity in time dt is contained in a spherical shell with radius r->inf. and thickness dr=c dt. So dL/dt is obtained by integrating the angular momentum density over this shell volume and "dividing" by dt.

Furthermore, I should use the time-averaged Poynting vector 1/2 ExH*, since the answer in fact is the average flow of angular momentum to infinity.

Is this understanding correct? Thanks again.