1. The problem statement, all variables and given/known data 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 [tex] \frac{d\mathbf{L}}{dt}=\frac{k^3}{12\pi\epsilon_0}\textrm{Im}[\mathbf{p^*\times p}] [/tex] Hint: The electromagnetic angular momentum density comes from more than the transverse (radiation zone) components of the field. 2. Relevant equations Dipole fields (9.18): [tex] \mathbf{H}=\frac{ck^2}{4\pi}(\mathbf{n\times p}) \frac{e^{ikr}}{r}(1-\frac{1}{ikr}) [/tex] [tex] \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}) [/tex] (n is the unit vector in direction x) Electromagnetic momentum density (6.118) [tex] \mathbf{g}=\frac{1}{c^2}(\mathbf{E\times H}) [/tex] 3. The attempt at a solution So I guess the angular momentum density is [tex] \mathbf{x\times g} [/tex] which with the fields in (9.18) simplifies to [tex] \frac{ik^2}{8\pi^2 \epsilon_0}\mathbf{(n\cdot p)(n \times p^*)}(\frac{k}{r^2}+\frac{1}{ikr^4}) [/tex] if I use the complex Poynting vector [tex] \mathbf{E\times H^*}. [/tex] 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 [tex] \mathbf{L}=\frac{ik^3}{6 c \pi \epsilon_0}(\mathbf{p^*\times p})R [/tex] 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!
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.