Electric Dipole Radiation

Now that we have that under our belts we can address the multipolar expansion of the vector potential intelligently. To begin with, we will write the general solution for the vector potential in terms of the multipolar expansion for the outgoing wave Green's function defined above:

$$\mbox{\boldmath$A$}(\mbox{\boldmath$r$}) = ik \sum_L \left [ J_L(\mbox{\boldmath$r$}) \int_r^\infty \mu_0 ... ...ath$J$}(\mbox{\boldmath$r$}') H_L(\mbox{\boldmath$r$}')^{(\ast)} d^3r' \right .$$

$$+ \left. H^+_L(\mbox{\boldmath$r$}) \int_0^r \mu_0 \mbox{\boldmath$J$}(\mbox{\boldmath$r$}') J_L(\mbox{\boldmath$r$}')^{(\ast)} d^3r' \right ]$$ (11.113)

where, by convention, $(\ast)$ means that the $Y_L(\hat{r})$ is conjugated but the bessel/neumann/hankel function is not. This is because the only point of the conjugation is to construct $P_\ell(\Theta)$ from the $m$-sum for each $\ell$ via the addition theorem for spherical harmonics. We certainly don't want to change $h^+$ into $h^-$, which changes the time dependent behavior of the solution^11.9. Note that the integral over all space is broken up in such a way that the Green's function expansions above always converge. This solution is exact everywhere in space including inside the source itself!

We can therefore simplify our notation by defining certain functions of the radial variable:

$$\mbox{\boldmath$A$}(\mbox{\boldmath$r$}) = \sum_L i k\left ... ...oldmath$r$}) + S_L(r) H^+_L(\mbox{\boldmath$r$}) \right ].$$ (11.114)

In this equation,

$$C_L(r) = \int_r^\infty \mu_0 \mbox{\boldmath$J$}(\mbox{\boldmath$r$}') H_L(\mbox{\boldmath$r$}')^{(\ast)} d^3r'$$ (11.115)

$$S_L(r) = \int_0^r \mu_0 \mbox{\boldmath$J$}(\mbox{\boldmath$r$}') J_L(\mbox{\boldmath$r$}')^{(\ast)} d^3r'.$$ (11.116)

Clearly $S_L(0) = 0$ and for $r > d$, $C_L(r) = 0$. At the origin the solution is completely regular and stationary. Outside the bounding sphere of the source distribution the solution behaves like a linear combination of outgoing spherical multipolar waves. From now on we will concentrate on the latter case, since it is the one relevant to the zones.