Connection to Old (Approximate) Multipole Moments

To conclude our discussion of multipole fields, let us relate the multipole moments defined and used above (which are exact) to the ``usual'' static, long wavelength moments we deduced in our earlier studies. Well,

$$n_L = \int {\bf J} \cdot {\bf N}_L^{0 \ast} d^3r$$ (13.83)

and

$$\mbox{\boldmath$N$}_L = \frac{1}{\sqrt{\ell(\ell+1)}} \frac{1}{k} \mbox{\boldmath$\nabla$... ...r$}\times \mbox{\boldmath$\nabla$}) (f_\ell(kr) Y_L(\hat{\mbox{\boldmath$r$}}))$$

$$= \frac{1}{\sqrt{\ell(\ell+1)}} \frac{1}{k} \left[ \mbox{\boldmath$... ...l }{\partial r} + 1 \right) \right] (f_\ell(kr) Y_L(\hat{\mbox{\boldmath$r$}}))$$ (13.84)

(using the vector identity

$$\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$L$}= i \left... ...la$}\left(r \frac{\partial }{\partial r} + 1 \right) \right]$$ (13.85)

to simplify). Then

$$n_L = \frac{-1}{k\sqrt{\ell(\ell+1})} \left\{ k^2 \int (\mbox{\boldmath... ...box{\boldmath$J$}) j_\ell(kr) Y_L^\ast(\hat{\mbox{\boldmath$r$}}) d^3r\right. +$$

$$\quad \quad \left. \int (\mbox{\boldmath$J$}\cdot \mbox{\boldmath... ...boldmath$r$}}) \frac{\partial }{\partial r} (r j_\ell(kr))\right] d^3r \right\}$$ (13.86)

Now, (from the continuity equation)

$$\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$J$}= i \omega \rho$$ (13.87)

so when we (sigh) integrate the second term by parts, (by using

$$\mbox{\boldmath$\nabla$}\cdot (a \mbox{\boldmath$B$}) = \mb... ...abla$}a + a \mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$B$}$$ (13.88)

so that

$$(\mbox{\boldmath$J$}\cdot \mbox{\boldmath$\nabla$})\left[ Y... ...eft[ \mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$J$}\right]$$ (13.89)

and the divergence theorem on the first term,

$$\int_V \mbox{\boldmath$\nabla$}\cdot \left[\mbox{\boldmath$J$}Y_L... ...\hat{\mbox{\boldmath$r$}}) \frac{\partial }{\partial r}(r j_\ell(kr))\right] dV = \int_{\partial V \to \infty} \hat{n} \cdot \left[\mbox{\boldmath$... ...\hat{\mbox{\boldmath$r$}}) \frac{\partial }{\partial r}(r j_\ell(kr))\right] dA$$

$$= 0$$ (13.90)

for sources with compact support to do the integration) we get

$$n_L = \frac{-1}{k\sqrt{\ell(\ell+1})} \left\{ k^2 \int (\mbox{\boldmath... ...box{\boldmath$J$}) j_\ell(kr) Y_L^\ast(\hat{\mbox{\boldmath$r$}}) d^3r\right. -$$

$$\quad \quad \left. \int (i\omega\rho(\mbox{\boldmath$r$}))\left[ ... ...boldmath$r$}}) \frac{\partial }{\partial r} (r j_\ell(kr))\right] d^3r \right\}$$

$$= \frac{ic}{ \sqrt{ \ell(\ell+1) } } \int \rho(\mbox{\boldmath$r$})... ...at{\mbox{\boldmath$r$}})\frac{\partial }{\partial r} (r j_\ell(kr))\right] d^3r$$

$$\quad \quad - \frac{k}{ \sqrt{\ell(\ell+1)} } \int (\mbox{\boldma... ...}\cdot \mbox{\boldmath$J$}) j_\ell(kr) Y_L^\ast(\hat{\mbox{\boldmath$r$}}) d^3r$$ (13.91)

The electric multipole moment thus consists of two terms. The first term appears to arise from oscillations of the charge density itself, and might be expected to correspond to our usual definition. The second term is the contribution to the radiation from the radial oscillation of the current density. (Note that it is the axial or transverse current density oscillations that give rise to the magnetic multipoles.)

Only if the wavelength is much larger than the source is the second term of lesser order (by a factor of $\frac{ik}{c}$). In that case we can write

$$n_L \approx \frac{i c}{\sqrt{\ell(\ell+1)}} \int \rho Y_L^\ast \frac{\partial }{\partial r}(r j_\ell(kr)) d^3r .$$ (13.92)

Finally, using the long wavelength approximation on the bessel functions,

$$n_L \approx \frac{i c}{(2\ell+1)!!} \sqrt{\frac{\ell+1}{\ell}} k^\ell \int \rho r^\ell Y_L^\ast d^3r$$ (13.93)

$$\approx \frac{i c}{(2\ell+1)!!} \sqrt{\frac{\ell+1}{\ell}} k^\ell q_{\ell,m}$$ (13.94)

and the connection with the static electric multipole moments $q_{\ell,m}$ is complete. In a similar manner one can establish the long wavelength connection between the $m_L$ and the magnetic moments of earlier chapters. Also note well that the relationship is not equality. The ``approximate'' multipoles need to be renormalized in order to fit together properly with the Hansen functions to reconstruct the EM field.