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,

(15.44) |

and _L & = & 1&ell#ell;(&ell#ell;+1) 1k × (×) (f_&ell#ell;(kr) Y_L())

& = & 1&ell#ell;(&ell#ell;+1) 1k [ &nabla#nabla;^2 - ( r r + 1 ) ] (f_&ell#ell;(kr) Y_L()) (using the vector identity ×= i [&nabla#nabla;^2 - (r r + 1 ) ] to simplify). Then n_L & = & -1k&ell#ell;(&ell#ell;+1) { k^2 &int#int;(· ) j_&ell#ell;(kr) Y_L^&ast#ast;() d^3r. +

& & . &int#int;(·) [ Y_L^&ast#ast;() r (r j_&ell#ell;(kr))] d^3r }

Now, (from the continuity equation)
= i &omega#omega;&rho#rho;
so when we (sigh) integrate the second term by parts, (by using
(a ) = ·a + a ·
so that
(·)[ Y_L^&ast#ast;() r(r j_&ell#ell;(kr))
] = ·[ Y_L^&ast#ast;() r(r j_&ell#ell;(kr))
] - Y_L^&ast#ast;() r(r j_&ell#ell;(kr))
[ ·]
and the divergence theorem on the first term,
&int#int;_V [Y_L^&ast#ast;() r(r j_&ell#ell;(kr))]
dV & = & &int#int;_&part#partial;V &rarr#to;
&infin#infty; n ·[Y_L^&ast#ast;() r(r
j_&ell#ell;(kr))] dA

& = & 0
for sources with compact support to do the integration) we get
n_L & = & -1k&ell#ell;(&ell#ell;+1) { k^2 &int#int;(·
) j_&ell#ell;(kr) Y_L^&ast#ast;() d^3r. -

& & . &int#int;(i&omega#omega;&rho#rho;())[ Y_L^&ast#ast;()
r (r j_&ell#ell;(kr))] d^3r }

& = & ic &ell#ell;(&ell#ell;+1) &int#int;&rho#rho;() [
Y_L^&ast#ast;()r (r j_&ell#ell;(kr))] d^3r

& & - k &ell#ell;(&ell#ell;+1)
&int#int;(·) j_&ell#ell;(kr) Y_L^&ast#ast;() d^3r

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
). In that case we can write

(15.45) |

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

(15.46) | |||

(15.47) |

and the connection with the static electric multipole moments is complete. In a similar manner one can establish the long wavelength connection between the and the magnetic moments of earlier chapters. Also note well that the relationship is