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Next: Solution of Inhomogeneous Helmholtz Up: Multipole Expansion Previous: Multipole Expansion of Vector

Properties of Multipole Fields

Let us examine some of the properties of the multipole fields (1470)-(1471) and (1474)-(1475). Consider, first of all, the so-called near zone, in which $ k\,r\ll 1$ . In this region, $ f_l(k\,r)$ is dominated by $ y_l(k\,r)$ , which blows up as $ k\,r\rightarrow 0$ , and which has the asymptotic expansion (1429), unless the coefficient of $ y_l(k\,r)$ vanishes identically. Excluding this possibility, the limiting behavior of the magnetic field for an electric $ l, m$ multipole is

$\displaystyle c\,{\bf B}_{lm}^{(E)} \rightarrow -\frac{k}{l}\, {\bf L} \,\frac{Y_{lm}}{r^{\,l+1}},$ (1483)

where the proportionality constant is chosen for later convenience. To find the corresponding electric field, we must take the curl of the right-hand side of the above equation. The following operator identity is useful

$\displaystyle {\rm i}\,\nabla\times{\bf L} \equiv {\bf r}\,\nabla^{\,2} -\nabla\!\left( 1+ r\,\frac{\partial}{\partial r}\right).$ (1484)

The electric field (1475) can be written

$\displaystyle {\bf E}_{lm}^{(E)} \rightarrow \frac{-{\rm i}}{l}\,\nabla\times {\bf L} \left(\frac{Y_{lm}}{r^{\,l+1}}\right).$ (1485)

Because $ Y_{lm}/r^{\,l+1}$ is a solution of Laplace's equation, it is annihilated by the first term on the right-hand side of (1486). Consequently, for an electric $ l, m$ multipole, the electric field in the near zone becomes

$\displaystyle {\bf E}_{lm}^{(E)} \rightarrow -\nabla\left(\frac{Y_{lm}}{r^{\,l+1}}\right).$ (1486)

This, of course, is an electrostatic multipole field. Such a field can be obtained in a more straightforward manner by observing that $ {\bf E}\rightarrow
-\nabla\phi$ , where $ \nabla^{\,2}\phi = 0$ , in the near zone. Solving Laplace's equation by separation of variables in spherical coordinates, and demanding that $ \phi$ be well behaved as $ \vert{\bf r}\vert\rightarrow\infty$ , yields

$\displaystyle \phi(r,\theta,\varphi) = \sum_{l,m}\frac{Y_{lm}(\theta,\varphi)}{r^{\,l+1}}.$ (1487)

Note that ($ c$ times) the magnetic field (1485) is smaller than the electric field (1488) by a factor of order $ k\,r$ . Thus, in the near zone, ($ c$ times) the magnetic field associated with an electric multipole is much smaller than the corresponding electric field. For magnetic multipole fields, it is evident from Equations (1470)-(1471) and (1474)-(1475) that the roles of $ {\bf E}$ and $ c\,{\bf B}$ are interchanged according to the transformation

$\displaystyle {\bf E}^{(E)}$ $\displaystyle \rightarrow -c\,{\bf B}^{(M)},$ (1488)
$\displaystyle c\,{\bf B}^{(E)}$ $\displaystyle \rightarrow {\bf E}^{(M)}.$ (1489)

In the so-called far zone, or radiation zone, in which $ k\,r\gg 1$ , the multipole fields depend on the boundary conditions imposed at infinity. For definiteness, let us consider the case of outgoing waves at infinity, which is appropriate to radiation by a localized source. For this case, the radial function $ f_l(k\,r)$ contains only the spherical Hankel function $ h_l^{(1)}(k\,r)$ . From the asymptotic form (1432), it is clear that in the radiation zone the magnetic field of an electric $ l, m$ multipole varies as

$\displaystyle c\,{\bf B}_{lm}^{(E)} \rightarrow (-{\rm i})^{\,l+1} \frac{{\rm e}^{\,{\rm i}\,k\,r}} {k\,r}\, {\bf L} \,Y_{lm}.$ (1490)

Using Equation (1475), the corresponding electric field can be written

$\displaystyle {\bf E}_{lm}^{(E)} = \frac{(-{\rm i})^{\,l}}{k^{\,2}}\left[\nabla...
...} +\frac{{\rm e}^{\,{\rm i}\,k\,r}}{r} \,\nabla\times {\bf L} \,Y_{lm} \right].$ (1491)

Neglecting terms that fall off faster than $ 1/r$ , the above expression reduces to

$\displaystyle {\bf E}_{lm}^{(E)} = -(-{\rm i})^{\,l+1} \frac{{\rm e}^{\,{\rm i}...
...\times {\bf L}\,Y_{lm}-\frac{1}{k}({\bf r}\,\nabla^{\,2}-\nabla) Y_{lm}\right],$ (1492)

where use has been made of the identity (1486), and $ {\bf n} = {\bf r}/r$ is a unit vector pointing in the radial direction. The second term in square brackets is smaller than the first term by a factor of order $ 1/(k\,r)$ , and can, therefore, be neglected in the limit $ k\,r\gg 1$ . Thus, we find that the electric field in the radiation zone takes the form

$\displaystyle {\bf E}_{lm}^{(E)} = c\,{\bf B}_{lm}^{(E)}\times{\bf n},$ (1493)

where $ c\,{\bf B}_{lm}^{(E)}$ is given by Equation (1492). These fields are typical radiation fields: that is, they are transverse to the radius vector, mutually orthogonal, fall off like $ 1/r$ , and are such that $ \vert{\bf E}\vert=c\,\vert{\bf B}\vert$ . To obtain expansions for magnetic multipoles, we merely make the transformation (1490)-(1491).

Consider a linear superposition of electric $ l, m$ multipoles with different $ m$ values that all possess a common $ l$ value. Suppose that all multipoles correspond to outgoing waves at infinity. It follows from Equations (1474)-(1476) that

$\displaystyle c\,{\bf B}_l$ $\displaystyle = \sum_l a_E(l,m) \, h_l^{(1)}(k\,r)\,{\rm e}^{-{\rm i}{ \,\omega \,t}}\,{\bf X}_{lm},$ (1494)
$\displaystyle {\bf E}_l$ $\displaystyle = \frac{\rm i}{k}\,\nabla\times c\, {\bf B}_{l}.$ (1495)

For harmonically varying fields, the time-averaged energy density is given by

$\displaystyle u = \frac{\epsilon_0}{4}\,({\bf E}\cdot{\bf E}^{\,\ast} + c\,{\bf B}\cdot c\,{\bf B}^{\,\ast}).$ (1496)

In the radiation zone, the two terms on the right-hand side of the above equation are equal. It follows that the energy contained in a spherical shell lying between radii $ r$ and $ r+dr$ is

$\displaystyle dU = \frac{\epsilon_0\,dr}{2\,k^{\,2}}\sum_{m,m'} a_E^\ast(l,m') \,a_E(l,m) \oint {\bf X}_{lm'}^\ast\!\cdot{\bf X}_{lm} \,d{\mit\Omega},$ (1497)

where use has been made of the asymptotic form (1432) of the spherical Hankel function $ h_l^{(1)}(z)$ . The orthogonality relation (1477) leads to

$\displaystyle \frac{dU}{dr} = \frac{\epsilon_0}{2\,k^{\,2}} \sum_m \vert a_E(l,m)\vert^{\,2},$ (1498)

which is clearly independent of the radius. For a general superposition of electric and magnetic multipoles, the sum over $ m$ becomes a sum over $ l$ and $ m$ , whereas $ \vert a_E\vert^{\,2}$ becomes $ \vert a_E\vert^{\,2}+\vert a_M\vert^{\,2}$ . Thus, the net energy in a spherical shell situated in the radiation zone is an incoherent sum over all multipoles.

The time-averaged angular momentum density of harmonically varying electromagnetic fields is given by

$\displaystyle {\bf m} = \frac{\epsilon_0}{2}\, {\rm Re}\, [{\bf r}\times({\bf E}\times {\bf B}^{\,\ast})].$ (1499)

For a superposition of electric multipoles, the triple product can be expanded, and the electric field (1497) substituted, to give

$\displaystyle {\bf m} = \frac{\epsilon_0 \,c}{2\,k} \,{\rm Re}\,[{\bf B}^{\,\ast}({\bf L}\cdot {\bf B})].$ (1500)

Thus, the net angular momentum contained in a spherical shell lying between radii $ r$ and $ r+dr$ (in the radiation zone) is

$\displaystyle d{\bf M} = \frac{\epsilon_0\, c \,dr} {2\,k^{\,3}} \,{\rm Re} \!\...
...l,m) \oint ({\bf L}\cdot{\bf X}_{lm'})^{\,\ast}\, {\bf X}_{lm} \,d{\mit\Omega}.$ (1501)

It follows from Equations (1436) and (1476) that

$\displaystyle \frac{d{\bf M}}{dr} = \frac{\epsilon_0 \,c}{2\,k^{\,3}} \,{\rm Re...
..._E^\ast(l,m')\,a_E(l,m) \oint Y_{lm'}^{\,\ast}\,{\bf L}\,Y_{lm}\,d{\mit\Omega}.$ (1502)

According to Equations (1439)-(1441), the Cartesian components of $ d{\bf M}/dr$ can be written:

$\displaystyle \frac{d M_x}{dr}$ $\displaystyle =\frac{\epsilon_0 \,c}{4\, k^{\,3}} \,{\rm Re}\!\sum_m\left[\sqrt{(l-m)\,(l+m+1)}\, a_E^{\,\ast}(l, m+1)\right.$    
  $\displaystyle \left.+ \sqrt{(l+m)\,(l-m+1)} \,a_E^{\,\ast}(l,m-1)\right]a_E(l,m),$ (1503)
$\displaystyle \frac{d M_y}{dr}$ $\displaystyle = \frac{\epsilon_0 \,c}{4 \,k^{\,3}} \,{\rm Im}\!\sum_m\left[\sqrt{(l-m)\,(l+m+1)}\, a_E^{\,\ast}(l, m+1)\right.$    
  $\displaystyle \left. - \sqrt{(l+m)\,(l-m+1)} \,a_E^{\,\ast}(l,m-1)\right]a_E(l,m),$ (1504)
$\displaystyle \frac{d M_z}{dr}$ $\displaystyle = \frac{\epsilon_0 \,c}{2 \,k^{\,3}} \sum_m m\,\vert a_E(l,m)\vert^{\,2}.$ (1505)

Thus, for a general $ l$ th order electric multipole that consists of a superposition of different $ m$ values, only the $ z$ component of the angular momentum takes a relatively simple form.

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Next: Solution of Inhomogeneous Helmholtz Up: Multipole Expansion Previous: Multipole Expansion of Vector
Richard Fitzpatrick 2014-06-27