Properties of multipole fields

Let us examine some of the properties of the multipole fields (7.48), (7.49), and (7.51). Consider, first of all, the so-called near zone, for which $kr\ll 1$. In this region $f_l(kr)$ is proportional to $y_l(kr)$, given by the asymptotic expansion (7.14b), unless its coefficient vanishes identically. Excluding this possibility, the limiting behaviour of the magnetic field for an electric $(l, m)$ multipole is

$$c{\bfm B}_{lm}^{(E)} \rightarrow -\frac{k}{l}\, {\bfm L} \,\frac{Y_{lm}}{r^{l+1}},$$ (1239)

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

$${\rm i}\,\nabla\wedge{\bfm L} = {\bfm r}\,\nabla^2 -\nabla\!\left( 1+ r\frac{\partial}{\partial r}\right).$$ (1240)

The electric field (7.51b) is

$${\bfm E}_{lm}^{(E)} \rightarrow \frac{-{\rm i}}{l}\,\nabla\wedge {\bfm L} \left(\frac{Y_{lm}}{r^{l+1}}\right).$$ (1241)

Since $Y_{lm}/r^{l+1}$ is a solution of Laplace's equation, the first term in (7.59) vanishes. Consequently, the electric field at close distances for an electric $(l, m)$ multipole is

$${\bfm E}_{lm}^{(E)} \rightarrow -\nabla\!\left(\frac{Y_{lm}}{r^{l+1}}\right).$$ (1242)

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

$$\phi(r,\theta,\varphi) = \sum_{l,m}\frac{Y_{lm}(\theta,\varphi)}{r^{l+1}}.$$ (1243)

Note that the magnetic field (7.58) (normalized with respect to $c^{-1}$) is smaller than the electric field (7.61) by a factor of order $kr$. Thus, in the near zone the magnetic field associated with an electric multipole is always much smaller than the corresponding electric field. For magnetic multipole fields it is evident from Eqs. (7.48), (7.49), and (7.51) that the roles of ${\bfm E}$ and ${\bfm B}$ are interchanged according to the transformation

$${\bfm E}^{(E)} \rightarrow -c{\bfm B}^{(M)},$$ (1244)

$$c{\bfm B}^{(E)} \rightarrow {\bfm E}^{(M)}.$$ (1245)

In the so-called far zone or radiation zone, for which $kr\gg 1$, 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 $f_l(kr)$ is proportional to the spherical Hankel function $h_l^{(1)}(kr)$. From the asymptotic form (7.16), it is clear that in the radiation zone the magnetic field of an electric $(l, m)$ multipole goes as

$$c{\bfm B}_{lm}^{(E)} \rightarrow (-{\rm i})^{l+1} \frac{{\rm e}^{\,{\rm i}\,kr}} {kr}\, {\bfm L} \,Y_{lm}.$$ (1246)

Using Eq. (7.51b), the electric field can be written

$${\bfm E}_{lm}^{(E)} = \frac{(-{\rm i})^l}{k^2}\left[\nabla\!... ...^{\,{\rm i}\,kr}}{r} \,\nabla\wedge {\bfm L} \,Y_{lm} \right].$$ (1247)

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

$${\bfm E}_{lm}^{(E)} = -(-{\rm i})^{l+1} \frac{{\rm e}^{\,{\r... ...L}\,Y_{lm}-\frac{1}{k}({\bfm r}\nabla^2-\nabla) Y_{lm}\right],$$ (1248)

where use has been made of the identity (7.59), and ${\bfm n} = {\bfm r}/r$ 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 $1/kr$, and can therefore be neglected in the limit $kr\gg 1$. Thus, we find that the electric field in the radiation zone is given by

$${\bfm E}_{lm}^{(E)} = c{\bfm B}_{lm}^{(E)}\wedge{\bfm n},$$ (1249)

where ${\bfm B}_{lm}^{(E)}$ is given by Eq. (7.64). These fields are typical radiation fields; i.e., they are transverse to the radius vector, mutually orthogonal, and fall off like $1/r$. For magnetic multipoles we merely make the transformation (7.63).

Consider a linear superposition of electric $(l, m)$ multipoles with different $m$ values, but all possessing a common $l$ value. It follows from Eqs. (7.54) that

$$c{\bfm B}_l = \sum_l a_E(l,m) \,{\bfm X}_{lm}\, h_l^{(1)}(kr)\,{\rm e}^{-{\rm i} \,\omega t},$$ (1250)

$${\bfm E}_l = \frac{\rm i}{k}\nabla\wedge c {\bfm B}_{l}.$$ (1251)

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

$$u = \frac{\epsilon_0}{4}\,({\bfm E}\!\cdot\!{\bfm E}^\ast + c{\bfm B}\!\cdot\!c{\bfm B}^\ast).$$ (1252)

In the radiation zone the two terms are equal. It follows that the energy density contained in a spherical shell between radii $r$ and $r+dr$ is

$$dU = \frac{\epsilon_0\,dr}{2k^2}\sum_{m,m'} a_E^\ast(l,m') \,a_E(l,m) \int {\bfm X}_{lm'}^\ast\!\cdot{\bfm X}_{lm} \,d\Omega,$$ (1253)

where the asymptotic form (7.16) of the spherical Hankel function has been used. Making use of the orthogonality relation (7.53a), we obtain

$$\frac{dU}{dr} = \frac{\epsilon_0}{2k^2} \sum_m \vert a_E(l,m)\vert^2,$$ (1254)

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 $m$, and $\vert a_E\vert^2$ becomes $\vert a_E\vert^2+\vert a_M\vert^2$. Thus, the total energy in a spherical shell 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

$${\bfm m} = \frac{\epsilon_0}{2}\, {\rm Re}\, [{\bfm r}\wedge({\bfm E}\wedge {\bfm B}^\ast)].$$ (1255)

For a superposition of electric multipoles the triple product can be expanded and the electric field (7.68b) substituted, to give

$${\bfm m} = \frac{\epsilon_0 c}{2k} \,{\rm Re}\,[{\bfm B}^\ast({\bfm L}\!\cdot \!{\bfm B})].$$ (1256)

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

$$d{\bfm M} = \frac{\epsilon_0 c \,dr} {2k^3} \,{\rm Re} \!\su... ...{\bfm L}\!\cdot\!{\bfm X}_{lm'})^\ast {\bfm X}_{lm} \,d\Omega.$$ (1257)

It follows from Eqs. (7.27) and (7.52) that

$$\frac{d{\bfm M}}{dr} = \frac{\epsilon_0 c}{2k^3} \,{\rm Re} ... ...(l,m')\,a_E(l,m) \int Y_{lm'}^\ast\,{\bfm L}\,Y_{lm}\,d\Omega.$$ (1258)

According to Eqs. (7.33), the Cartesian components of $d{\bfm M}/dr$ can be written:

$$\frac{d M_x}{dr} = \frac{\epsilon_0 c}{4 k^3} \,{\rm Re}\!\sum_m\left[\sqrt{(l-m)(l+m+1)}\, a_E^\ast(l, m+1)\right.$$

$$\left.+ \sqrt{(l+m)(l-m+1)} \,a_E^\ast(l,m-1)\right]a_E(l,m),$$ (1259)

$$\frac{d M_y}{dr} = \frac{\epsilon_0 c}{4 k^3} \,{\rm Im}\!\sum_m\left[\sqrt{(l-m)(l+m+1)}\, a_E^\ast(l, m+1)\right.$$

$$\left. - \sqrt{(l+m)(l-m+1)} \,a_E^\ast(l,m-1)\right]a_E(l,m),$$ (1260)

$$\frac{d M_z}{dr} = \frac{\epsilon_0 c}{2 k^3} \sum_m m\,\vert a_E(l,m)\vert^2.$$ (1261)

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.