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Next: Radiation from Linear Centre-Fed Up: Multipole Expansion Previous: Solution of Inhomogeneous Helmholtz

Sources of Multipole Radiation

Let us now examine the connection between multipole fields and their sources. Suppose that there exist localized distributions of electric change, $ \rho({\bf r},t)$ , true current, $ {\bf j}({\bf r},t)$ , and magnetization, $ {\bf M}({\bf r}, t)$ . We assume that any time dependence can be analyzed into its Fourier components, and we therefore only consider harmonically varying sources, $ \rho({\bf r}) \,{\rm e}^{-{\rm i}\,\omega\, t}$ , $ {\bf j}({\bf r}) \,{\rm e}^{-{\rm i}\,\omega\, t}$ , and $ {\bf M}({\bf r}) \,{\rm e}^{-{\rm i}\,\omega\, t}$ , where it is understood that we take the real parts of complex quantities.

Maxwell's equations can be written

$\displaystyle \nabla\cdot {\bf E}$ $\displaystyle =\frac{\rho}{\epsilon_0},$ (1516)
$\displaystyle \nabla\cdot{\bf B}$ $\displaystyle =0,$ (1517)
$\displaystyle \nabla\times{\bf E} -{\rm i}\,k \,c\,{\bf B}$ $\displaystyle ={\bf0},$ (1518)
$\displaystyle \nabla\times c\,{\bf B} + {\rm i}\,k \,{\bf E}$ $\displaystyle = \mu_0 \,c \,({\bf j} + \nabla\times {\bf M}),$ (1519)

whereas the charge continuity equation takes the form

$\displaystyle {\rm i}\,\omega \,\rho = \nabla\cdot{\bf j}.$ (1520)

It is convenient to deal only with divergence-free fields. Thus, we use as our field variables, $ {\bf B}$ and

$\displaystyle {\bf E}' = {\bf E} + \frac{\rm i}{\epsilon_0 \,\omega} \,{\bf j}.$ (1521)

In the region external to the sources, $ {\bf E}'$ reduces to $ {\bf E}$ . When expressed in terms of these fields, Maxwell's equations become

$\displaystyle \nabla\cdot{\bf E}'$ $\displaystyle =0,$ (1522)
$\displaystyle \nabla\cdot{\bf B}$ $\displaystyle =0,$ (1523)
$\displaystyle \nabla\times{\bf E}' -{\rm i}\,k \,c\,{\bf B}$ $\displaystyle = \frac{\rm i}{\epsilon_0\, \omega} \,\nabla\times{\bf j},$ (1524)
$\displaystyle \nabla\times c\,{\bf B} + {\rm i}\,k \,{\bf E}'$ $\displaystyle = \mu_0\, c \,\nabla\times {\bf M}.$ (1525)

The curl equations can be combined to give two inhomogeneous Helmholtz wave equations:

$\displaystyle (\nabla^{\,2} + k^{\,2})\,c\,{\bf B} = -\mu_0\, c\, \nabla\times({\bf j}+\nabla\times{\bf M}),$ (1526)


$\displaystyle (\nabla^{\,2}+k^{\,2})\,{\bf E}' = -{\rm i}\,k\,\mu_0 \,c\,\nabla\times\left( {\bf M} + \frac{\nabla\times{\bf j}}{k^{\,2}}\right).$ (1527)

These equations, together with $ \nabla\cdot{\bf B} = 0$ , and $ \nabla\cdot{\bf E}' = 0$ , as well as the curl equations giving $ {\bf E}'$ in terms of $ {\bf B}$ , and vice versa, are the generalizations of Equations (1455)-(1460) when sources are present.

Because the multipole coefficients in Equations (1479)-(1480) are determined, via Equations (1483)-(1484), from the scalars $ {\bf r}\cdot c\,{\bf B}$ and $ {\bf r}\cdot{\bf E}'$ , it is sufficient to consider wave equations for these quantities, rather than the vector fields $ {\bf B}$ and $ {\bf E}'$ . From Equations (1462), (1528), (1529), and the identity

$\displaystyle {\bf r}\cdot(\nabla\times{\bf A}) = ({\bf r}\times\nabla)\cdot {\bf A} = {\rm i}\,{\bf L}\cdot{\bf A},$ (1528)

which holds for any vector field $ {\bf A}$ , we obtain the inhomogeneous wave equations

$\displaystyle (\nabla^{\,2} + k^{\,2})\,{\bf r}\cdot c\,{\bf B}$ $\displaystyle = -{\rm i}\,\mu_0 \,c\,{\bf L}\cdot({\bf j} + \nabla\times{\bf M}),$ (1529)
$\displaystyle (\nabla^{\,2}+k^{\,2})\,{\bf r}\cdot{\bf E}'$ $\displaystyle =k\,\mu_0 \,c\, {\bf L}\cdot\left({\bf M} + \frac{\nabla\times{\bf j}}{k^{\,2}}\right).$ (1530)

Now, the Green's function for the inhomogeneous Helmholtz equation, subject to the boundary condition of outgoing waves at infinity, is given by Equation (1509). It follows that Equations (1531)-(1532) can be inverted to give

$\displaystyle {\bf r}\cdot c\,{\bf B}({\bf r})$ $\displaystyle =\frac{{\rm i}\,\mu_0\, c}{4\pi} \int\frac{{\rm e}^{\,{\rm i}\,k\...
...\bf L}'\cdot\left[{\bf j}({\bf r}') +\nabla'\times {\bf M}({\bf r}')\right]dV',$ (1531)
$\displaystyle {\bf r}\cdot {\bf E}'({\bf r})$ $\displaystyle = -\frac{k\,\mu_0 \, c}{4\pi}\int \frac{{\rm e}^{\,{\rm i}\,k\,\v...
...t\left[{\bf M}({\bf r}')+\frac{\nabla'\times {\bf j}({\bf r}')}{k^2}\right]dV'.$ (1532)

In order to evaluate the multipole coefficients by means of Equations (1483)-(1484), we first observe that the requirement of outgoing waves at infinity implies that $ A_l^{(2)} = 0$ in Equation (1467). Thus, in Equations (1479)-(1480), we choose $ f_l(k\,r) =g_l(k\,r) = h_l^{(1)}(k\,r)$ as the radial eigenfunctions of $ {\bf E}$ and $ {\bf B}$ in the source-free region. Next, let us consider the expansion (1517) of the Green's function for the inhomogeneous Helmholtz equation. We assume that the point $ {\bf r}$ lies outside some spherical shell that completely encloses the sources. It follows that $ r_< = r'$ and $ r_>=r$ in all of the integrations. Making use of the orthogonality property of the spherical harmonics, it follows from Equation (1517) that

$\displaystyle \oint Y^{\,\ast}_{lm}(\theta,\varphi)\,\frac{{\rm e}^{\,{\rm i}\,...
...={\rm i}\,k\, h_l^{(1)}(k\,r) \,j_l(k\,r')\, Y_{lm}^{\,\ast}(\theta',\varphi').$ (1533)

Finally, Equations (1483)-(1484), and (1533)-(1535) yield

$\displaystyle a_E(l,m)$ $\displaystyle = \frac{\mu_0 \,c\,{\rm i}\,k^{\,3}}{\sqrt{l\,(l+1)}}\int j_l(k\,...
...m}\,{\bf L}\cdot\left({\bf M} +\frac{\nabla\times {\bf j}}{k^{\,2}}\right)\,dV,$ (1534)
$\displaystyle a_M(l,m)$ $\displaystyle = -\frac{\mu_0\, c\,k^{\,2}}{\sqrt{l\,(l+1)}}\int j_l(k\,r)\,Y_{lm}^{\,\ast}\, {\bf L}\cdot({\bf j} +\nabla\times{\bf M})\, dV.$ (1535)

The previous two equations allow us to calculate the strengths of the various multipole fields, external to the source region, in terms of integrals over the source densities, $ {\bf j}$ and $ {\bf M}$ . These equations can be transformed into more useful forms by means of the following arguments. The results

$\displaystyle {\bf L}\cdot{\bf A}$ $\displaystyle = {\rm i}\,\nabla\!\cdot\!({\bf r}\times {\bf A}),$ (1536)
$\displaystyle {\bf L}\cdot(\nabla\times{\bf A})$ $\displaystyle = {\rm i}\,\nabla^{\,2}({\bf r} \cdot{\bf A}) -{\rm i}\,\frac{1}{r}\frac{\partial (r^{\,2} \,\nabla\cdot {\bf A})}{\partial r}$ (1537)

follow from the definition of $ {\bf L}$ [see (1438)], and simple vector identities. Substituting into Equation (1536), we obtain

$\displaystyle a_E(l,m) = - \frac{\mu_0\, c\,\,k^{\,3}}{\sqrt{l\,(l+1)}} \int j_...
...}} -{\rm i}\,\frac{c}{k\,r} \frac{\partial(r^{\,2} \rho)}{\partial r}\right]dV,$ (1538)

where use has been made of Equation (1522). Use of Green's theorem on the second term in square brackets allows us to replace $ \nabla^{\,2}$ by $ -k^{\,2}$ (because we can neglect surface terms, and $ j_l(k\,r)\,Y_{lm}^{\,\ast}$ is a solution of the Helmholtz equation). A radial integration by parts on the third term (again neglecting surface terms) cause the radial derivate to operate on the spherical Bessel function. The resulting expression for the electric multipole coefficient is

$\displaystyle a_E(l,m) = \frac{\mu_0\, c\,k^{\,2}}{{\rm i}\,\sqrt{l\,(l+1)}} \i...
...,j_l(k\,r)-{\rm i}\,k\,\nabla\cdot({\bf r}\times{\bf M})\,j_l(k\,r)\right]\,dV.$ (1539)

Similarly, Equation (1537) leads to the following expression for the magnetic multipole coefficient:

$\displaystyle a_M(l,m) = \frac{\mu_0\, c\,k^{\,2}}{{\rm i}\,\sqrt{l\,(l+1)}} \i...
...\frac{d [r\,j_l(k\,r)]}{d r}-k^2\,({\bf r}\cdot{\bf M})\,j_l(k\,r) \right]\,dV.$ (1540)

Both of the previous results are exact, and are valid for arbitrary wavelength and source size.

In the limit in which the source dimensions are small compared to a wavelength (i.e., $ k\,r\ll 1$ ), the above expressions for the multipole coefficients can be considerably simplified. Using the asymptotic form (1428), and retaining only lowest powers in $ k\,r$ for terms involving $ \rho$ , $ {\bf j}$ , and $ {\bf M}$ , we obtain the approximate electric multipole coefficient

$\displaystyle a_E(l,m) \simeq \frac{\mu_0\, c\, k^{\,l+2}}{{\rm i}\, (2\,l+1)!!}\,\left(\frac{l+1}{l}\right)^{1/2} (Q_{lm}+Q_{lm}'),$ (1541)

where the multipole moments are

$\displaystyle Q_{lm}$ $\displaystyle = \int r^{\,l} \,Y_{lm}^{\,\ast}\,c\,\rho\,dV,$ (1542)
$\displaystyle Q_{lm}'$ $\displaystyle = \frac{-{\rm i}\,k}{l+1} \int r^{\,l} \,Y_{lm}^\ast\, \nabla\cdot({\bf r}\times{\bf M})\,dV.$ (1543)

The moment $ Q_{lm}$ has the same form as a conventional electrostatic multipole moment. The moment $ Q_{lm}'$ is an induced electric multipole moment due to the magnetization. The latter moment is generally a factor $ k\,r$ smaller than the former. For the magnetic multipole coefficient $ a_M(l,m)$ , the corresponding long wavelength approximation is

$\displaystyle a_M(l,m) \simeq \frac{\mu_0\, c\, \,{\rm i}\, k^{\,l+2}}{ (2\,l+1)!!}\,\left(\frac{l+1}{l}\right)^{1/2} ({\cal M}_{lm}+{\cal M}_{lm}'),$ (1544)

where the magnetic multipole moments are

$\displaystyle {\cal M}_{lm}$ $\displaystyle = -\frac{1}{l+1} \int r^{\,l}\,Y_{lm}^{\,\ast}\, \nabla\cdot({\bf r}\times{\bf j})\,dV,$ (1545)
$\displaystyle {\cal M}_{lm}'$ $\displaystyle = -\int r^{\,l}\,Y_{lm}^{\,\ast}\, \nabla\cdot{\bf M} \,dV.$ (1546)

Note that for a system with intrinsic magnetization, the magnetic moments $ {\cal M}_{lm}$ and $ {\cal M}_{lm}'$ are generally of the same order of magnitude. We conclude that, in the long wavelength limit, the electric multipole fields are determined by the charge density, $ \rho$ , whereas the magnetic multipole fields are determined by the magnetic moment densities, $ {\bf r}\times{\bf j}$ and $ {\bf M}$ .

next up previous
Next: Radiation from Linear Centre-Fed Up: Multipole Expansion Previous: Solution of Inhomogeneous Helmholtz
Richard Fitzpatrick 2014-06-27