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Axisymmetric Charge Distributions

For the case of an axisymmetric charge distribution (i.e., a charge distribution that is independent of the azimuthal angle $ \varphi$ ), we can neglect the spherical harmonics of non-zero order (i.e., the non-axisymmetric harmonics) in Equation (335), which reduces to the following expression for the general axisymmetric Green's function:

$\displaystyle G({\bf r},{\bf r}') = -\frac{1}{4\pi} \sum_{l=0,\infty}\frac{r_<^{\,l}}{r_>^{\,l+1}}\,P_l(\cos\theta')\,P_l(\cos\theta).$ (364)

Here, use have been made of the fact that [see Equation (309)]

$\displaystyle Y_{l,0}(\theta,\varphi) = \left(\frac{2\,l+1}{4\pi}\right)^{1/2} P_l(\cos\theta).$ (365)

In this case, the general solution to Poisson's equation, (337), reduces to

$\displaystyle \phi({\bf r}) = \frac{1}{4\pi\,\epsilon_0}\sum_{l=0,\infty}\left[r^{\,l}\,p_{l}(r)+ \frac{q_{l}(r)}{r^{\,l+1}}\right]P_l(\cos\theta),$ (366)

where

$\displaystyle p_{l}(r)$ $\displaystyle = \int_r^\infty \int_0^\pi \frac{1}{r'^{\,l+1}} \,\rho(r',\theta)\,\,P_l(\cos\theta)\,2\pi\,r'^{\,2}\,\sin\theta\,d\theta\,dr',$ (367)
$\displaystyle q_{l}(r)$ $\displaystyle = \int_0^r \int_0^\pi r'^{\,l}\, \rho(r',\theta)\,P_l(\cos\theta)\,2\pi\,r'^{\,2}\,\sin\theta\,d\theta\,dr'.$ (368)

Consider the potential generated by a charge $ q$ distributed uniformly in a thin ring of radius $ a$ that lies in the $ x$ -$ y$ plane, and is centered at the origin. It follows that

$\displaystyle \rho(r,\theta)\,2\pi\,r^{\,2}\,\sin\theta\,d\theta\,dr \rightarrow q\,\delta(r-a)\,\delta(\theta-\pi/2)\,d\theta\,dr.$ (369)

Hence, for $ r<a$ we obtain $ q_l=0$ and $ p_l=q\,P_l(0)/a^{\,l+1}$ . On the other hand, for $ r>a$ we get $ p_l=0$ and $ q_l=q\,a^{\,l}\,P_l(0)$ . Thus,

$\displaystyle \phi(r,\theta) = \frac{q}{4\pi\,\epsilon_0}\sum_{l=0,\infty} \frac{r_<^{\,l}}{r_>^{\,l+1}}\,P_l(0)\,P_l(\cos\theta),$ (370)

where $ r_<$ represents the lesser of $ r$ and $ a$ , whereas $ r_>$ represents the greater.


next up previous
Next: Dirichlet Problem in Spherical Up: Potential Theory Previous: Multipole Expansion
Richard Fitzpatrick 2014-06-27