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Dielectric Sphere in Field of a Point Charge

by Apteronotus
Tags: dielectric sphere, point charge
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Apteronotus
#1
Sep8-08, 01:31 PM
P: 203
Hi,
I have derived the electric potential equations inside and outside the sphere due to a point charge [tex]q[/tex] placed a distance [tex]b[/tex] way from the sphere's center. The potentials are given by:
[tex]\Phi_{in}(r,\theta) = \sum^{\infty}_{n=0} A_{n}r^{n}P_{n}(cos\theta)
[/tex]
and
[tex]\Phi_{out}(r,\theta) = \sum^{\infty}_{n=0} \frac{kr^{n}}{b^{n+1}} + \sum^{\infty}_{n=0}\frac{B_{n}}{r^{n+1}}P_{n}(cos\theta)
[/tex]

where
[tex]k=\frac{q}{4\pi\epsilon_{0}}[/tex] and [tex]P_{n}[/tex] - are the Legendre polynomials

I have calculated the the constants [tex]A_{n}[/tex] and [tex]B_{n}[/tex] according to the usual boundary conditions. Unfortunately, almost non of them are equal to zero unlike the the case of a 'sphere in a uniform field'. Is there any way of truncating these infinite sums to end up with something nice and clean?
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clem
#2
Sep8-08, 05:04 PM
Sci Advisor
P: 1,256
No. It is an infinite series.


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