Dielectric Sphere in Field of a Point Charge

[Apteronotus]

Hi,
I have derived the electric potential equations inside and outside the sphere due to a point charge $$q$$ placed a distance $$b$$ way from the sphere's center. The potentials are given by:
$$\Phi_{in}(r,\theta) = \sum^{\infty}_{n=0} A_{n}r^{n}P_{n}(cos\theta)$$
and
$$\Phi_{out}(r,\theta) = \sum^{\infty}_{n=0} \frac{kr^{n}}{b^{n+1}} + \sum^{\infty}_{n=0}\frac{B_{n}}{r^{n+1}}P_{n}(cos\theta)$$

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

I have calculated the the constants $$A_{n}$$ and $$B_{n}$$ 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?

 

[Meir Achuz]

No. It is an infinite series.