Dielectric Sphere in Field of a Point Charge

  1. 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]\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]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?
  2. jcsd
  3. clem

    clem 1,276
    Science Advisor

    No. It is an infinite series.
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