- #1
Apteronotus
- 201
- 0
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)<br />
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)<br />
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?
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)<br />
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)<br />
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?
