Find the first non-zero multipole expansion term for voltage far away from a sphere of radius \( R \) and surface charge \( \sigma(\theta)=\sigma_{0} P_{2}(\cos \theta) \) where \( P_{2} \) is the second order Legendre Polynomial.
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Suppose the potential $V_{0}(\theta)$ at the surface of a sphere is specified, and there is no charge inside or outside the sphere. Show that the charge density on the sphere is given by $$ \sigma(\theta)=\frac{\epsilon_{0}}{2 R} \sum_{l=0}^{\infty}(2 l+1)^{2} C_{l} P_{l}(\cos \theta) $$ where $$ C_{l}=\int_{0}^{\pi} V_{0}(\theta) P_{l}(\cos \theta) \sin \theta d \theta $$
A charge $+2 q$ is situated at the origin and charges of $-q$ are situated at distances $\pm a$ from it along the polar axis. By relating it to the generating function for the Legendre polynomials, show that the electrostatic potential $\Phi$ at a point $(r, \theta, \phi)$ with $r>a$ is given by $$ \Phi(r, \theta, \phi)=\frac{2 q}{4 \pi \epsilon_{0} r} \sum^{\infty}\left(\frac{a}{r}\right)^{2 s} P_{2 s}(\cos \theta) $$
Use the expression for the field of a disk of radius R and charge density...
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