- #1
andre220
- 75
- 1
Homework Statement
Assume that the electrostatic potential of a point charge ##Q## is $$ \Phi(r) = \frac{1}{4\pi\epsilon_0}\frac{Q}{r^{1+\delta}},$$
such that ##\delta \ll 1##.
(a) Determine ##\Phi(r)## at any point inside and outside a spherical shell of radius ##R## with a uniform surface charge ##\sigma##.
(b) If two concentric spherical conducting shells of radii ##a## and ##b## are connected by a thin wire, a charge ##q_a## resides on the outer shell and charge ##q_b## resides on the inner shell. Determine the ratio of charges ##\frac{q_a}{q_b}## to the first order in ##\delta##.
Homework Equations
$$E=-\vec{\nabla}\Phi$$
$$Q = \sigma A = 4\pi R^2\sigma$$
$$V = -\int \vec{E}\cdot \vec{dl}$$
The Attempt at a Solution
In the case when there is no ##\delta##: $$V(r>R) = -\int\limits_\infty^r\frac{1}{4\pi\epsilon}\frac{Q}{r^2}dr = \frac{1}{4\pi\epsilon_0}\frac{Q}{r}$$
$$V(r<R) = -\int\limits_\infty^R\frac{1}{4\pi\epsilon}\frac{Q}{r^2}dr = \frac{1}{4\pi\epsilon_0}\frac{Q}{R}$$
But...
I have no idea what to do here, since if we were given the equation for ##E## I think it would make more sense.
Any help is appreciated.