# Alternative electrostatic potential

## 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.

DEvens
In the case of a charge distribution I would integrate. So then I would just evaluate: $$\Phi = \frac{1}{4\pi\epsilon_0}\int \frac{dQ}{r^{1+\delta}}$$ Inside we would have: $$\Phi(r<R) = \frac{1}{4\pi\epsilon_0}\int\limits_0^r \frac{dQ}{r^{1+\delta}}$$ and outside
$$\Phi(r>R) = \frac{1}{4\pi\epsilon_0}\int\limits_r^\infty \frac{dQ}{r^{1+\delta}}$$