Potential on the inside of a hollow insulating sphere

[valarking]

Homework Statement

We have a hollow insulating spherical shell of inner radius a and outer radius b. While it can be treated as a point for r > b, find the electric potential on the inner surface of the shell.

Homework Equations

$$\phi = k_e\frac{q}{r}$$

The Attempt at a Solution

My best idea is that I'm going to need to set up a triple integral. If this was from any point in the center it would be trivial, but it's a little harder from the inner surface. Any suggestions on where to start with this integral?

 

[Delphi51]

You'll need to know the charge on the shell and how it is distributed.
If uniformly, you should be able to get it with Gauss' law.

 

[nlightner]

As a scientist, your attempt at a solution is a good start. However, there are a few things to consider before jumping into a triple integral. First, it's important to understand the concept of electric potential and how it relates to electric fields. The electric potential on the inner surface of a hollow insulating sphere is determined by the amount of charge contained within the shell and its distribution. This can be calculated using the equation \phi = k_e\frac{q}{r}, where k_e is the Coulomb constant, q is the charge contained within the shell, and r is the distance from the center of the shell.

In this case, since the shell is insulating, we can assume that the charge is evenly distributed on the inner surface. This means that the potential at any point on the inner surface will be the same, and we do not need to use a triple integral. Instead, we can simply use the equation \phi = k_e\frac{q}{r} with r = a, the inner radius of the shell.

Another important consideration is the boundary conditions. Since the shell is hollow, there is no charge within the shell, and therefore the potential at the inner surface will be zero. This is because the potential at any point within the shell is determined by the charge outside of it, and in this case there is no charge outside of the shell.

In summary, to find the electric potential on the inner surface of a hollow insulating sphere, you simply need to use the equation \phi = k_e\frac{q}{r} with r = a, where q is the charge contained within the shell. No triple integral is necessary.