Question

A sphere of radius R has a radius dependent charge density ρ = B · r3 in terms of R and B.

Calculate the potential as a function of r from the center of the sphere.

Answer 1

The equation for electric potential is:

   (equation 1)

It means, the change in electric potential from the initial value Vi to the final value Vf equals minus the integral of the electric field E along a curve connecting the initial position to the final position.

So you need the electric field to find the electric potential. In this case, the electric field is different inside and outside the sphere, so you have to split the problem into two regions. The charge density has a spherical symmetry, because it's onlye a function of r, and not the angle. This means the electric field will have the same symmetry, in other words, the electric field is a function of r, and has no dependency on the angle.

Let's start with the region outside the sphere:

To find the electric field in this region, you have to use Gauss's law.

   (equation 2)

The left side of this equation is the flux of the electric field trough a surface, on the right side, Q is the total charge enclosed by this surface.

Since the charge density is a function of R, you need to calculate the total charge as a function of r:

(equation 3)

That would be the charge of a sphere inside the full charged sphere. If you want the total charge, you have to use r=R to get:

    (equation 4)

For r in the region outside the sphere, the surface you have to use is a sphere greater then the charged sphere:

Using Gauss's law you get:

Replacing the formulas for area of a sphere and the total charge from equation (4), you have:

Now you can use equation (1) to find the potential. The points you have to use are infinity and r. The potential at infinity is zero.

So:    when

For the region inside the charged sphere, where , applying Gauss's law:

Using the charge from equation (3):

So, for the potential, you have to use R as the initial position , and r as the final position. The potential at r is found using the result from the other region:

Using equation (1) again:

   is the potential for