Potential of a spherical shell (non-uniform charge density)

[SquidgyGuff]

Homework Statement

Given a spherical shell of radius R and the surface charge density

(

being the angle from the top of the sphere

and

being a constant) find the electric potential

and the electric field

inside and outside the sphere. Check that both the potential is continuous inside and outside the sphere and that

inside and out. I made this little diagram to illustrate.

2. Homework Equations

The Attempt at a Solution

I can't even understand this problem. I feel like the point of interest shown in the diagram should be on the shell so that

can return a meaningful value, but I need to find the potential and electric field everywhere, so it can't be on the sphere.

 

[BvU]

Hello Guff,

I notice that your problem statement mentions a charge distribution on the surface of the sphere, but that there is no equation where charge or charge density plays a role. Is there something you can add to your 'toolbox' ?

 

[SquidgyGuff]

Hello Guff,

I notice that your problem statement mentions a charge distribution on the surface of the sphere, but that there is no equation where charge or charge density plays a role. Is there something you can add to your 'toolbox' ?

If I looked at the charge distribution a point charge the field would look like this

, but I know that isn't correct because I believe the total charge is zero. because half of the values of

are negative and the other half are positive.

 

[BvU]

I still don't see how you want to find the field or the potential at a point in space.
Where exactly on the sphere is $\cos\theta$ negative ?