Electric potential at a distance r from a non-conducting sphere

Wheelwalker

1. The problem statement, all variables and given/known data

Determine the electric potential at a distance r from a non-conducting sphere of radius a and non-uniform charge density Br (where B is a constant) for each of the following cases:

i. r>a
ii. 0<r<a

2. Relevant equations

Electric field outside of the sphere: (k*B*pi*(a^4))/(r^2)
Electric field inside of the sphere: (k*pi*B*r^2)
V=-integral(E*dl)


3. The attempt at a solution

For part 1, I integrated in from infinity to r to determine the potential as a function of r...

V=-integral from infinity to r of (k*B*pi*(a^4))/(r^2) dr and ended up with the answer V=(k*B*pi*(a^4))/(r).

I'm not sure if my bounds were correct for that integral. I'm fairly certain I need to integrate in from infinity assuming the potential is zero at infinity.

Also, for the next part I am not sure if I need to integrate in from infinity to the outer edge of the sphere, then add that to another integral inside of the sphere (I remember doing that with conducting concentric spheres). Any help would be much appreciated. I am not looking for an answer, just some help and/or pointers. I am mainly concerned about my bounds and whether or not I need to integrate twice for the second part. Thanks!
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution

Sleepy_time

Yes, just take the integral of the electric field outside the sphere from ∞ to a, and add the integral of the electric field inside the sphere from a to r and don't forget the overall minus sign.

Wheelwalker

Okay, got it. Thank you for the quick and helpful reply. It's much appreciated!

Wheelwalker

Actually, I have one more question. For the first part, I took the integral from ∞ to r. Is that incorrect? If I take the integral from ∞ to a, I get a constant number. Adding this integral to the integral of the electric field inside the sphere should work fine for the second part of the question but for the first, there won't be any change in potential outside of the sphere. Is that right?

SammyS

That would be true if the electric field outside the sphere were zero, which of course, is not true.

Wheelwalker

Yeah, that's what I was thinking. So that must be incorrect. Would it be correct to integrate in from infinity to r instead of a then? That's what I originally did and came up with the answer (k*B*pi*(a^4))/(r).

SammyS

I could do the integration --- but I'm too lazy.

What's the total charge of the sphere ?

Added in Edit:

OK, I integrated to find the total charge on the sphere.

It's [itex]Q=B\,\pi\,a^4\ .[/itex]

So, your answer looks like it's correct !

Wheelwalker

Thanks Sammy!