Determining Electric Potential

[Kennedy Poch]

Homework Statement

Consider a nonconducting sphere of radius r₂. It has a concentric spherical cavity of r₁. The material between r₁ and r₂ has a charge density p (C/m³). Take V=0 and r=infinity. Determine the electrical potential V as a function of the distance r from the center for (a) r>r₂, (b) r₁<r<r₂, and (c)r<r1

Homework Equations

ρ=Q/V

The Attempt at a Solution

a) I have no idea where to start. Would my volume be just (4/3)πr³??

b) Total volume enclosed = (4/3)(π)(r₂)³ - (4/3)(π)(r₁)³

= (4/3)(π)(r₂³-r₁³
ρ=Q/V
=3Q/4(π)(r₂³-r₁³)

dQ= 3Qr²/(r₂³-r₁³)

V= (1/(4πε)∫dQ/r

Having trouble doing the integral...

c) There should be no electric potential inside

 

[Simon Bridge]

Welcome to PF;

a) I have no idea where to start. Would my volume be just (4/3)πr3??

"no idea" is not acceptable here - you have just done a section of coursework that should have given you some ideas ... go back and look again.

The volume of the sphere radius r is going to be $\frac{4}{3}\pi r^3$ all right - but what is the volume you are supposed to find? Your next answer suggests you are confused about this.

Lets say you had a ball of charge radius R and charge Q with uniform density - you want to find the potential at radius r < R from the center ... the rule (for spherical distributions of charge) is that the potential is proportional to the total charge inside the radius and inversely proportional to the distance from the center.

So you should not be thinking in terms of volumes at all, but in terms of how much charge there is.
You'll end up doing as volume integral but only because the maths is easier that way.

 

[Kennedy Poch]

Outside of the sphere would we not just use V= ((1/(4πε))(Q/r) ?

 

[Simon Bridge]

That's outside the sphere.
For r > R, the total charge inside r is Q, so V=kQ/r
For r< R, what is the total charge inside r?