00:01
Okay, so we've got a question about a spherical conductor with a cavity in the middle.
00:06
So the radius of the conductor as a whole is a, and the radius of the cavity, which is the hole in the middle, is b.
00:14
And then it's given a charge q.
00:17
Now, because it's a conductor, the charges are free to move about within it, but obviously like charges are going to repel each other.
00:24
So they're going to sort of get as far away as possible so that they're in an equilibrium.
00:27
And this is going to be distributed on the outer surface.
00:31
Okay? like this.
00:35
So this is what our sphere looks like.
00:37
They could be negative.
00:37
I'm just drawing them as positive for now.
00:39
We're not actually told which one it is, but it doesn't really matter.
00:42
Okay.
00:43
So then it wants us to find the electric field everywhere to start off with.
00:49
Now, to do that, we're going to use gauces law.
00:58
Now, the key thing here is that for r less than a, so if we take our gauce surface to be a sphere of radius r, centered on the center of the cavity hole, q enclosed is zero, right? all the charge lies on the outside of the sphere.
01:24
And so if this r is less than a, our gauce sphere is going to lie entirely within the cylinder.
01:31
And so there's going to be no enclosed charge.
01:33
And so there's going to be no electric field.
01:37
If r is bigger than a, then the electric field is radiating out radially, which is the same direction as the area vector.
01:51
And so gals's law just becomes e times the area of the gals sphere, which is 4 pi little r squared, is equal to the enclosed charge, which for r greater than a, is just going to be all of the charge q divided by epsilon.
02:07
0.
02:08
And so we find that e, other the function of the distance from the centre, r, is equal to q over 4 pi epsilon 0, r squared.
02:20
Okay, it then asks us for the potential.
02:25
Now, we can find the potential difference.
02:29
Remember, the potential difference is just the integral of the radial part of the electric field.
02:41
And so, so say this is the potential difference between points a and b, then this would be our formula.
02:52
But remember, we always take the potential at infinity to be zero.
02:57
And so if we do vr minus v infinity, sorry, there should be a minus sign here, this is equal to minus the integral from infinity to r of e.
03:13
D .r.
03:15
Now, we'll take, so for the minute, take r to be greater than a.
03:21
So we're talking about points outside the sphere.
03:26
Then we know the expression for the electric field.
03:30
It's just given by what we wrote here up here.
03:40
So we've got the integral from infinity to r of q over 4 pi epsilon naught r squared...