00:01
All right, so let's say we have a sphere that has a radius of r and a charge q on it.
00:06
We want to find the energy stored in the sphere due to the electromagnetic energy.
00:14
So let's call the charge density row, and this is going to be, sorry, not capital q, lowercase q over 4 pi over 3rq, which we can move the 3 in the denominator.
00:26
So the electric field inside, according to gauss's law, is going to be, you know, the enclosed charge.
00:34
So this is going to be row times four thirds pi little r cubed over epsilon knot, which we can write as q times little r cubed over capital r cubed over epsilon not.
00:50
And so the electric field inside goes as 1 over 4 pi epsilon knot.
00:57
Times q times r over capital r cubed.
01:01
This is for r inside.
01:04
And then outside, as we know, it just falls off like a point charge.
01:07
This is 1 over 4 pi, epsilon knot, q over little r squared.
01:12
So what we can do is just integrate the energy density.
01:16
So our total energy is going to be the integral of 1 half epsilon not times the electric field squared.
01:24
And we'll integrate this over all space.
01:27
So this is going to be the integral from 0 to r of the interior part.
01:32
So this is epsilon knot over 2.
01:34
When we square e, we're going to get 1 over 4 pi epsilon not squared times q squared over r to the fourth.
01:42
And then this is multiplied by 4 pi r squared d r.
01:47
And then plus epsilon not over 2 times the integral from r to infinity of 1 over 4 pi epsilon 0 squared, q squared over, and this is going to be r squared times dr squared, so it's really like dr.
02:07
And sorry, i actually have the outer limits written first...