00:01
Here's a situation involving two concentric charged spheres, and we are to find an electric field and electric potential at three different locations relative to those spheres.
00:12
Let's find the electric fields first.
00:16
We can use gauss's law to do that, since we have a situation with high symmetry.
00:21
Let's use a gaussian surface inside of this inner sphere, and that our gaussian surface also be a sphere.
00:31
Now, by symmetry, we know that the electric field, if any, in there must be radial, and it must be equal in magnitude at a given distance away from the center.
00:50
So we know then that the electric flux through our gaussian surface, let's say ea, call it, times four pi r, a squared r a radius of our gaucian surface that's the electric flux there according to gauce's law then that should equal the charge inside of my sphere divided by epsilon not but the charge inside of that gaussian is zero there's no charge in there and so therefore the electric field at point a must equal zero.
01:33
Now we can do the same thing, but now we need a different gaussian surface.
01:36
We'll use one that is outside of the inner sphere, the inside of the outer sphere.
01:42
So we use a gaussian surface here, also spherical.
01:46
And then we can say, the electric flux there has the same symmetry.
01:51
So eb times 4 pi rb squared must equal the charge containment there.
02:02
I'm going to call that q1, that's the charge that's on that inner sphere divided by epsilon not.
02:11
Well, and q2, the charging outer sphere doesn't matter at all because that's outside of the gaussian.
02:19
So now we have eb as being equal to 1 over 4 pi epsilon times the charge inside, which is q1 divided by r b squared.
02:38
If you plug in those numbers, what you end up with for eb is equal to 6 ,481 neutrons per coulon.
02:52
E .c.
02:53
Is going to be the same thing, except now we're going to use a gaussian surface that goes around both spheres.
03:01
And it's going to have the same symmetry, though.
03:04
So now we're going to end up with a c as being equal to 1 over 4 pi epsilon not, times the charge inside, which is q1 plus q2 divided by rc squared, and that ends up being 7604 neutrons per coulomb.
03:27
So those are our three electric fields, 0 ,641, and 764 newtons per coulomb.
03:34
Now, to get the little potential, you could use integration.
03:37
You can start all the way out of infinity and integrate to each point separately.
03:42
But when you do that, the way the limits of integration work, it's going to end up being you're going to be adding potential due to the inner charge, getting to the potential at the surface of the outer charge, and then you're going to subtract it off when you do it.
04:01
You can actually do it a little bit easier way, though, if you think about it, the potential from a spherically symmetric arrangement of charge, if you're on a sphere here, you know that the potential, if you're outside of that sphere, is just equal to 1 over 4, perhaps or not, which you can start calling k now, times the charge on the sphere, divided by r or r is the distance you are away from the center of that sphere.
04:38
And inside of it, well, there's no electric field inside of that charge...