00:01
Here we're going to look at finding the total charge in a region using the charge density.
00:08
And the idea is very simple.
00:10
Density times volume is equal to the total.
00:15
And of course we could have a function of position.
00:21
So what we really want to do is take density, function of position, and integrate it over an...
00:32
Entire volume in which the charge resides over whatever space is holding onto that charge.
00:42
So we're going to start off with a simple example, or a simple case.
00:53
We are taking the density of charge inside of a hydrogen atom, and i am going to make a friendly amendment to what was written down.
01:05
What we should have is charge per cubic meter, where, of course, whatever unit you would like to use for position is included.
01:26
I'm just saying meter because that is the nice si unit.
01:33
So a friendly amendment that it should be an a -cubed in the denominator.
01:41
So that you have columns per cubic meter.
01:44
And of course, pi is a natural quantity with no units.
01:51
So what we will need to do is to integrate in order to find the total charge.
01:57
We have to integrate over all space.
02:01
And we're going to do this in spherical coordinates.
02:04
Now, because there is no phi or theta in the density, we can pop out the integral over theta and phi is 2 times 2 pi.
02:21
We won't go through that.
02:23
We can pull out the constants, q over pi 8 cubed.
02:29
And then we are left with the integral from r equals zero all the way out to infinity because the charge is smeared out over all of the radii possible.
02:44
And then we want the radial part of that.
02:50
So a reminder that if you are working with spherical coordinates, the integral over the volume is equal to integral of, let me expand that out a little bit.
03:09
Let's see, d -fi, sine of theta, d -theta, d -theta, and then r -squared, dr.
03:20
And phi typically goes through 0 to 2 pi, theta goes from 0 to pi, and r, if it's all space, goes from 0 to infinity.
03:39
Now it's probably better to look up the integral.
03:45
You know, it can be done in closed form, and basically what it gives is 8 cubed over 4.
04:05
Okay, and so we can see.
04:07
That everything cancels, the pies cancel.
04:12
The fours cancel, the eight cubes cancel, and not surprisingly, q total is equal to q.
04:23
And the one caveat that i would say is this should be a negative.
04:37
Okay, so when they said that q is the positive charge in the nucleus, really the, the q in there could be positive or negative.
04:51
And here, if it's an electron cloud, it should be negative.
04:59
Okay, example two is going to get into some of the electrostatics because we're being given an electric field, and we are asked to find the density from the electric field.
05:17
And this is going to involve galseslaw.
05:20
We are going to use it in terms.
05:23
Differential form as well as integral form.
05:30
Gals's law in differential form gives us the density, the charge density, in terms of the divergence of the electric field.
05:56
We can go look up the divergence for a spherical system in which we just have a radial component...