00:01
In this prime, we're going to be looking at divergence of this vector function, r head over r squared.
00:07
Here's a sketch of it for a few points, but it's always radially outward.
00:12
And as you get farther away, r squared is getting larger, so the magnitude of the vector function is getting smaller.
00:23
This obviously exists at all points.
00:25
This is just a small collection.
00:27
It can be written in cartesian form like this, and i'll use that to do my divergence.
00:33
Here it is divergence in spherical coordinates, but i'll do it with the cartesian.
00:38
Now, the problem isn't so much anything difficult in terms of the calculation of the divergence.
00:46
It's a straightforward calculation, very straightforward.
00:48
The idea is, though, it's going to lead to something we don't that isn't right.
00:54
There's a paradox.
00:55
If you think about divergence, it really is a, so flux density.
01:04
If you're looking out here, what comes, you're looking at the infinitesimal region around a point here.
01:11
What comes in, goes out.
01:14
So zero flux, zero divergence.
01:17
Same thing.
01:18
For our purposes, for our purposes.
01:20
But if you go infinitesimal region around the article zero point, everything.
01:25
So out here, you have in and you have out.
01:28
So zero.
01:28
Here, though, only got out.
01:30
So you say, wait a second.
01:30
And there is a flux, there is divergence.
01:37
So something is not going to be right in our calculation.
01:41
We're going to have to address that.
01:42
So let's do the calculation.
01:44
So let me actually write down what i just said.
01:48
So divergence is equal to zero, r not equal to zero, and it's not equal to zero, r equals zero.
02:03
That's what we, just from understanding what we mean by flux, flux density technically to be precise, and divergence.
02:16
Divergence is the flux density.
02:17
I'll mention that in the formula a little later, show you how that comes about.
02:23
Okay, but let's do the divergence right now.
02:25
So let me do the x term.
02:30
And the other two terms would be trivial from this.
02:33
So let's take a derivative of x first.
02:35
So it gives me one over, and this is, yes, 1 over x squared, y squared plus c squared, 3 halves, nothing happens to it.
02:45
Now we'll take the derivative of the denominator plus x minus three halves.
02:52
Now we've got minus three halves, minus one, so minus five halves.
02:59
So x squared, y squared, plus c squared, five halves.
03:04
And then we've got to take the derivative of the inside to x.
03:08
So, we're going back to using r.
03:11
This is 1 over r cubed minus 3 x squared over r to the 5th.
03:22
So that's what we have.
03:24
Now let's put all the other two terms together.
03:29
So a divergence of v, 3 over r cubed minus 3, x squared, y squared, plus c squared, over r to the 5th.
03:43
Well, but what's x squared plus y squared plus c squared? that's r squared.
03:47
R squared over r to the fifth.
03:48
It gives r to the third on the bottom.
03:50
This is zero.
03:52
So, obviously, we see all as well elsewhere, but not the origin.
03:58
So something's, we got to do something.
04:00
Where did we go wrong? well, the only thing is we're applying a formula that does not apply at the origin.
04:09
We're dividing by zero.
04:12
So this does not apply.
04:13
Does not apply at origin.
04:31
So it's okay everywhere else, we're not the origin, so we're going to do something else.
04:35
So what we're going to do is the divergence theorem.
04:39
You may or may not have seen it or had it derived for you yet.
04:44
I'll use it.
04:46
Here is a volume integral of the divergence.
04:53
That's the volume element.
04:56
And this is going to be a sphere of capital radius r around the origin.
05:02
And here's a divergent.
05:03
Convergence theorem.
05:04
Volume integration replaced by surface integration where that surface is the surface of that volume.
05:11
A surface of sphere, a radius, so that's what we're going to be.
05:28
That's, so our volume is a sphere of radius r and this integration is going to be over its surface.
05:35
So let me put in our formula again.
05:38
Now d .a is the outward point normal.
05:40
If you remember your goss's law from introductory physics.
05:43
So there's our formulation.
05:45
Now with, now before i do the rest, remember i mentioned about this being like a flux density? but what is this? isn't this scalcis law that you're used to? your e .da type idea.
06:02
Okay.
06:03
But we call this whole thing, electric flux, didn't we just the flux? so this is the, this is the flux, infinitesimal flux.
06:12
This is the flux through an infinitesimal volume.
06:18
And so on.
06:18
So this quantity here is flux density.
06:22
That's how we think about the divergences of flux density.
06:27
Okay, so with this being r, i can bring out one over r squared, and all i'm left with r .r.
06:36
Is one, is area integral.
06:39
So this gives me 4 pi r squared over r squared.
06:45
So the r squareds go away, so i'm just left with 4 pi.
06:50
But now let's talk about this capital r.
06:52
It can be any size i want.
06:55
I can shrink it and shrink it and shrink it even to infinitesimo around the origin and i get four pi is my answer.
07:05
So this tells me that the only point in this integration here there's only one point, one infinitesimal region and that is around r equals zero.
07:18
So i now know that the divergence of v and r equals zero is four pi.
07:26
That's what i get.
07:28
That's what i get.
07:31
Now, you might say, how do you reconcile all this together? how do you put it all together? well, we introduce the direct delta function.
07:45
It's a weird little function, i've got to say.
07:48
Very unusual properties.
07:50
It's not a classic function as you're used to having.
07:54
Its definition, it is zero, r not equal to zero.
08:04
Now, at r equal zero, becomes extremely steep, effectively infinite, but in such a manner that the area under the curve is equal to one.
08:22
Like i said, a very unusual function.
08:25
It's not a classic function...