00:01
So here i'll show how you get the magnetic field of a very large sheet of current.
00:06
And to do so, the symmetry is such that you may use ampere's law for magnetic field, sorry, not flux, which says that the line integral of the magnetic field dotted into the path around a closed path is proportional to the enclosed current.
00:26
So here i've drawn the current coming out of the page.
00:30
We'll just set a coordinate system with y up and x to the right.
00:36
So we're talking about z coming out of the page.
00:41
And so the current sheet points into the page.
00:45
It's a little bit hard to draw, but i could kind of show the idea that we have a sheet of current that's extending into the page.
00:58
But the loop that you would draw, the amperian loop, we'll draw that in green.
01:06
We'll cut through the wires into and perpendicular to the sheet and kind of make a rectangular shape.
01:16
And i usually like to circulate in the way i would expect the magnetic field to circulate around a single wire.
01:25
So the magnetic field will, using the right -hand rule, circulate around in a counterclockwise sense.
01:35
So we are going to take our empyrian loop also circulating in the counterclockwise direction.
01:48
And the magnetic field will occur just along the sides of the loop, either pointing up or down in the j direction, up on one side, down on the other.
02:04
And due to the fact that, yeah, we have these wires continuing on and on and on, the ends will have the magnetic field projection cancel on those two ends.
02:18
So we have basically b times l on the right -hand side plus b times l on the left -hand side is equal to b .dl.
02:34
We're assuming that it's constant along that contour, and that has got to equal to mu not.
02:44
The ion closed is the linear current density, sometimes called k, which is the current per unit length.
02:56
So we're talking about the current k, the total will be k times l.
03:09
Okay, we can think about that sheet made up of individual wires, as we'll see an example.
03:16
So what we have is basically twice bl is equal to mu not kl, and if we cancel out the l, that means the magnetic field has magnitude mu not k over 2.
03:38
And it points in two different directions on one side.
03:43
So on the right -hand side, b points in the positive y direction.
03:50
And by the way, this is very similar to the electric field of a uniform sheet.
03:57
So it is uniform and points in opposite directions on either side of the sheet.
04:06
Okay, so let's take a look at some examples.
04:11
Let's first calculate the magnetic field due to a long string of actual wires that are carrying the current.
04:28
And so k is the current per wire times the number of wires per unit length.
04:38
We'll just call that n.
04:39
And equals number of wires per unit length.
04:51
And we'll calculate the magnitude of the magnetic field, in this case, if i is 0 .17 amps and n is equal to 940 wires per meter.
05:10
Okay, so that's fairly easy to do.
05:13
Mu -naut is 4 pi times 10 to the minus 7, tesla times meter per amp.
05:21
That's mu -not.
05:22
And the i enclosed comes from a product of 0 .17 amps times 940 wires per meter.
05:39
And we could put in amps per wire and so many wires per meter.
05:43
The unit of wire will cancel.
05:47
And then we have to divide by two.
05:53
Okay.
05:53
So that magnitude is substantial.
06:03
And that comes out to be 1 .00 times 10 to the minus 4 tesla.
06:12
Now let's take a look at some examples with superposition.
06:19
We'll assume that we have the same arrangement of wires, but we'll take the case with the same arrangement...