00:01
In this problem, you have two currents, one left coming out towards you, the right going into the screen, away from you.
00:10
And we're told we want to find the direction of the magnetic field, the net magnetic field, at this point mark x, which is directly above the right current, right wire.
00:20
They give us the individual currents, the ds and the h, the geometric quantities involved.
00:26
Notice h is written in terms of d, so don't forget that.
00:30
And so our job is to find individually each of the magnetic fields due to each wire and then add them up as vectors.
00:37
That's what we're going to be doing.
00:39
Now, before we do that, though, there is one quantity i can get right off the bat geometric quantity that we're going to need.
00:48
And it's going to be this angle theta.
00:56
So i'll just do that over here and then we'll done with it.
00:59
Tangent theta would be h over d.
01:04
So theta is n.
01:06
Inverse tangent of 0 .27 d over d.
01:14
So it's just basically inverse tangent of 0 .27.
01:16
And it gives us 15 .1 degrees.
01:21
We'll need that later on.
01:23
But i'll get it.
01:24
See in the diagram was nice and open to us here.
01:27
I'll take care of it now.
01:29
Okay.
01:30
So let's start working on the individual magnetic fields.
01:33
Let's start with the right one because it's the most straightforward, because the point x is right above it.
01:40
Actually, let me give myself, okay.
01:56
So there's our current.
01:58
Here's our point x.
02:00
This distance will be h for future use.
02:04
Now, right -hand rule.
02:06
Thumb the direction of the current.
02:08
So into the screen, fingers give the arrows on the magnetic field lines, which would be clockwise in this case.
02:15
So here are your arrows on the magnetic field line.
02:22
Now, you draw the magnetic field tangential, the point you care about.
02:27
So it's going to be along the, in this case, along the horizontal, one way or another.
02:32
But you've got to follow the general sense of the arrows.
02:35
The arrows are clockwise, so here the arrow would be pointing to the right.
02:42
So this is b.
02:45
Sabar.
02:46
This is the magnetic field due to the right current.
02:52
And we'll worry about the vector nature when we have the second magnetic field to add them up.
02:57
But let's not worry about that just yet.
03:00
We've taken care of enough for now for that one.
03:02
But let's calculate the magnitude.
03:04
We could leave everything to the very end, but this will work out fine too.
03:11
So mu not for a long straight wire, mu not i over 2 pi times r, the perpendicular distance between the wire, the long wire, and the point we care about, which in this case is h.
03:27
And now let me just do one last thing.
03:29
Munot, ir 2 pi 0 .27 times d.
03:39
I'll get it in terms of d.
03:40
It's a minor point for what i did.
03:43
All right, so let's put in our values.
03:45
4 pi times 10 to the minus 7 tesla meters per amp.
03:55
That's mu not.
03:56
Current is 4 .72 amps.
04:03
2 pi, 0 .27, 0 .23 meters.
04:14
Okay.
04:18
Pies go away, 2 goes away, but don't forget, there's still 2 up here.
04:24
And when you calculate that out, you get 1 .52 times 10 to minus 5 tesla.
04:33
That is the magnitude of the magnetic field due to the right wire.
04:39
So we've done everything we can at the moment for the right wire.
04:42
There's nothing more to do.
04:44
All i'll be done later is the adding up.
04:47
Okay.
04:48
Let me give myself another magnetic field line.
05:02
Move it up a little.
05:04
Okay.
05:07
And now this one is for the left.
05:13
So that one's coming out toward us.
05:18
So thumb out of the screen.
05:21
Fingers are wrapping around counterclockwise.
05:30
So there's are the arrows.
05:31
Now, here is the point x.
05:34
Here's the point x.
05:38
Now, here is, well, let me, we draw the magnetic field, then i'll do more geometry.
05:47
We draw tangential always.
05:49
So it's going to be along this line here, following the sense of the arrows, which is counterclockwise.
05:57
So we have to, our vector, i can draw halfway decent.
06:02
That's reasonable enough.
06:04
So this is bl.
06:06
That's the magnetic field due to the left -hand current.
06:12
Now, let's do a little bit of work here in terms of what we have.
06:26
Again, not the scale compared to the other diagrams.
06:29
This is d.
06:30
This is h.
06:33
This is theta.
06:39
This then also be theta.
06:42
This is 90 degrees.
06:43
It's tangential.
06:47
So this angle here, which i'll call phi, the whole thing's gotta be 180 when you add it up.
06:58
So phi is equal to 90 degrees minus theta.
07:06
Add it all up.
07:07
90 minus theta plus 90 plus 90 plus theta, the 90s add up to 180, the theta's cancel out...