00:01
In this problem, you have a metal cube, basically a conductor, that's moving to the right, which is the positive y direction.
00:07
And that's at constant velocity.
00:11
And you have a uniform magnetic field that is pointing straight up, which is in the positive z direction.
00:18
The x -axis is coming out towards you.
00:20
And the goal of the problem is to find the induced magnetic field in part a, when the free charges are electrons, that's metal, conductor.
00:29
And in part b, when now that free charge, the charge is free to move, is a positive charge cube.
00:36
So, and we want to compare those cases.
00:38
So we'll take each one, one at a time.
00:43
All right, let's, now what i'm going to do now is i'm going to take a top view of the cube.
00:48
I'm looking down on this top surface.
00:50
So that means i'm looking down on the magnetic field.
00:53
I'm looking down, i'm seeing the tip of the arrow, the magnetic field.
00:57
So magnetic field is coming out towards me now.
00:59
Now again i'm up here looking down.
01:04
The v, the velocity of the cube is still to the right.
01:08
My axis, x, y.
01:13
So again i'm looking down.
01:15
This top surface, this top portion is this here and the bottom, the lower portion of my drawing is this here.
01:27
Okay now let's look at a little negative charge electron.
01:34
Now remember, in any material, the pluses and minuses all bounce off.
01:41
So he's got a companion somewhere, but we're just going to look at him and we'll see what happens to him.
01:50
But the companion can't move, so whatever happens to him, that means they are going to separate in some manner.
01:56
Now the formula for the magnetic force on a moving charge is qv cross v.
02:04
Now in our case the charge is got is an electron so it's minus e for its q now putting our expressions for v and b remember constant velocity and uniform magnetic field so everything's there's no changing in time or anything of that nature so this is minus e v naught b naught j cross k now you can use your right hand rule for getting the cross product or here's a show you a little trick draw circle put around it going counterclockwise i j and k if the product you're asked for the cross project you ask for you have to go around counterclockwise gives you a plus cross product result you've got to go around clockwise minus cross product result so j cross k going around counterclockwise this is equal to i so this becomes minus e v naught b naught i hat that's the magnetic force so that is in the negative x direction so that's here now when you do this he's only there's no electric field he's only feeling this as his net force that means he's going to move to this this surface here here, this top surface in my top view, which actually in the original picture is the back surface.
03:49
But let's not worry about that.
03:50
Let's not confuse the issue.
03:51
He's going to move to here, leaving his companion, that plus charge, alone now.
03:57
They're no longer balanced.
04:00
So you're going to get a plus charge here, and he's going to end up up here.
04:06
Any time, though, you have a charge separation, you get an electric field.
04:09
But originally as you do this though the electric field is not going to be strong when you just get when you first have charges moving to the surface negative charges it's got to grow equilibrium will be when the electric force and the magnetic force are equal magnitude in opposite directions so let me draw a later point so now we have more negative charges having grown been deposited on this surface and leaving their companions.
04:39
In a way what's happening is that really the whole electron c is moving and it's leaving all these guys behind.
04:50
So it may not be that this charge here is it was the original match with him but doesn't matter it's an electron leaving as these all these electrons move upward they're leaving these guys unmatched.
05:05
Now electric field is drawn starts at a plus charge and ends at a negative charge.
05:13
So from that alone, these are electric field lines.
05:20
Or we can remember the definition of electric field.
05:23
The direction of the electric field gives you the direction of the force on a positive charge.
05:29
Okay, if you put a positive charge right here, it's a repulsion.
05:32
It's going to go that way.
05:33
So that also tells me electric field is pointing in the negative x direction.
05:39
So however you do it.
05:41
Now, the electric field gives me the direction, so this is e, gives me the direction of the force on a positive charge.
05:48
That means the force on a negative charge is the exact opposite direction.
05:53
Think about it.
05:54
If you put a negative charge here, it's a traction.
05:57
It's the opposite direction of the electric field.
06:00
So the force, the electric force, is in that direction, in the positive x direction.
06:06
Direction.
06:07
But notice now we can have the same magnitude and this remember to be in equilibrium we have to have the same magnitude and the opposite directions.
06:17
They could certainly be say magnitude pointing the same direction.
06:20
Well that's not going to balance out now is it? that's not going to stop the transfer.
06:25
Without this, this transfer continues and continues.
06:29
It will stop later on but we're not you know in terms of a build -up but that's a different issue.
06:36
Let's not concentrate on our problem at hand.
06:40
But we need, for equilibrium, we got to have zero net.
06:44
That means the force must be magnitude in opposite direction.
06:49
Okay.
06:51
So equilibrium when fb is equal to fe in opposite directions...