00:01
In this problem we have a conducting rod that is moving across conducting rails and we have a magnetic field that is out of the screen.
00:13
The first part of the problem wants to know why is it that the induced current flows clockwise.
00:20
So that is what we have to look at.
00:22
So what we want to do first is concentrate on the moving rod.
00:27
So let me redraw that here.
00:31
Now let's look at a positive charge.
00:34
What happens to it because of this motion? now the magnetic force on a moving charge is given by q v cross.
00:43
Now we don't have any values.
00:50
Q is positive, it's a positive charge.
00:52
So whatever the direction of v cross b is, that's the direction of the force.
00:55
That's all we care about here.
00:58
So i'm going to show you how to get the direction two ways.
01:02
First right hand rule.
01:03
Fingers are in the direction of the first vector which is v.
01:05
So your fingers are pointing to the right.
01:09
Palm is facing the direction of the second vector b which is out of the screen in such a manner so that your fingers can turn in that direction.
01:18
Thumb is down.
01:20
So that is the direction of the magnetic force downward.
01:27
Now i'll have more to say about that in a second.
01:29
But now say you don't like that.
01:30
You want to do it with unit vectors.
01:32
Alright, let's look at this.
01:33
V is in the positive y direction.
01:36
Such j hat.
01:39
The b is out of the screen.
01:42
That is in the positive x direction.
01:45
That's i hat.
01:47
J cross i is minus k.
01:50
And so that's in the down direction.
01:56
Whichever way you're comfortable with.
01:59
Knowing multiple ways is always good.
02:01
You can check yourself.
02:03
You can check yourself.
02:05
So now what happens here with this vector diagram? with this force, it's making positive charges move down this way.
02:15
It's leaving behind negative charges.
02:18
Don't worry about what actually moves.
02:22
We think of conventional current as positive charge even though we know in say a copper wire that it actually is electrons that are moving.
02:30
It doesn't matter.
02:32
One moving one way is the same as the other moving the other way.
02:36
So positive charges are moving this way.
02:39
Now any time that, so there will be a separation of charges.
02:41
There's going to be a buildup of plus here.
02:43
And there's going to be then a buildup effectively of negative there if you like.
02:46
Or being left behind.
02:48
Being left unmatched.
02:49
So i'm going to get plus here, minus here.
02:54
Now as soon as this separation of charge arises, there is going to be an electric field that's going to be the negative or the minus.
03:03
So that will oppose this.
03:05
But that field has to grow as you get more and more charge built up.
03:11
That electric field is growing with that buildup.
03:15
So it starts out to be zero electric field up.
03:18
That grows a little.
03:19
Finally it will reach the same value of fb.
03:23
And then all this is off.
03:25
And you are stable at that point.
03:28
You are stable.
03:30
That's how you get your formula for motion emf.
03:32
But we don't need to worry about that here.
03:34
I'm just giving you some additional information.
03:39
So this separation, what does this kind of look like to you? you got pluses on the bottom and minuses on top.
03:45
That means i can really redraw my circuit like this.
04:00
There is my plus.
04:02
There is my minus.
04:03
If you had that in a circuit homework problem, where would you say the current flows? counterclockwise or clockwise? you'd say there's what you wanted to prove.
04:21
Just by knowing the direction of the force causing that separation.
04:25
You can see the positive charges.
04:26
I'm moving this way.
04:29
And remember, the rails are still there.
04:32
Rails are still there.
04:36
So that's just like following the current in the battery.
04:40
Going around.
04:42
Same exact thing.
04:47
Now, that's one way of getting it.
04:49
Let's talk about flux now.
04:53
This is the most basic definition, b dot a.
04:55
You can use this formula when the enclosed area by the circuit, this area here, is flat.
05:03
Which, unless you're doing something, well, i won't.
05:09
For our situations and stuff, they're flat.
05:12
But in reality, it has to be flat and the magnetic field must be uniform over that area.
05:23
It's got to be constant magnitude, constant direction over that area.
05:27
If it's not, then you've got to break up that bigger area into smaller areas where it would be true that the magnetic field would be constant over that area, that sub -piece.
05:38
But again, that's a different problem.
05:41
Now the area vector.
05:44
Magnitude is the area, the enclosed area.
05:47
Its direction is perpendicular to the area.
05:50
So it's either going to be out of the screen or into the screen.
05:52
Your choice.
05:55
And i'm going to choose out of screen, out of the page.
06:04
My choice.
06:05
Anything, as long as you're consistent, anything can be made to work.
06:07
It's going to lead you to the same answer.
06:12
So it's going to be ba cosine of the angle when they're tail to tail.
06:16
Well, they're in the same direction.
06:17
Cosine is zero degrees.
06:19
So the flux is just ba.
06:21
If i'd chosen a to be into the screen, then i'd have cosine 1a, they'd have a minus sign here.
06:29
Does that cause any problems? no.
06:32
You just have to be consistent with it.
06:34
Any way you look at it, the flux is increasing, as i'm going to write now.
06:40
So a to b increases due to a increasing.
06:53
The rod is moving to the right.
06:55
A is getting bigger...