So last lecture was arguably the most important of all my lectures.
We saw how a changing magnetic field can produce a current, an induced electric field, an induced EMF.
And Faraday expressed that in his famous law, his famous equation which we see there on the blackboard.
You select a closed loop in your circuit.
Any loop is OK.
You attach an open surface to that closed loop.
Any open surface is OK.
And you then get an EMF in the loop, and that's the time derivative of the magnetic flux through that surface.
And the minus sign indicates that the induced current itself produces a magnetic flux that opposes the flux change, and that we refer to as Lenz's Law.
Today, I will expand on this a lot further.
So let's start with a conducting loop and a magnetic field.
This is a conducting loop.
Let the dimensions be Y, X and let- I have a uniform magnetic field.
Magnetic field B is like so.
And I choose as the perpendicular vector to my surface, this is the surface that I attach to that closed loop, I choose it pointing up.
And so the angle between dA and B, say theta, but B is uniform.
So the flux, phi B, is defined as the integral of B dot dA, over this open surface.
Flux is a scalar.
It's plus or it's minus or it's 0.
Flux has no direction.
So the flux in this case would be XY, which is the area of this loop since the magnetic field is uniform.
That's a very easy integral and then I get the magnetic field B, and then I get the cosine of the angle.
So now according to Faraday, it is the time derivative of this quantity that determines the EMF.
And, you can do that in several ways.
You can have dB/dT, the change in the magnetic field.
This is the area A of the loop.
You can change the area.
You can have a dA/dt.
But you can also change theta.
You can have a d theta/dt.
And I will look at those today.
This number here, the way I have chosen my dA, is a positive number.
If somehow this number increases in positive value, the induced current that is going to run will try to create a magnetic field to oppose the change.
So in that case if the flux, which is now positive, is getting larger positive, then the current that's going to run will be in this direction.
That's Lenz for you.
So it creates by itself, this current will create a magnetic field in this direction.
And if the magnetic flux, which is now positive the way I've defined it, were decreasing, then the current would go the other way around.
Last time, I did several demonstrations whereby we changed B.
We had dB/dT's.
And there was one particular demonstration that blew your mind and that you will tell your grandchildren about and that you will always remember, I hope.
Today, I'm going to change theta and I'm going to change the area, which will also give me then induced EMF's and therefore induced currents into a closed conducting loop.
So let me make another drawing of the closed conducting loop.
This has length Y and width X, and I'm going to rotate this.
My idea is that you can see this three-dimensionally.
I'm going to rotate this about this axis with angular frequency omega.
Omega is 2 pi divided by the period.
The period is the time of one rotation.
Normally we choose for that capital T.
I don't want to do that today because T can confuse you with Tesla.
And so I'm going to rotate this around so the angle theta that you have there, theta then becomes theta 0 plus omega T, going back to 8.01.
And I choose this theta 0 such that at T 0, I choose my theta to be 0, and so I have nothing to do with theta 0.
So what now is the magnetic flux?
This is my loop.
I have to commit myself to a surface.
Well, I will just choose this flat surface, just like I did there.
I chose that flat surface.
I'm free to choose any surface, why not taking the flat one.
And so the flux through that flat surface is then the area which is X times A, X times Y, that's the area of this loop.
And then I have the magnetic field.
And then I have cosine omega T.
Maxwell tells me it's not the flux that matters.
It is the change in the flux that matters.
OK, so d phi/dt.
I've got the A, the area, I've got the magnetic field.
An omega pops out, and I get a sine of omega T and I get a minus sign.
Normally I don't care about minus signs, because I'm only interested in the magnitude of the induced EMF.
I always know in which direction the current will flow, I really do, because I know Lenz's law.
So you should never have too many hang-ups on those minus signs, but since I'm getting a minus sign out of this now here, it would be a little foolish not to put a minus here and make this into a plus because that, then, according to Faraday is immediately the EMF and that EMF is changing with time because you have this sine omega T in here.
And so the current that is going to flow, the induced current, which will also be time-dependent, is the EMF divided by the resistance in the loop, and this is the total resistance of that entire network.
There could be light bulbs in there, there could be resistances in there.
It's the total resistance.
And this current, when I rotate this loop, is going to alternate in a sinusoidal fashion.
And we call that alternating current, AC.
That's what's coming out of the wall, AC.
Suppose this loop was double, and what I mean by double is the following, that it works like this.
Follow my picture closely.
I will go slowly.
It's like this, like this, like this, so, back, and I close it here, so it's one closed loop, but I have two windings.
I have to attach a surface to this closed loop.
Farado- Faraday insists I attach an open surface to this closed loop.
What would it look like?
Well, I advise you to take that, dip it in soap, and look at it, and what you will see then, because the soap will attach everywhere to the closed loop, you're going to see one surface.
It's not two separate surface.
You don't have two separate loops.
It's one surface but sort of two layers.
One is lower and the other one comes on top.
And so, the magnetic flux will double now, because you're going to see that this magnetic field penetrates both this soap film and the one that is below, and so you get twice the EMF and if you have N windings in one closed loop, capital N, then the EMF that you get would be N times larger and you can make N 1000.
There is no problem with that.
I'm going to do a demonstration for you whereby I'm going to use the earth's magnetic field and a loop that you see here that has 42 windings.
So my capital N is 42.
Not just two like here, but 42.
And it is circular.
It has a radius.
I think it's about thirty centimeters.
Here you have it.
It's about thirty centimeters.
So the area, pi r squared, which is my capital A, pi r squared is about 0.28 square meters.
You may want to check that.
I use the Earth's magnetic field, which is about half a Gauss, so that's about 5 times 10 to the -5 Tesla, if we work in SI units.
And I'm going to rotate it around with a period, period of about 1 second.
That means omega, 2 pi divided by the period, is then about 6 radians per second.
2 pi -- I call that 6 for now.
And so what is the EMF that I'm going to get when I rotate it once around per second?
Well, the EMF will change as a function of time.
We're going to get 42, that's N.
We're going to get A, that is 0.28.
We're going to get B, that is 5 times 10 to the -5, and then we're going to get omega, that is 6, and then we get this sine of 6 T.
You see the equation there.
The only difference is we have a capital N out here because we have N windings in the closed loop.
And this number here in front of the sine 6 T, you should check that, is about 3.5 millivolts.
3.5 times 10 to the -3 times the sine of 6 T, and that now is in volts.
So you get an alternating EMF, positive, negative, and the maximum value that you would get is 3.5 millivolts.
If I look at the EMF as a function of time, it would be something like this.
And from here to here, would then be 1 second if I really rotated around in 1 second.
And so the current, the induced EMF, according to Ohm's Law, is always the induced current times the resistance of the whole loop, so the induced current will also have this shape, of course.
And how high that is depends on how large R is.
The EMF is independent of capital R.
The EMF follows exclusively from those numbers.
It's the current that depends on what the resistance is.
Suppose now I rotate twice as fast.
I double omega.
Two things are changing now.
For one thing, that the full period now goes from here to here, only in half a second.
But there's something else that changes.
The EMF now doubles, because look at my equation.
It's hiding behind the blackboard, I think.
There is an omega in there.
It's linearly proportional to omega, because it's d phi/dt that matters.
See, the omega pops out, and so you now get double the EMF, so the 3.5 millivolts maximum would become 7, and so if I try to make a drawing of that twice as high here, twice as low here, then you would get something like this, and so this omega is now twice this one.
You get double the maximum value of the EMF.
I'm going to show that here.
I'm going to improve on my lights.
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