Our Energy Sources, Electricity

I’d like to thank you for your evaluations. They were very useful to me. I already sent e-mail to about fifty students and I had some interesting exchanges

with some of you. Many of you are very happy

with their recitation instructors. That’s great. Many are moderately happy. Maybe that’s OK. But there are quite a few who are very unhappy

with their recitation instructors. If you are very unhappy with

your recitation instructor, you are complete idiots

if you stay in that recitation. We have thirteen recitation instructors,

and can assure you that it will be very easy to find one

that agrees with you and you can come and see me

if that helps. Some are better than others. That’s the way it goes in life. Some students would like to see more

cut-and-dried problem solving in my lectures. I think that’s really the domain of recitations. Lectures and recitations are complementary. In lectures, I prefer to go

over the concepts and I always give numerical examples

to support the concepts, in a way that’s problem solving,

and I show demonstrations to further support the concept,

because seeing, obviously, is believing. I try to make you see through

the dumb equations and admittedly my methods

are sometimes somewhat different from what you’re used to here at MIT. I try to inspire you and at times I try to

make you wonder and think. And I want to keep it that way. I believe that hardcore probling- problem-solving

is really the domain of the recitations. Many of you found the exam too easy

and many of you found the exam too hard. Some complained it was too hard

because it was too easy. [audience laughter] Quite ironic, isn’t it? They say, “We want more math,

we want more standard problems.” Look, who wants more math? I’m teaching physics. I test you physics,

I don’t test you math abilities. If you digest the homework and that’s very important

that you make the homework part of your culture,

that you study the solutions. The solutions that we put on the web,

today, four fifteen, solutions through number four

will go on the Web. Believe me, they are truly excellent solutions,

not cut and dry. They give you a lot of background. If you digest those solutions

then the concepts will sink in. And now, at your fifty minute test,

do you really want problems which are complicated maths? Clearly, not. I could try that,

during next exam, but then I may have to buy myself

a bullet-proof vest to be safe. Concepts is what matters. When I gave my exam review here,

I highlighted the concept. Each little problem that I did here

was extremely simple. Conceptually, they were not so simple. But from a math point of view, trivial. Clearly, I can not cover all the subjects

in a fifteen minute exam. I have to make a choice, so your preferred

topic may not be there. Some of you think that the pace

of this course is too slow. Some of you think it’s too fast. The score, the average score,

was three point eight. Four point zero would have been ideal. What do you want me to do? I can’t accommodate all of you. Those who think it’s too slow

and those who think it’s too fast. Three point eight is close enough to ideal

for me, four point zero. And so I’ll have to leave it the way it is. Besides that, keep in mind you are now at

MIT. You’re no longer in high school. Now the good news. There were quite a few students who said

the homework is too long. Not a single person said it was too short. I can fix that. I will reduce all future assignments by about

twenty-five percent, effective tomorrow. I have already taken off assignment

number five, two problems. You’re down now to seven and I will do that,

all assignments that are coming up. [applause] My pleasure. Today, I’m going to cover

with you something that conceptually is the most difficult

of all of 802. And you will need time to digest it. And if you think that what you’re going to

see is crazy, then you’re not alone. The only good news is that conceptually,

it’s not going to become more difficult. Remember that Oersted

in 1819 discovered that a steady current produces

a steady magnetic field, and that connected electricity

with magnetism. A little later,

Faraday therefore suggested that maybe a steady magnetic field

produces a steady current. And he did many experiments

to show that. Turned out to not to be so. And one way he tried that is as follows. He had here battery, with a switch

and here he had a solenoid. He closes the switch. A current will flow and that creates

a magnetic field in the solenoid and that magnetic field, maybe it runs like so,

depends on the direction of the current. And so now, he put around this solenoid

a loop. Let’s call this loop number two

and it was around the solenoid and let’s call this loop number one,

of which the solenoid is part. Whenever there was a current

in number one, he never managed to see

a current in number two. If there is a current going in number one,

there is a magnetic field and that magnetic field is seen, of course,

by the conductor number two by that loop. Never any current. And so he concluded that a steady magnetic field

as produced by the solenoids, circuit one, does not produce a steady current

in number two. But then one day he noticed, that as he closed

the switch he saw a current in number two, and when he opened the switch

again he saw a current in number two and therefore he now concluded that a changing

magnetic field is causing a current. Not a steady magnetic field,

but a changing magnetic field. And this was a profound discovery which changed

our world and it contributed largely to the technological revolution of the late nineteenth

and early twenty century. A current, therefore an electric field,

can be produced by a changing magnetic field, and that phenomenon is called

electromagnetic induction, and that phenomenon runs our economy,

as you will see in the next few lectures. I have here a conducting wire, a square. I could’ve chosen any other shape. Try to make you see three dimensionally. And I approach this conducting wire

with a bar magnet. The bar magnet has a magnetic field

running like so. As I approach that loop, that conducting wire,

moving the bar magnet, that’s essential. I can’t hold it still. I have to move it. If I come down from above

and I move it down, you’re going to see a current

going through this loop. And that current will go

into such a direction that it opposes the change

of the magnetic field. The magnetic field is

in down direction and it is increasing as I move

the bar magnet in. Then this current loop will produce a magnetic

field which is in this direction, and when you look from below the current

will go clockwise, producing a magnetic field in this direction. If you move the bar magnetic out,

then the magnetic field is going down here, then the current will reverse. The current wants to oppose the change

in the magnetic field and that’s called Lenz’s Law. It is the most human law in physics,

because there’s inertia in all of us. We all fight change at some level. Lenz’s Law is extremely powerful in always

determining in which direction these induced currents will run. It is not a quantitative law. You can not calculate how strong

the current will be, but it’s very useful as you will see today

to know the direction of that current that gets you out of all kinds of problems

with minus signs. I now want to do a demonstration which is

very much like what you see here. I have here a loop. That is the square that you see there

except that it’s not– not one loop, but it is many of them. Hundreds, doesn’t matter. And what we’re going to show you is

an amp meter that is connected, so there is somewhere in this circuit

an amp meter. I have a bar magnet and I’m going to approach

this conducting loop with a bar magnet and you’re going to see a current

running in one direction and when I pull it out it will be running

in the opposite direction, and when I hold my hand still so that

the magnetic field is not changing– no current. You’re going to see the current meter there,

and here is my bar magnet. I come close to this conducting loop. Notice we see a current. I pull back,

the current is in the other direction. Now I will go faster, so that the change of

the magnetic field per unit time is stronger. [whistle] More current. I go out fast. [whistle] More current. So you see it’s the change

of the magnetic field that matters. If I come in very slowly, which I do now,

very slowly, we almost see nothing. Right now the entire magnetic field

is inside this loop. The strongest I can have it. Nothing happens because there is no change

in the magnetic field. It’s only when I do this

that you see the current. So an induced current is clearly

the result of a driving force. There must be, just like we had with batteries

in the past, there must be an EMF. There must be an electric field that is produced

in this conducting loop. And so I create now an induced EMF–

we used that word EMF earlier for batteries, so now we have an induced EMF, which is

the result of this changing magnetic field, and that therefore is the induced current times

the resistance of that entire closed conductor, whatever is in there. In this case, the total resistance of all

these windings, of all the copper wire. That’s Ohm’s Law. So the induced EMF is always the induced current

times the resistance. Faraday did a lot of experiments,

and one of the experiments that he did was that he produced a magnetic field,

so he ran a current through a loop of some kind, let’s say he ran a current going around,

creating therefore a magnetic field and he was switching the current in and out

so that he could change the current and so it produces a magnetic field

and this magnetic field changes when you– close and open the– the switch. And then here, he had his second

conducting wire, just like we had there, and he measured in there the current. And what he found, experimentally,

is that the EMF that is generated in here, which I will call EMF generated

in my conducting loop number two, is proportional to the magnetic field change

produced by number one, so the field goes through number two

and this field is changing, so he knows that if the change is faster,

as you just saw, you get a higher EMF. He also noticed that E two is proportional

to this area, so to the area of number two. And that gave him the idea that the EMF

really is the result of the change of the magnetic flux through this surface

of number two. And I want to refresh your memory

on the idea of magnetic flux. We do know, or we remember

what electric flux is. And magnetic flux, very similar. If this is a surface and the local vector

perpendicular to the surface is like so, of course it could be in a different direction

and the local magnetic field is for instance like so, then a magnetic flux

through this surface is defined. We call it phi of B, is the integral

over an open surface. This is an open surface of B dot dA. And the electric fields we defined in exactly

the same way, electric flux, except we had an E here. There was nothing there. So if this magnetic flux is changing,

Faraday concluded, that then you have an EMF

in this conducting wire. So essential is the change

of the magnetic flux. If we take some kind

of a conducting wire, like so, let’s make it in the blackboard

for now to make it easy. And I attach to this wire a surface

because the moment that you talk about flux you must always specify your surface. A flux can only go through a surface,

so this is my surface now for simplicity. And there is a magnetic field coming out

of the blackboard at me– and it is growing. It is increasing. I will now get an EMF, a current,

flowing in this direction. Lenz’s law. If the magnetic field is increasing,

then the current will be in such a direction that it opposes the change. It doesn’t want that magnetic field to increase. And so it goes around like this,

the current, so that it produces a magnetic field

that is in the blackboard. And so it is the flux change of that magnetic

field through this flat surface that determines the EMF. So the EMF is then the flux change,

d phi dt, through that surface. To express Lenz’s Law that it is always opposing

the change of the magnetic flux, we have a minus sign here. But minus signs will never bother you,

believe me, because you’ll always know

in which direction the EMF is. It’s clear that the EMF is going to be

in this direction. That’s the direction in which it will make

the current flow. But we have to put it there

to be mathematically correct. That’s really Lenz’s Law. You’re looking at Lenz’s Law here. So you can also write down for this:

minus the surface integral of B dot dA over that open–

whoo, I hope you didn’t see this. Over this open surface. That’s the [tape slows down] Oops, look what I did. I forgot the DDT in front of

the integral sign. Sorry for that. [tape speeds up to normal speed] If you put yourself inside that conductor, and you marched around in the direction

of the current, you will see everywhere in the

wire an electric field, of course. Otherwise, there would be no current flowing. And so if you go once around

this whole circuit, then that EMF must of course also be

E dot dl over the closed loop. So you’re marching inside the wire,

you find everywhere an electric field and these little sections I dl. E and dl are always in the same direction

if you stay in the wire and so this should be the same

and this is a closed loop. So this is all if you want

what we call Faraday’s Law. We never see it in so much detail. I will abbreviate it a little bit

on the board there. But I want you to appreciate

that there is no battery in this circuit. There is only a change in the magnetic flux

through a surface that I have attached

to the conducting wire and then I get an induced EMF

and the induced EMF will produce a current given by Ohm’s Law. So I want to write down now

on that blackboard there, Faraday’s law in a somewhat abbreviated way because we have all

Maxwell’s equations here and so we now have

that the closed loop integral, closed loop of E dot dl–

that’s that induced EMF. You can take minus d phi dt or the time

derivative of the integral B dot dA. That’s the one I will take. Integral of B dot dA and this is over an

open surface. And that open surface has to be attached

to this loop and that is Faraday. We have Gauss’s Law,

we have Ampere’s Law. We have this one which tells you

that magnetic monopoles don’t exist. This would only not be zero if you had a magnetic

monopole and put it in a closed surface. Come and see me if you find one. And this now is Faraday’s Law, so you think that all four Maxwell’s equations

are now complete. Not quite. We’re going to change this one shortly. So we can’t celebrate yet. We have to wait. It’s the big party. There’s always a little bit of an issue about

the direction of dA and I will explain to you how the convention goes

but it really is not so crucial because Lenz’s Law always helps you to find

the direction of the EMF, but if we are trying to be a purist, if this is my conducting loop and if I attach

a flat surface to this, if I did that, and if I go around

closed loop integral E dot dl, Faraday doesn’t tell me

which way I have to go. I can go clockwise. I can go counterclockwise. We will then do the same thing that we did

before with Ampere’s Law, apply the right-hand corkscrew rule

and that is that if you march around clockwise, then dA will be in the blackboard,

perpendicular to the blackboard, perpendicular to this surface

and if you go counterclockwise then dA will come towards you. The surface doesn’t have to be flat. It can be flat. There’s nothing wrong with it. But there can also be a bag attached to it,

as we also had earlier. I have here a closed conducting wire

and I could put a surface right here but I can also make it a [inaudible],

like this, perfectly fine. Nothing wrong with that. That’s a open surface attached to this loop. That’s fine. You have a choice and the convention

with dA is then exactly the same, that if you go clockwise then the dA

would be in this direction using the right-hand corkscrew

locally here. If you went counterclockwise,

the DA would flip over. So what is now the recipe

that you have to follow? You have a circuit, electric circuit

that determines then your loops, of course. You can take loops anywhere in space,

but that’s not too meaningful, so you take them into your circuits,

and so you define the loop first. Then you define the direction in which you

want to march around that circuit. You attach an open surface to that

closed loop and you can determine on that entire surface

the integral of B dot dA. Everywhere on that surface locally

you know the dA, locally you know the B, you do the integration

and you get your magnetic flux and then if you know the time change

of that magnetic flux, then you know the EMF. If you go around in this conducting circuit, and you measure everywhere

the electric fields, then the integral of E dot dl,

if you go around the loop will give you the same answer

and that connects the two. The magnetic flux change is connected with

the integral of E dot dl when you go around. And you have to take that minus sign

into account. How come it doesn’t matter whether you choose

a flat surface or whether you choose a bag? Well, think of magnetic field lines

as a flow of water or spaghetti, if you like that, or a flow of air. It is clear that if there is some kind of

a flow of air through this opening, that it’s got to come out somewhere,

so it always comes out of this surface. And therefore,

you’re really free to choose that surface, so you always pick a surface

that is the best one for you. Now, all this looks very complicated. But in practice, it really isn’t, because your loop is always

a conducting wire in your circuit and the minus sign is never an issue

because you always know with Lenz’s law in which direction the EMF is. In fact, when I solve these problems,

I don’t even look at the minus sign. I ignore it completely. I def– I calculate the magnetic flux change

and then I always know in which direction the current is,

so I don’t even look at the minus sign. Now I want to show you a demonstration which

is very much like what Faraday tried to do. I have here a solenoid. We’ve seen this one before. We can generate quite a strong magnetic field

with that. And we’re going to put around this solenoid

one loop, like we had here, like Faraday did and then we’re going to close the switch and so we’re going to build up

this magnetic field and we’re going to see the current

in that loop. And so if we look– if we make a cross-section

straight through here, then it will look as follows. Then you see here the solenoid, so the magnetic

field is really confined to the solenoid. Magnetic field outside the solenoid

as we discussed earlier is almost zero, so there’s only a magnetic field right here. Keep that in mind in what follows. And now we’re going to put a wire around it,

with an amp meter in there. If the magnetic field comes out of the board,

and is growing, is increasing, the current will flow in this direction. Lenz’s Law. If it is decreasing, the current will go in

the opposite direction. Now keep in mind that the magnetic flux

through this surface, that is, my surface which I attach

to this closed loop, that that magnetic flux remains the same

whether I make the loop this big or whether I make the loop

very crooked like so, because the magnetic flux is only confined

to the inner portion of the solenoid and that’s not changing. And so when I change the shape

of this outer loop, you will not see any change in the current. I hope you– that doesn’t confuse you. I’m going to purposely change the size of

the loop and so I’m going to do that now. You’re going to see there

a very sensitive amp meter– and you’re going to see here this loop, the big wire and I’m going to just put it

over this solenoid. Let me first make sure that my amp meter

which is extremely sensitive, I can zero it. It’s sign sensitive. If the current goes in one direction, you

will see the needle go in one direction. If the current goes in the other direction,

you will see the change. And so now I put this loop around here,

crazy shape this loop. So it’s around this solenoid once,

so the magnetic field is inside the solenoids and so think of a surface which is

attached to this crazy loop and now I’m going to turn the current on and only while the current is changing

will there be a changing magnetic flux. Only during that portion

will you see a current flow. Three, two, one, zero. I will break the current,

three, two, one, zero. Went the other direction. If I change the size of the loop, I’m making

it now different, much smaller. Makes no difference,

for reasons that I explained to you, because the magnetic flux is not determined

in this case by the size of my loop but is determined by the solenoids,

so if I do it again now, with a very different shape of

the loop– let me zero this again. Three, two, one, zero. Three, two, one, zero. No change. Almost the same which you saw before. Now comes something that may not be

so intuitive to you. I’m now going to wrap this wire three times

around. And so this outer loop, this outer conducting

wire, is now like this. One, two, three. Something like that. Now I have to attach in my head a surface

to this closed loop. My god, what does it look like? What a ridiculous surface. Well, that’s your problem, not Faraday’s problem. How can you imagine that there is a surface

attached to this loop? Well, take the whole thing and dip it in soap. Take it out and see what you see. The soap will attach everywhere

on the conducting loop. And if this loop were like this,

going up like a spiral staircase, you’re going to get a surface

that goes up like this. But the magnetic fields go through

all three of them. Therefore, the changing magnetic flux will go

three times through the surface now and so Faraday says, fine, than

you’re going to see three times the EMF that you would see if there were only one loop. And if you go thousand times around, you get

thousand times the EMF of one loop. Not so intuitive. So I’m around now once. I go around twice. And I go around a third time. I have three loops around it now. I can zero that, but that’s not so important. Three, two, one, zero and you saw

a much larger current. It’s about three times larger

because the EMF is three times larger. I break the current. We see it three times larger. And this is the idea behind transformers. You can get any EMF in that wire that you

want to, by having many, many loops. You can get it up to thousands of volts

and that’s not so intuitive. So Faraday’s law is very non-intuitive. Kirchoff’s Rule was very intuitive. Kirchoff said when you go around a circuit the closed-loop integral of E dot dl

is always zero. Not true is you have a changing magnetic flux. If you have a changing magnetic flux,

the electric fields inside the conducting wires now become non-conservative. Kirchoff’s Rule only holds as long

as the electric fields are conservative. If an electric field is conservative

and you go from point one to point two, the integral E dot dl is independent of the path. That’s the potential difference between two

points, that’s uniquely defined. That’s no longer the case. If you go around once with this experiment,

you get a certain EMF, you go three times around,

you get a different value. Your path is now different

and that’s very non-intuitive, because you’re dealing with non-conservative

fields for which we have very little feeling. Now, I’m going to blow your mind. I’m going to make you see something

that you won’t believe– and so try to follow step-by-step– leading up to this unbelievable

and very non-intuitive result. I have here a battery and the battery

has an EMF of one volt. Here is a resistor, R one,

which is hundred ohms. And here is a resistor, R two,

which is nine hundred ohms. And I’m asking you what is the current

that is flowing around. And you will laugh at me. You will say that’s almost an insult. I wish you had given that problem

at the first exam, because E equals the current that is going to run, divided by R one plus R two. [tape slows down] Oh, my goodness, what did I do. [Lewin laughs] I forgot Ohm’s Law. E equals IR, remember, not I over R. So R one plus R two should go upstairs. And everything that follows is correct,

so you don’t have to worry about that. This was just a big slip of the pen. [tape back to normal speed] And so the current I is ten to the minus three

amperes. One milliampere. Big deal. Easy. Current is going to flow like this. Fine. Let’s call this point D

and call this point A. And I asked you what is the potential difference

between D and A. You will be equally insulted. VD minus VA, you apply Ohm’s Law,

you say that’s this current times R two. Absolutely. I times R two. But that is plus oh point nine volts. Now I say to you, well,

suppose you had gone this way, then you would’ve said, “Well,

I find the same thing, of course.” Kirchoff’s Rule. So indeed, if you go VD minus VA,

and you go this way, then notice this battery,

this point is one volt above this point. But you have in the resistor here, you have

a voltage drop according to Ohm’s Law, and the current times hundred ohms gives you

a one-tenth voltage drop here, so VD minus VA is the one volt from the battery minus I times R one, and that is plus oh point nine volts. What a waste of time that we did it twice

and we found the same result. So I connect here a voltmeter. The voltmeter is connected to point D

and to point A. And I asked you what are you going to see. The answer is plus oh point nine volts and you will provided that the plus side

of the voltmeter is connected here and the minus side of the voltmeter there. Voltmeters are polarity sensitive. This is fine. Kirchhov’s rule works. The closed loop integral from E dot dl

going from D back to D is zero. So far, so good. Now, hold on to your chairs. I’m going to take the battery out. Who needs the battery. I’m going to replace the battery by a solenoid– which you see right here,

and this solenoid when I switch it on is creating an increasing magnetic field. Only here, and let’s assume that that increasing

magnetic field is coming out of the board, and that it is increasing. Lenz’s Law will immediately tell you

in what direction the current is. If this magnetic field is increasing towards you,

the current will be in this direction. The magnetic flux change, d phi dt, at a particular

moment in time, happens to be one volt. An amazing coincidence, isn’t it. E induced at a moment in time is one volt. Now, I ask you, what is the current? Well, you’ll be surprised that I even have

the courage to ask you that, because Ohm’s Law holds. The induced EMF is one volt and R one

plus R two is still a thousand ohms, so ten to the minus three amperes. I really make a nuisance of myself when I

say, “What is VD minus VA?” and you get annoyed at me and you say, “Look,

the current I through R two, Ohm’s Law, V equals IR, plus oh point nine volts.” And then I say, but now suppose we go the other–

the other side, and we want to know now what VD minus VA is, and now it’s not so simple,

because there’s no battery. And so now when I go from D to A,

I don’t have this one, and therefore I now find

minus oh point one volts. I find a totally different answer. I attach a voltmeter here. That voltmeter will show me

plus oh point nine volts. Now I attach a voltmeter here, the same one. I flip it over. It’s connected between point D and point A. It will read minus oh point one volts. This voltmeter, which is connected between

D and A, reads plus oh point nine. This voltmeter which is connected to D and A

reads minus oh point one. The two values are different

and I placed on the web a lecture supplement which goes

through the derivation step-by-step, which will convince you that indeed

this is what is happening. Why we can’t digest this so easily is we don’t

know how to handle non-conservative fields. If you have a non-conservative field,

then if you go from A to D of E dot dl or from D to A for that matter, doesn’t matter,

the answer depends on the path. It’s no longer independent of the path. And so if here is D and here is A,

and you go this way, you find oh point nine volts,

plus if you go this way– you find minus oh point one volts. Faraday has no problems with that. Kirchoff has a problem with that,

but who cares about Kirchoff? Faraday is the law that matters,

because Faraday’s Law always holds, because if d phi dt is zero,

then you get Kirchoff’s. Kirchoff’s rule is simply a special case

of Faraday’s Law, and Faraday’s Law always holds,

so Kirchoff is for the birds and Faraday is not. Suppose you go from D to A and back to D. Well, we know that VD minus VA,

if we go through this– if we go this way, through R two, we know that

VD minus VA is plus oh point nine volts. Now we are at A and we go through

the left side back to D. So we now have VA minus VD. That of course is now plus oh point one volts,

because remember, if VD minus VA is minus oh point one,

then VA minus VD is plus. And so we add them up and we find that

VD minus VD is plus one volts. Kirchoff said, has to be zero, because I’m

back at the same potential where I was before. Faraday says, uh-uh, I’m sorry,

you can’t do that. That one volt is exactly that EMF of one volt. That is the closed loop integral of

E dot dl around that loop. It’s no longer zero. And therefore,

whenever you define potential difference, if you do that in the way

of the integral of E dot dl, keep in mind that with non-conservative fields,

it depends on the path and that is very non-intuitive. And I’m going to demonstrate this now to you. I have a circuit which is exactly

what you have here. I have nine hundred ohms in a conducting

copper wire here and I have a hundred ohms here

and here is the solenoid. We can switch the current on in the solenoid

and get a blast of magnetic field coming up and the system is going to react by driving

a current in the direction that you see there. And I’d like to be even a little bit more

quantitative, so that you get a little bit

more for your money. The magnetic field takes a little bit of time

to reach the maximum value. In this course, we will be able to calculate

the time that it takes for the magnetic field to build up. We didn’t get to that yet,

so forget that part. It’s not so important. I just want you to appreciate the fact

that the magnetic field as a function of time will come up like this

and will then reach a maximum. It’s no longer changing. It’s constant, it’s a maximum value. It’s very high, seven, eight hundred Gauss

or so for this unit. We are not interested in a magnetic field. We are interested in the change

of the magnetic field, so the change of the magnetic field, dB dt,

is going to be something like this, that’s the derivative of this curve. And that is proportional with the induced

EMF and that’s in por– pro– proportional with the current,

through Ohm’s law. So if we now plot the voltage

as a function of– let me do that here, the voltage as a function of time,

then that voltmeter on the right side, I call that V two, will do this. This is V two, which is I times R two

at the maximum value. If those values were correct it would be oh

point nine volts and V one would go like this. V one equals minus I times R one. That gives me the minus oh point one volts. So the question now is what is the largest

value of dB dt that we can expect. We also have to know the surface area of the

solenoids so we can convert it to a flux change. Well, the change in magnetic field

is roughly at the fastest here is about hundred Gauss in one millisecond. Very roughly. So that would mean a field change, dB dt. That’s the maximum value possible only in

the beginning of about ten Tesla per second. And the surface area, which is that inner

circle there through which the flux is changing, the fact that my surface has to be attached

to that loop doesn’t change the magnetic flux. The magnetic flux is only determined, of course,

by that inner portion and so if the inner portion has an area

of say ten square centimeters, which is ten to the minus two square meters, then d phi dt will be approximately

ten times ten to the minus two, so that’s about oh point one and that’s volts. That’s EMF. I don’t care about the direction,

because I know Lenz’s Law. So you’re going to see an experiment which

is almost identical to what I have there, except all values are down

by a factor of ten. But that’s all. And you’re going to see

that demonstration there. And a few years ago, when I first did

this experiment in 26-100, there were several of my colleagues,

professors of both the physics department and EE department in my audience. And some did not believe what they saw. In fact, it was so bad that after my lecture

they came to me and some accused me for having cheated on the demonstration. This tells you something about them. [people in the audience chuckle] Imagine, professors in physics and professors

in electrical engineering department who did not believe what they were seeing. That tells you how non-intuitive this is. The simple fact that we had one voltmeter

connected to point D and A and another voltmeter

connected to the same point, they were unwilling to accept

that the two voltmeters read a totally different value. They were not used to non-conservative fields. Their brains couldn’t handle it. But that’s the way it is, and I’m going to

show this to you now. You’re going to see it there and when you

see this demonstration, it will be probably the only time in your life

that you will ever see this and I want you to remember this. You’re going to see something

that is very strange and I want you to tell

your grandchildren about it, that you have actually seen it

with your own eyes. You’re going to see there on the left side,

you’re going to see V one and on the right side you’re going to see V two. The vertical scale is such that very roughly

from here to here is about oh point one volts. And a horizontal unit is about five milliseconds and the whole voltage pulse lasts about

ten milliseconds, because from here to here is about

ten milliseconds. And the value that you expect for V two

will be nine times higher than V one and the polarities will be reversed. If you’re ready for this big moment

in your life– three, two, one, zero. Look on the left. There’s V one. Notice, it’s negative. Look on the right. There’s V two. It’s about nine times larger than V one. Don’t pay any attention to this wiggle. It has to do with the voltage that we apply,

which is not exactly flat. And notice that the whole pulse goes from

here to here, lasts about ten milliseconds. The moment that the magnetic field reaches

a maximum and remains constant, there is no longer any induced current. Think about this. Give this some thought. This is not easy. And have a good weekend.

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