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.
