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 happy 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 I 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 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.
My pleasure. Today, I'm going to cover with
you something that conceptually is the most difficult of all of
eight oh two. 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 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. 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. The 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 current -- 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 the 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 we 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
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,
sort of, 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 [break in tape] Oops, look what I did.
I forgot the DDT in front of the integral sign.
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 IDL. 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 through 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 a 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 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 the 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 solved 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
solenoids, 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,
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 though 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 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.
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,
fah- that 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.
Oh, my goodness, what did I do.
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.
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.
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 an 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 we 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 volts in a conducting copper wire here and
I have a hundred volts here and
here is the solenoid. W 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, just 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
fields 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 twenty-six one hundred,
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.
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 you
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 the 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
[inaudible]. 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.