WEBVTT

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Prof: Let's start.

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I told you about Ampere's law,
which is key to what we are

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going to do next,
so I'll remind you what the

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deal is.

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It says, if you have a closed
loop and around that loop you

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integrate the magnetic field
that is equal to

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μ_0 times all the
currents that penetrate this

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loop.

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So some current would be going
in, some current would be coming

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out.

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The guys who are outside the
loop don't contribute.

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Guys who are trapped inside
this, they make a contribution.

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That's Ampere's law.

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Now you have to know the
convention.

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The convention is that you do
the line integral

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counterclockwise,
and if your fingers are

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following the line integral,
your thumb points towards

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outside the board,
and that current is positive.

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Anything going into the board
therefore will be counted as

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negative in this one.

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So this has to be counted with
sign, do you understand?

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If the contour goes one way,
all currents coming out of the

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board are positive,
all those going in are

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negative.

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In particular,
if these 4 currents add up

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algebraically to 0,
one is 2 amp,

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another is -2 amp and so on,
then this integral will vanish,

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even though there are currents.

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So it counts not the absolute
value but the algebraic sum of

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all the currents.

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Now there's another subtlety I
didn't get into last time,

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which is the following -
remember, this is the current

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penetrating a surface,
bounded by this loop.

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You guys with me on what that
means?

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Now as long as I force you to
put this loop and to live in the

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plane of the blackboard,
there's only one surface with

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that loop as the boundary.

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But in real life,
we are doing stuff in three

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dimensions.

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Imagine pulling this loop out
of the blackboard.

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A lot of currents are coming
and going out.

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What is the surface of which
this loop is the boundary?

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The answer is not unique.

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There are many,
many surfaces that have the

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same loop as the boundary,
right?

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Take a piece of wire and wrap
it in the form of this boundary

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and dip it in soap,
and you pull it out,

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you'll get a soap filament.

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That soap filament has as its
rim, or the boundary,

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this loop.

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That may not even be planar.

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And if you blow on it,
it's going to bulge outside,

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right?

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Still this rim will be the
boundary of that bulging

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surface.

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So actually,
even if the surface is coming

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out of the blackboard,
the theorem is still valid.

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So let me draw another picture
that may be helpful.

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Suppose this is the surface and
you've got all these currents,

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some going in and some coming
out.

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You may have imagined all this
time I meant a flat planar

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surface.

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I don't.

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You can imagine now a curvy
surface, like maybe a

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hemisphere, of which this rim is
still the boundary.

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You can ask,
"How can that be?

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Where do you want me to count
the currents,

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here or on that surface?"

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The answer is,
it doesn't matter.

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It doesn't matter because any
current coming in here has to

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get out also through the other
one.

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The only--or it may come in and
decide to come back here,

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in which case it doesn't leave
the other one,

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but it will make 0 contribution
to this contour because it's

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coming in once and going out
once.

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Because currents don't
terminate, just like magnetic

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field lines, whatever crosses
this surface will also cross any

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surface with the same boundary.

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That's a fact that will become
important later on.

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So you follow that?

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The law, if you want to write
it formally,

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I'm going to write it as some
surface,

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and this is the surface
integral of the current density,

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which is the current per unit
area.

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And this loop is the boundary
of that surface,

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and I told you that the
boundary of any surface is

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written as partial S.

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This is the correct way to
write Ampere's law.

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And the point I'm making is,
this could be any surface with

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that as the boundary.

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Now once you've got Ampere's
law, I was trying to show you

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how you can use it to solve
certain problems that might have

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otherwise been difficult.

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Just like Gauss's law.

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Remember, Gauss's law was very
helpful when you took a solid

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ball of charge.

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The electric field of that is
usually hard to find,

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but from spherical symmetry,
you know the field is radially

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outwards.

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You draw a Gaussian surface and
you know the field's magnitude

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is constant on a sphere.

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How big is it?

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You get that from Gauss's law.

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So certain problems which are
highly symmetric,

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you can get using Ampere's law
also.

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So one example I did,
I want to repeat it,

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because one thing I said wasn't
right.

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There's this current going from
- to infinity,

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and we want to find the field
around it.

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If you do an honest
calculation, you will say,

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let me take a segment,
let me take a point,

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let me join the segment to that
point and take the cross product

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of the
dl with the R,

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etc.

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and so on, right?

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And it will give you a
contribution coming out of the

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blackboard, but then you have to
integrate it.

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Now you don't have to do all
that if you use symmetry.

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The symmetry argument would
tell you first of all that any

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field configuration you have
will be the same if you rotate

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it around a circle.

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Because if you look at the wire
end on and you turn the wire,

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it looks the same so the field
configuration should look the

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same.

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And I said there are only a
couple of things that have the

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property.

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Here's a good one.

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Maybe it does that.

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And the other one is this.

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Here is the wire.

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Notice if you turn the wire,
the wire looks the same,

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configuration looks the same.

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If you turn this wire around
its own axis,

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wire looks the same and the
configuration looks the same.

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Then I said we rule this guy
out, because magnetic field

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lines cannot come and end at a
point.

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Then I gave another
explanation, which is that if

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you reverse the current,
the field lines are supposed to

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go outwards,
but instead of reversing the

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current,
if I grab it and turn it by 180

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degrees around this axis,
then the lines will still go in

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but the current would be
reversed.

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So for whatever reason,
this is out and this is in.

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Then as to the direction of the
field,

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we also know that it has to be
perpendicular to the blackboard,

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because the two vectors whose
cross product determines it lie

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in the plane of the blackboard,
so the field has to come out.

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It has to come out and the
amount by which it comes out

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cannot change as you go around
the loop.

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And the answer can only depend
on how far you are from the

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wire.

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Then I write Ampere's law as
2ΠR times B at

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the radius r =
μ_0 times the

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current.

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In other words,
I look at the wire from the

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end, and I take any contour I
like of radius r.

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Then since B is
tangential to the circle,

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the line integral is simply
2Πr times B,

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and that's the current
enclosed, so you find B.

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If you like,
B_

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Φ,
meaning in the azimuthal

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direction.

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This angle is called phi.

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B_Φ
= μ_0

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I/2Πr.

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That's the result we got from
integration, but you can also

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get it from Ampere's law.

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Then I did another variation.

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I definitely don't want to
repeat that, which is,

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what if the wire is thick.

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Not a point wire but got finite
thickness with current coming

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out of the blackboard.

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What is going to be the
magnetic field?

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Then the answer depends on
whether you pick a contour like

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this or a contour inside the
wire.

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And if you do the two things,
you'll find the field actually

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grows linearly inside,
then falls like 1 over r

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outside.

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Anyway, this is stuff I did
near the end.

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But now I want to consider two
new problems which I didn't

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study last time.

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The first one is going to be
very important,

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is the solenoid.

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If you imagine on a tube you
wrap some wire,

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and the wire is wrapped like
this many, many times.

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And you want to find the
magnetic field.

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Now one loop we know produces a
field that goes like this.

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You stack them up,
you expect the field to go like

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that inside the coil,
then sort of loop around and

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come back outside the coil.

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What we want to find out is the
strength of the field inside the

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coil, and also outside.

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So the way you handle this
problem, you've got to take an

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Amperian loop that looks like
this.

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If this is the cardboard on
which you wound that thing,

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see what this wire looks like
when you slice it vertically.

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That's the tube on which you
wrapped it.

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I want you to slice it
vertically.

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On the left hand side,
I've current coming in.

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On the right hand side--coming
out of the board and right,

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it's going on,
you guys see that?

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Wrap it around the solenoid and
slice it down the middle,

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then on this end,
the current's coming out,

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on this end,
the current's going in.

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So I'm showing you the cross
section of the tube.

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It looks like this.

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So first thing I want to argue
is that the field outside does

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not vary with distance,
because if I took a loop like

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this--it's not a real loop;
it's a mathematical loop on

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which I'm going to use Ampere's
law.

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I know the field outside is
going to be coming down,

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but maybe it's weaker here and
stronger here and that's what

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I'm going to rule out.

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You know how we can rule that
out?

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What would go wrong if the
field was weaker here than here?

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Something to do with Ampere's
law.

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Well, let me explain.

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Suppose you go around this loop.

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Here B and dl are
perpendicular,

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so there's no contribution to
B⋅dl.

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Likewise here.

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Here you have a B going
one way and you have the same

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B but the path is going
the opposite way.

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Now these two have to cancel.

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These two have to cancel
because there is no current

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coming out of this.

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Because there's no current
coming out,

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this line integral has
vanished, therefore B

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times that length and B
times that length going the

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opposite way must be equal in
magnitude.

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That means the B here
must be as strong as the

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B there.

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So B doesn't vary when
you leave the wire.

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Well, I can keep on repeating
this argument until I go all the

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way to infinity,
but I know B is 0.

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Therefore coming from infinity
inwards, B is going to be

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0 everywhere here.

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That's only true for an
infinite solenoid.

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If you take a finite solenoid,
there will be lines coming in

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here.

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For an infinite solenoid,
is when the lines always go

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perpendicularly.

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See, in a finite solenoid,
you sort of know the lines will

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do this, right?

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They'll be perpendicular here,
but they could be at an angle

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there.

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But for an infinite solenoid,
every portion looks like you

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are here.

12:42.480 --> 12:45.360
The lines are only vertical,
they have no choice but to

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vanish.

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So the flux actually goes out
and returns on a sphere of

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infinity.

12:51.950 --> 12:56.180
So we only have to find the
magnetic field inside.

12:56.178 --> 13:00.388
So then you've got to do the
right hand rule and the right

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hand rule tells you the magnetic
field looks like that.

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And the question is,
how big is it?

13:09.590 --> 13:14.370
Once again, I want to argue
that the magnetic field is

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constant inside the solenoid.

13:17.168 --> 13:20.928
Now atleast now you should be
able to guess what argument I

13:20.927 --> 13:21.637
will use.

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The argument will be,
take an Amperian loop like this

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one.

13:28.350 --> 13:29.740
The two sides,
this and that,

13:29.740 --> 13:32.920
do not contribute because they
are perpendicular to the field.

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These two better make
cancelling contributions.

13:37.048 --> 13:38.978
And that's going up and that's
going down.

13:38.980 --> 13:41.350
For them to cancel,
the strength of the field here

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must be the same as the strength
here.

13:43.279 --> 13:46.979
So we have a result of the
magnetic field as some number

13:46.976 --> 13:51.206
inside and 0 outside and we're
trying to find the one number.

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This is going to be the case
whenever you use Ampere's law to

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get anything.

13:55.090 --> 13:59.730
In the end, Ampere's law is a
single statement about the

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magnetic field.

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You can get only one piece of
information from it.

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If you reduced your problem to
the point where there's only one

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thing you do not know,
you can find that one thing.

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Here I've talked my way out of
the field in this region and

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that region, and here it's
constant, and it's got a known

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direction.

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Only thing I don't know is,
how big is it?

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To find how big it is,
you can take a loop like this.

14:27.830 --> 14:31.150
Let me take a loop like this
one.

14:31.149 --> 14:32.609
Let me repeat that for you.

14:32.610 --> 14:40.960
So here is the current coming
out and I want to take a loop

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like this.

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I want the loop to have length
l.

14:50.389 --> 14:53.339
So the magnetic field is some
number B there which I

14:53.339 --> 14:53.999
don't know.

14:54.000 --> 14:58.040
It is 0 on the outside and I
never care about these sides

14:58.039 --> 15:00.349
because they're perpendicular.

15:00.350 --> 15:04.520
So the line integral will be
B times l.going

15:04.524 --> 15:08.114
up, nothing there,
0 times l.going down and

15:08.114 --> 15:09.364
nothing here.

15:09.360 --> 15:11.980
That's the whole line integral,
B⋅

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dl..

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That = μ_0 times
the current crossing the loop.

15:19.028 --> 15:22.538
Now here's where you should not
make the mistake of saying

15:22.544 --> 15:25.574
there's a current I
flowing in the wire and

15:25.567 --> 15:27.847
therefore it's equal to
I.

15:27.850 --> 15:35.320
Can you tell me what's wrong
with that logic?

15:35.320 --> 15:36.990
Yes?

15:36.990 --> 15:41.620
Student:  It's coiling
around so you have to count the

15:41.620 --> 15:43.010
number of coils.

15:43.009 --> 15:45.129
Prof: You understand
that?

15:45.129 --> 15:48.179
First of all,
if you just walked into the

15:48.178 --> 15:51.608
class now, first of all,
you're in trouble.

15:51.610 --> 15:54.040
Secondly, if you just walked
into the class and you saw this

15:54.035 --> 15:55.965
picture and you knew all about
Ampere's law,

15:55.970 --> 15:57.220
did not know anything about
solenoid,

15:57.220 --> 15:59.320
nothing, what will you do?

15:59.320 --> 16:01.910
You'll say, "Here's a
loop, current's coming out.

16:01.908 --> 16:03.798
I just count the current coming
in."

16:03.798 --> 16:06.928
The fact that these guys all
belong to a big solenoid and

16:06.932 --> 16:08.782
they wrap around is irrelevant.

16:08.778 --> 16:12.008
All that matters for the
Ampere's loop is how much amps

16:12.014 --> 16:14.474
are coming at me or going away
from me.

16:14.470 --> 16:17.060
So the fact that they're all
part of one gigantic loop is

16:17.057 --> 16:17.657
irrelevant.

16:17.658 --> 16:21.098
So you've got to count the
number of turns that got trapped

16:21.097 --> 16:22.637
in this length l.

16:22.639 --> 16:28.749
So we use the symbol small
n, which is the number of

16:28.750 --> 16:31.280
turns per unit length.

16:31.278 --> 16:36.678
Then it's going to be number of
turns per unit length times

16:36.683 --> 16:37.713
l.

16:37.710 --> 16:40.140
So that depends on how tightly
you wound the coil.

16:40.139 --> 16:43.839
If you wound 100 turns per
centimeter, well,

16:43.844 --> 16:47.294
it's whatever,
10,000 turns per meter.

16:47.288 --> 16:50.878
Now one nice thing is that this
l cancels.

16:50.879 --> 16:56.249
Again, I want you to think
about why l should

16:56.250 --> 16:57.200
cancel.

16:57.200 --> 17:04.160
Why shouldn't the answer depend
on l?

17:04.160 --> 17:07.750
This happens even when you do
Gauss's law, when you take some

17:07.749 --> 17:11.099
surface, the details of the
surface cancel out always.

17:11.099 --> 17:11.709
Yes?

17:11.710 --> 17:19.080
Student:
>

17:19.078 --> 17:20.838
Prof: More turns of the
wire.

17:20.838 --> 17:24.718
But the more important thing
is, the question that was given

17:24.724 --> 17:26.244
to us was a solenoid.

17:26.240 --> 17:29.930
It had some number of turns.

17:29.930 --> 17:31.800
It had a certain current.

17:31.798 --> 17:34.918
And the answer should only
depend on the current and how

17:34.921 --> 17:36.171
densely it is wound.

17:36.170 --> 17:37.130
This l.

17:37.134 --> 17:38.584
was my own artifact.

17:38.578 --> 17:41.858
The Amperian loop is not a real
loop.

17:41.859 --> 17:43.129
I made it up.

17:43.130 --> 17:45.540
So the answer I get for
B cannot depend on

17:45.538 --> 17:48.198
whether I took that loop or that
loop or that loop.

17:48.200 --> 17:49.500
They should all give the same
answer.

17:49.500 --> 17:53.110
That's why the properties of
the loop should cancel out and

17:53.114 --> 17:56.734
you should get in the end a
result that depends only on the

17:56.729 --> 17:59.159
intrinsic variables in the
problem.

17:59.160 --> 18:02.380
So this is a very useful result.

18:02.380 --> 18:04.820
That's why we draw a box around
it.

18:04.819 --> 18:06.409
That means important.

18:06.410 --> 18:10.890
The field inside a solenoid is
μ_0 times the

18:10.890 --> 18:15.690
number of turns per unit length,
times the current flowing to

18:15.690 --> 18:17.050
the solenoid.

18:17.048 --> 18:19.238
I'll be invoking this result
all the time.

18:19.240 --> 18:21.370
It's one of the few results I
carry in my head.

18:21.368 --> 18:25.418
Normally I encourage you not to
carry too much stuff in your

18:25.420 --> 18:28.030
head, but this guy's pretty
helpful.

18:28.028 --> 18:30.918
So here's another solenoid that
people have, which is better

18:30.919 --> 18:32.829
than this one in the following
sense.

18:32.828 --> 18:36.498
As long as you make a finite
solenoid--in real life,

18:36.496 --> 18:40.446
everything is finite--you will
have this return flux.

18:40.450 --> 18:44.780
There's a trick by which you
can keep all the flux hidden

18:44.779 --> 18:48.099
inside the tube,
by wrapping the tube into a

18:48.104 --> 18:49.114
doughnut.

18:49.108 --> 18:54.608
So take the long tube and glue
the ends together,

18:54.612 --> 19:01.612
then you have a solenoid which
is like a doughnut on which you

19:01.605 --> 19:03.665
wrap the wire.

19:03.670 --> 19:05.280
I've shown you a few turns.

19:05.278 --> 19:08.938
So you wrap it all the way
round and you come back and you

19:08.942 --> 19:10.872
drive the current from here.

19:10.869 --> 19:11.579
Are you guys with me?

19:11.578 --> 19:15.398
The cross section of the
doughnut can be circular,

19:15.398 --> 19:16.878
it can be square.

19:16.880 --> 19:18.340
It doesn't matter what it is.

19:18.338 --> 19:20.608
Sometimes it even contains some
metal or iron,

19:20.607 --> 19:22.017
which I'm not showing you.

19:22.019 --> 19:22.599
It's fine.

19:22.602 --> 19:25.922
Even if it's an empty tube,
you wrap the wire around it.

19:25.920 --> 19:34.360
But now the flux lines go like
this along the tube.

19:34.358 --> 19:39.158
So let me take a problem where
the cross section is actually a

19:39.164 --> 19:42.714
square, in other words,
it looks like this.

19:42.710 --> 19:48.840
I cut the doughnut right there
and I slice it and I look at it,

19:48.843 --> 19:52.903
and you have the flux lines
doing that.

19:52.900 --> 19:54.620
That's the cross section.

19:54.619 --> 19:55.939
So which way is the flux going?

19:55.940 --> 19:58.010
Well, you've got to figure out
again with the right hand rule,

19:58.009 --> 20:01.989
the way I've shown the
currents, if you do this around

20:01.989 --> 20:06.409
the wire,
the flux is actually going like

20:06.410 --> 20:07.660
that here.

20:07.660 --> 20:12.940
Our job is to find out what is
the magnetic field inside that

20:12.941 --> 20:13.911
doughnut.

20:13.910 --> 20:18.470
For that, I'm going to slice
the doughnut the way you

20:18.465 --> 20:20.915
normally slice a doughnut.

20:20.920 --> 20:25.220
You normally slice a doughnut
so it looks like this.

20:25.220 --> 20:28.550
This is half the doughnut, okay?

20:28.548 --> 20:32.718
You cut it, then let's say the
wire looks like that one.

20:32.720 --> 20:36.910
That guy is coming in,
coming out of the board,

20:36.911 --> 20:41.011
going into the board,
coming out of the board,

20:41.010 --> 20:43.290
going into the board.

20:43.289 --> 20:44.749
Yes? Can you see that?

20:44.750 --> 20:48.160
If you take a doughnut with a
wire wrapped around it and you

20:48.163 --> 20:50.423
cut it, it's going to look like
this.

20:50.420 --> 20:54.180
Now the important thing to
notice is that we haven't really

20:54.182 --> 20:55.482
cut that doughnut.

20:55.480 --> 20:57.540
If you really cut the doughnut,
there's going to be no current.

20:57.539 --> 21:00.519
This is a mental slice you make.

21:00.519 --> 21:05.589
Then you ask yourself,
now what do you think is going

21:05.586 --> 21:08.116
to be the Amperian loop?

21:08.118 --> 21:13.328
Can you make a guess what the
loop will look like?

21:13.329 --> 21:14.139
Yes?

21:14.140 --> 21:16.680
Student:  It will look
like a piece of pie.

21:16.680 --> 21:17.810
Prof: A piece of pie.

21:17.809 --> 21:18.949
We're all getting hungry now.

21:18.950 --> 21:20.730
I've talked about doughnuts.

21:20.730 --> 21:21.670
Which pie?

21:21.670 --> 21:23.940
Student:  It will have
it so that the sides

21:23.941 --> 21:26.731
perpendicular to the--
Prof: Oh, okay.

21:26.730 --> 21:31.970
Maybe you could do that,
but here is what the general

21:31.973 --> 21:37.323
consensus, 9 out of 10
physicists use this contour.

21:37.318 --> 21:41.948
This is going to be the
Amperian loop.

21:41.950 --> 21:44.220
Then what can you say?

21:44.220 --> 21:47.120
The loop has some radius
r,

21:47.118 --> 21:51.038
so once again,
2Πr times B is

21:51.039 --> 21:56.269
μ_0I times
the number of turns altogether

21:56.267 --> 21:58.007
in the solenoid.

21:58.009 --> 21:59.629
You understand?

21:59.630 --> 22:06.150
Because this loop cuts through
every turn of wire.

22:06.150 --> 22:10.450
As far as the direction is
concerned, it will go like this.

22:10.450 --> 22:16.420
The field will go this way.

22:16.420 --> 22:23.090
So the magnetic field here is
μ_0IN

22:23.086 --> 22:26.416
divided by 2Πr.

22:26.420 --> 22:29.980
So the field actually gets
weaker, so that if you cut this

22:29.980 --> 22:33.790
doughnut, the field on the inner
rim will be stronger than the

22:33.791 --> 22:35.481
field on the outer rim.

22:35.480 --> 22:38.450
And if you go inside the inner
rim, you'll get 0;

22:38.450 --> 22:42.070
outside the outer rim,
you'll get 0 because any loop

22:42.069 --> 22:46.049
you draw on the contour will
intersect an equal number of

22:46.045 --> 22:48.525
incoming and outgoing currents.

22:48.529 --> 22:50.739
So B is going to vanish
outside the doughnut,

22:50.740 --> 22:54.190
and it's going to be non-zero
right inside the heart of the

22:54.185 --> 22:59.715
doughnut,
and the strength will be this.

22:59.720 --> 23:03.250
Now if this doughnut had a
radius of say 1 light year,

23:03.252 --> 23:06.322
you can ask what will happen to
the formula.

23:06.318 --> 23:09.678
You should be ready for what
you expect.

23:09.680 --> 23:13.910
Take a doughnut 10 miles long,
10 miles in circumference.

23:13.910 --> 23:17.020
What do you hope the formula to
give?

23:17.019 --> 23:17.789
Yes.

23:17.788 --> 23:19.838
Student:  It should
approach the results of the

23:19.836 --> 23:20.606
infinite solenoid.

23:20.608 --> 23:21.598
Prof: For infinite
solenoid,

23:21.598 --> 23:24.818
because the circle of radius 10
zillion miles,

23:24.818 --> 23:26.938
for most purposes,
it's like a straight tube,

23:26.942 --> 23:27.332
right?

23:27.328 --> 23:29.588
You cannot tell it's going
around in a circle.

23:29.589 --> 23:30.659
Let's see if that happens.

23:30.660 --> 23:34.920
Well, you can see the field is
μ_0NI,

23:34.915 --> 23:39.165
divided by 2Πr is
just the circumference of the

23:39.173 --> 23:40.013
circle.

23:40.009 --> 23:42.659
So it's going to be
μ_0 little n

23:42.663 --> 23:48.823
times I,
because little n is big

23:48.818 --> 23:54.908
N divided by
2Πr.

23:54.910 --> 23:55.400
See that?

23:55.400 --> 23:57.210
This 2Πr will-- in
other words,

23:57.210 --> 23:59.660
you cut it here and you open it
into a solenoid,

23:59.660 --> 24:01.410
its length will be
2Πr,

24:01.410 --> 24:04.340
and the number of turns per
unit length will be big N

24:04.336 --> 24:05.376
over 2Πr.

24:05.380 --> 24:08.390
But that's an approximation,
because in a real finite

24:08.385 --> 24:10.925
system, the magnetic field is
not uniform.

24:10.930 --> 24:14.960
There's a slight variation from
this end to that end.

24:14.960 --> 24:18.630
But if this thickness is 2
inches and that radius is 37

24:18.625 --> 24:22.285
miles, that variation is
negligible and it becomes that

24:22.289 --> 24:23.919
other easier result.

24:23.920 --> 24:30.150
Student:  How did you
get the small n again?

24:30.150 --> 24:35.830
Prof: We got the small n
because this guy.

24:35.828 --> 24:38.978
This is the total number of
turns wrapped around the

24:38.983 --> 24:41.893
solenoid and 2Πr is
the circumference,

24:41.891 --> 24:42.511
right?

24:42.509 --> 24:47.079
So number of turns divided by
the circumference will be the

24:47.079 --> 24:49.759
number of turns per unit length.

24:49.759 --> 24:51.819
In other words,
if it was a huge solenoid,

24:51.818 --> 24:54.838
so huge that it looked like a
straight line to you from your

24:54.840 --> 24:57.530
range of observation,
the number of turns it would

24:57.528 --> 25:00.078
have per unit length would be
exactly this much.

25:00.078 --> 25:04.268
So it will look like a straight
solenoid to you,

25:04.271 --> 25:07.931
but it's curving around on a
big scale.

25:07.930 --> 25:13.640
All right, so this marks the
end of one topic.

25:13.640 --> 25:17.050
So I'm going to write down over
and over what we know,

25:17.048 --> 25:21.558
because I think I talked to
some of you people,

25:21.558 --> 25:24.408
and I asked you,
what's the part you find

25:24.413 --> 25:25.273
difficult.

25:25.269 --> 25:28.539
And one thing I heard,
which is very reasonable,

25:28.536 --> 25:32.076
is the amount of stuff you're
learning every day.

25:32.079 --> 25:34.539
This is just a lot of stuff.

25:34.538 --> 25:37.488
"Drinking out of the fire
hose" was an expression

25:37.487 --> 25:38.247
that came up.

25:38.250 --> 25:41.450
You drink and you drink and you
drink and I keep throwing stuff

25:41.445 --> 25:41.905
at you.

25:41.910 --> 25:44.060
I would like it to end.

25:44.058 --> 25:46.808
It's almost going to end,
but it's not over yet.

25:46.808 --> 25:49.448
But I'll tell you where things
stand now.

25:49.450 --> 25:53.800
All of electrostatics and
magnetostatics are summarized by

25:53.797 --> 25:55.777
the following equations.

25:55.779 --> 25:59.019
I'm going to write them again
and again.

25:59.019 --> 26:03.519
E⋅dA
is the charge inside.

26:03.519 --> 26:05.019
Line integral of
E⋅

26:05.021 --> 26:06.071
dl.

26:06.068 --> 26:09.878
or dr--I forgot what I
called it--that is 0.

26:09.880 --> 26:12.860
Surface integral of the
magnetic field is 0 because

26:12.861 --> 26:14.771
there are no magnetic charges.

26:14.769 --> 26:20.179
The line integral of the
magnetic field is mu 0 times all

26:20.180 --> 26:22.210
the current inside.

26:22.210 --> 26:23.690
That's it.

26:23.690 --> 26:26.800
And the force--after all,
who cares about E and

26:26.804 --> 26:29.124
B,
except for this great equation,

26:29.124 --> 26:31.884
which tells you that if a
charged particle goes into a

26:31.884 --> 26:34.234
region with electric and
magnetic fields,

26:34.230 --> 26:36.150
this will be the force acting
on it.

26:36.150 --> 26:38.080
That's very important.

26:38.078 --> 26:40.808
That's why we care about
E and B,

26:40.805 --> 26:42.815
because they make things
happen.

26:42.818 --> 26:46.418
So the cycle of physics will
proceed as follows - this is all

26:46.420 --> 26:47.260
for statics.

26:47.259 --> 26:57.729
That means J is constant
and rho is constant.

26:57.730 --> 27:00.410
So somebody has to give you the
currents and somebody has to

27:00.405 --> 27:01.445
give you the charges.

27:01.450 --> 27:04.050
Then it's a purely mathematical
problem to solve for the

27:04.051 --> 27:05.521
electric and magnetic fields.

27:05.519 --> 27:09.159
And I was telling you yesterday
a fact that may not be obvious,

27:09.160 --> 27:13.340
that these equations suffice to
determine electric and magnetic

27:13.336 --> 27:15.606
fields,
given the cause of electric and

27:15.609 --> 27:20.689
magnetic fields,
which is currents and charges.

27:20.690 --> 27:30.570
Okay, so now we are going to
do--let me ask you something.

27:30.568 --> 27:33.488
When do we keep on modifying
the equations?

27:33.490 --> 27:35.310
I think I explained that to you.

27:35.309 --> 27:38.519
Why don't we stop now?

27:38.519 --> 27:42.509
Any idea why I'm going to
modify the stuff?

27:42.509 --> 27:44.789
Do you know?

27:44.788 --> 27:49.888
Do you know when we modify
equations?

27:49.890 --> 27:51.580
What could be wrong with this
equation?

27:51.579 --> 27:55.459
Who can convince me it's wrong?

27:55.460 --> 27:59.970
Any idea?

27:59.970 --> 28:03.100
I'm not talking about a
particular thing,

28:03.101 --> 28:07.561
but I'm talking in general,
what makes people modify their

28:07.563 --> 28:08.663
equations.

28:08.660 --> 28:09.440
Pardon me?

28:09.440 --> 28:10.420
Student:  New experiment.

28:10.420 --> 28:11.390
Prof: New experiment.

28:11.390 --> 28:15.170
That's the only reason you
modify your results.

28:15.170 --> 28:17.810
If these things worked,
and it could have been a very

28:17.807 --> 28:20.797
consistent world in which it
worked, but it is not the whole

28:20.801 --> 28:21.311
story.

28:21.308 --> 28:24.078
At least one reason it's not
the whole story is,

28:24.083 --> 28:26.623
rho and J are not always
constant.

28:26.618 --> 28:29.798
You can have charges shaking
around and you can have

28:29.803 --> 28:30.493
currents.

28:30.490 --> 28:33.480
You turn them on and off as you
wish.

28:33.480 --> 28:37.260
One thing is to hope that this
continues to describe them even

28:37.259 --> 28:38.499
in this situation.

28:38.500 --> 28:41.180
The fact that we derive them
from steady currents and

28:41.178 --> 28:44.468
densities doesn't mean it has to
fail when they are changing with

28:44.473 --> 28:47.443
time,
but there is no guarantee that

28:47.438 --> 28:48.988
they won't change.

28:48.990 --> 28:54.000
So I'm going to now lead you
through the experiments that

28:53.997 --> 28:58.107
forced us to change some of
these equations.

28:58.108 --> 29:01.538
So these are called Maxwell's
equations, but they're not the

29:01.535 --> 29:04.545
real Maxwell equations,
because they're not valid for

29:04.554 --> 29:06.184
the time dependent case.

29:06.180 --> 29:10.100
I'm going to start the
phenomenon, then we'll see where

29:10.096 --> 29:11.616
the trouble begins.

29:11.618 --> 29:21.608
So let's take a problem where I
have a uniform magnetic field

29:21.609 --> 29:25.439
going into the board.

29:25.440 --> 29:29.170
This goes on forever,
but it doesn't go beyond this

29:29.172 --> 29:29.772
point.

29:29.769 --> 29:33.289
I've only shown you that much.

29:33.288 --> 29:41.288
Now I'm going to take a loop of
wire like this.

29:41.288 --> 29:48.248
It's got some width w and maybe
some length l..

29:48.250 --> 29:51.550
And I put a little light bulb
here.

29:51.548 --> 29:54.848
Then I drag this to the right
with some speed V. And

29:54.847 --> 30:02.447
this magnetic field
B going into the

30:02.448 --> 30:05.758
blackboard.

30:05.759 --> 30:08.009
Now if the loop is not moving,
nothing will happen.

30:08.009 --> 30:13.309
If the loop is moving,
something does happen.

30:13.308 --> 30:16.798
Now I'm going to ask you what
happens, because this is

30:16.798 --> 30:18.508
something you must know.

30:18.509 --> 30:19.749
What happens when I drag the
loop?

30:19.750 --> 30:20.330
Yes?

30:20.328 --> 30:20.948
Student:  The light bulb
will go on.

30:20.950 --> 30:21.450
Prof: Good.

30:21.450 --> 30:24.800
The light bulb will go on is
the answer that you've got to

30:24.798 --> 30:25.208
give.

30:25.210 --> 30:27.220
You may not know by how much
and whatnot.

30:27.220 --> 30:29.680
But first of all,
why would I draw a light bulb

30:29.679 --> 30:31.229
if it's not going to go on?

30:31.230 --> 30:32.220
That's how you should reason.

30:32.220 --> 30:34.190
That's how my kids did all
their SATs.

30:34.190 --> 30:36.370
They asked extraneous questions.

30:36.368 --> 30:39.838
One of them is,
you put the light bulb because

30:39.839 --> 30:41.459
it's going to glow.

30:41.460 --> 30:43.810
So you can ask yourself,
what's going on?

30:43.809 --> 30:45.309
Why is the light bulb glowing?

30:45.309 --> 30:47.289
Is it some new physics?

30:47.288 --> 30:50.598
The answer is,
whenever youwant a light bulb

30:50.595 --> 30:53.665
to glow, you're looking for a
battery.

30:53.670 --> 30:56.470
There is no battery in the
circuit.

30:56.470 --> 31:00.080
And yet there is an emf,
because every time a charge

31:00.078 --> 31:02.908
makes one full turn and comes
around,

31:02.910 --> 31:06.260
it has delivered some work,
because that's what makes this

31:06.263 --> 31:07.033
thing glow.

31:07.028 --> 31:09.858
You're constantly pumping
energy into the resistor inside

31:09.862 --> 31:10.472
your bulb.

31:10.470 --> 31:11.720
And who's providing the energy?

31:11.720 --> 31:16.410
These charges are going round
and round doing some work.

31:16.410 --> 31:21.400
That means something is pushing
them around this loop.

31:21.400 --> 31:26.370
We defined the emf to be the
line integral of any force per

31:26.373 --> 31:30.063
unit charge pushing them around
the loop.

31:30.059 --> 31:33.389
This is per unit charge.

31:33.390 --> 31:40.450
That's got to be not 0.

31:40.450 --> 31:45.290
So what is that force,
and what is the emf is what I

31:45.290 --> 31:46.620
want to ask.

31:46.618 --> 31:53.708
Do you know what the force
might be?

31:53.710 --> 31:55.970
And why does it kick in only
when I move the loop?

31:55.970 --> 31:56.870
Yes?

31:56.868 --> 31:59.788
Student:  The force is
the magnetic force that we

31:59.785 --> 32:00.205
wrote.

32:00.210 --> 32:01.630
Prof: The v x
B force.

32:01.630 --> 32:03.400
That's because there's a
v here,

32:03.396 --> 32:04.836
B into the blackboard.

32:04.838 --> 32:08.518
v x B looks like
that.

32:08.519 --> 32:12.069
The force is v x
B and it acts only here.

32:12.069 --> 32:16.129
Let's understand why.

32:16.130 --> 32:20.280
If you come to this portion,
if you take this v x

32:20.282 --> 32:22.392
B,
this v is this way,

32:22.393 --> 32:24.773
and B is into the board,
v x B is

32:24.772 --> 32:25.822
perpendicular to the wire.

32:25.818 --> 32:29.348
But charges cannot go--even if
there's a force perpendicular to

32:29.347 --> 32:32.707
the wire, it doesn't contribute
to the emf, which I found by

32:32.707 --> 32:34.127
going along the loop.

32:34.130 --> 32:36.190
So dr is along the loop.

32:36.190 --> 32:39.150
The only way to get that to
contribute is to have a force

32:39.151 --> 32:40.051
along the loop.

32:40.048 --> 32:43.368
That's only here and it's not
here.

32:43.368 --> 32:47.348
So the whole force is non-zero
only in that segment.

32:47.348 --> 32:52.968
So its line integral is equal
to simply the force,

32:52.970 --> 32:57.100
which is Bv times
w.

32:57.099 --> 33:00.919
That's the emf.

33:00.920 --> 33:05.250
Any time a charge goes around
the loop once,

33:05.247 --> 33:08.867
that amount of work is being
done.

33:08.869 --> 33:10.579
Okay, so that explains it.

33:10.578 --> 33:11.968
We're not in any kind of
trouble.

33:11.970 --> 33:15.350
In fact, we understand this
problem completely without

33:15.346 --> 33:17.126
bringing in any new stuff.

33:17.130 --> 33:19.890
But there's one paradox.

33:19.890 --> 33:23.560
We were told the magnetic field
doesn't do any work.

33:23.559 --> 33:25.279
You remember that?

33:25.278 --> 33:30.318
And yet here is a magnetic
field pushing these charges and

33:30.316 --> 33:32.256
getting things done.

33:32.259 --> 33:35.369
So what can be going on there?

33:35.368 --> 33:41.278
Student:  It takes work
to move the whole loop.

33:41.279 --> 33:42.539
Prof: It takes work to
move the loop.

33:42.538 --> 33:45.578
That is certainly true,
but how about the fact that the

33:45.580 --> 33:47.890
magnetic field manages to do
some work?

33:47.890 --> 33:54.560
Is that a problem for you or
not?

33:54.558 --> 33:57.698
Remember, the original argument
for why it doesn't do any work

33:57.701 --> 34:00.851
is, the force is
v x B and the

34:00.854 --> 34:04.724
power is the velocity times the
force and it's 0,

34:04.718 --> 34:09.548
because v x B is
perpendicular to v.

34:09.550 --> 34:12.100
But here, v x B
is this way, and the charge is

34:12.101 --> 34:12.881
moving this way.

34:12.880 --> 34:19.670
It looks like we got some work
done.

34:19.670 --> 34:24.480
Okay, so the answer is that
this is not the full--there are

34:24.481 --> 34:26.641
two kinds of velocities.

34:26.639 --> 34:31.059
If you have a current in the
wire, the charges are also

34:31.059 --> 34:34.659
moving in the wire at some speed
u.

34:34.659 --> 34:38.269
So the real velocity has got
the velocity because the whole

34:38.268 --> 34:40.508
loop is moving,
v, and the fact the

34:40.510 --> 34:43.250
current is moving this way,
u, and w,

34:43.251 --> 34:47.271
if you like,
is the true velocity.

34:47.268 --> 34:49.838
And v x B is
actually in that direction.

34:49.840 --> 34:52.120
That is v x B,
perpendicular to w.

34:52.119 --> 34:54.009
So let me draw the picture for
you.

34:54.010 --> 35:03.430
w and w x
B.

35:03.429 --> 35:04.659
w is made up of two
parts.

35:04.659 --> 35:12.039
It's got a part v here,
it's got a part u here.

35:12.039 --> 35:13.989
So the magnetic force is not
just along the wire.

35:13.989 --> 35:17.549
It's really at this strange
angle.

35:17.550 --> 35:21.750
And we can easily find out how
much of the force is this part

35:21.750 --> 35:24.620
and how much of the force is
this part.

35:24.619 --> 35:33.169
The force here will be--let me
see.

35:33.170 --> 35:37.730
It will depend on the other
component, so it will depend

35:37.733 --> 35:42.463
on--yes, the force here is
B times v and the

35:42.461 --> 35:45.201
power is that times u.

35:45.199 --> 35:49.599
The power is Bvu,
due to this portion.

35:49.599 --> 35:52.869
But in this side,
if you ask what's the power

35:52.873 --> 35:57.113
delivered by that component,
it is B times u

35:57.112 --> 35:59.942
times v,
but with a - sign.

35:59.940 --> 36:03.660
In other words,
if you resolve this w x

36:03.657 --> 36:08.457
B into two parts,
the vertical force is due to

36:08.456 --> 36:12.576
the horizontal velocity,
and the power is that times the

36:12.583 --> 36:14.163
vertical velocity u.

36:14.159 --> 36:16.459
The horizontal force is due to
the vertical velocity.

36:16.460 --> 36:18.160
That's how the cross product
works.

36:18.159 --> 36:20.019
So when you're done,
both are equal to B

36:20.023 --> 36:21.323
times u times v.

36:21.320 --> 36:23.770
But this is and this is -.

36:23.768 --> 36:26.238
In other words,
in the end, it would like to

36:26.244 --> 36:29.124
move the charge this way,
because that's the force,

36:29.121 --> 36:32.231
but it manages to move it along
the wire this way.

36:32.230 --> 36:36.840
It would like to move it this
way, but it's force to go

36:36.840 --> 36:37.950
oppositely.

36:37.949 --> 36:41.609
So a force trying to move this
way, this part it is doing work,

36:41.612 --> 36:43.802
but this part it is not doing
work.

36:43.800 --> 36:48.040
It has work done on it,
because we forced the electrons

36:48.041 --> 36:49.851
to move to the right.

36:49.849 --> 36:54.109
Of course, that energy comes
from the person pulling the rod,

36:54.110 --> 36:57.180
pulling this loop,
because every charge that has

36:57.175 --> 36:59.495
this force,
it won't just move unless you

36:59.501 --> 37:01.501
pull it,
and the force you apply is

37:01.498 --> 37:04.808
really coming from you that
balances the magnetic force.

37:04.809 --> 37:06.569
So there's yet another force.

37:06.570 --> 37:12.080
That's the force due to me,
and I'm pulling this to balance

37:12.076 --> 37:14.446
this part of the force.

37:14.449 --> 37:19.679
So what is the role of the
magnetic field?

37:19.679 --> 37:23.459
It doesn't do any work,
but you need it here,

37:23.463 --> 37:28.283
because one component of the
magnetic field does work.

37:28.280 --> 37:31.110
The other component has work
done on it.

37:31.110 --> 37:34.750
So it takes with one hand and
gives with the other hand,

37:34.746 --> 37:36.066
but you need that.

37:36.070 --> 37:38.950
It's no used saying,
"I'm not very impressed

37:38.954 --> 37:41.364
because I did all the hard
work."

37:41.360 --> 37:45.440
You did all the hard work by
grabbing this loop and pulling

37:45.438 --> 37:47.208
it,
but I challenge you to go and

37:47.213 --> 37:50.113
push the individual electrons so
they can move down the wire.

37:50.110 --> 37:52.730
You cannot even see those
little guys, right?

37:52.730 --> 37:55.630
But the magnetic field does
that for you.

37:55.630 --> 37:59.240
So it takes macroscopic
mechanical power,

37:59.237 --> 38:04.467
turns into microscopic energy
supplied to the electrons.

38:04.469 --> 38:06.909
Even though the field did not
profit in the end,

38:06.909 --> 38:09.669
it just delivered to the
electrons what you gave them,

38:09.670 --> 38:13.550
you need that field to do a
transfer from something you can

38:13.552 --> 38:16.502
see and pull to something you
cannot see,

38:16.500 --> 38:17.910
right?

38:17.909 --> 38:19.899
I mean, I give you a light bulb
and you've got all the muscles

38:19.900 --> 38:20.390
in your hand.

38:20.389 --> 38:22.669
Let's see you make it glow.

38:22.670 --> 38:24.720
You cannot say,
"I have the energy."

38:24.719 --> 38:26.639
There's no way to transfer it.

38:26.639 --> 38:29.949
That's the role of the magnetic
field, so you shouldn't say it's

38:29.947 --> 38:30.417
no use.

38:30.420 --> 38:32.710
Anyway, we got that answer.

38:32.710 --> 38:37.810
I'm going to give you another
equivalent way to explain the

38:37.811 --> 38:39.571
balance of energy.

38:39.570 --> 38:41.440
We all know that the energy is
coming,

38:41.440 --> 38:44.320
because once the loop carries a
current,

38:44.320 --> 38:47.010
it's going to be hard for you
to pull the loop,

38:47.010 --> 38:50.220
so you've got to do some work,
and that will explain the power

38:50.224 --> 38:50.914
usage here.

38:50.909 --> 38:54.379
So let's calculate the power
two ways.

38:54.380 --> 39:00.070
The power in the resistor = emf
squared over R.

39:00.070 --> 39:13.910
emf squared is
B^(2)v^(2)W^(2)/R.

39:13.909 --> 39:19.009
Now how about the power that I
provide?

39:19.010 --> 39:20.720
What's the power I provide?

39:20.719 --> 39:23.389
I'm pulling this leg of the
wire here.

39:23.389 --> 39:25.869
The force on anything,
you remember,

39:25.873 --> 39:28.713
is BwI,
and I pull it with a force

39:28.711 --> 39:31.481
v, with a velocity
v.

39:31.480 --> 39:35.370
That's the power.

39:35.369 --> 39:38.759
So it is Bwv times
I.

39:38.760 --> 39:41.750
I = emf divided by
R.

39:41.750 --> 39:48.420
But if you put the emf that we
just got, you will get this.

39:48.420 --> 39:50.880
So they will match.

39:50.880 --> 39:54.200
So do you understand how this
thing works?

39:54.199 --> 39:56.009
This is a generator,
if you like.

39:56.010 --> 39:58.880
If you want to light the bulb
in your house,

39:58.880 --> 40:01.680
one option is to set up a
magnetic field,

40:01.679 --> 40:04.719
take the light bulb and connect
it to a rectangle,

40:04.719 --> 40:06.919
and grab it and keep running.

40:06.920 --> 40:11.770
As long as you're running,
the light bulb will be glowing.

40:11.768 --> 40:12.928
When do you think it will stop
glowing?

40:12.929 --> 40:13.659
Yes?

40:13.659 --> 40:16.309
Student:  I'm sorry,
what does that word say?

40:16.309 --> 40:18.119
Prof: This one?

40:18.119 --> 40:28.989
The power that I provide.

40:28.989 --> 40:32.439
Okay, I'm just balancing the
power loss in the resistor.

40:32.440 --> 40:33.420
Are you with me now?

40:33.420 --> 40:35.410
You understand the two things?

40:35.409 --> 40:38.549
I take the force I apply on the
wire, multiply by the velocity

40:38.550 --> 40:39.580
with which I pull.

40:39.579 --> 40:40.639
That's the power.

40:40.639 --> 40:42.219
Likewise I take I^(2)R.

40:42.219 --> 40:44.059
That's the electric power
dissipated.

40:44.059 --> 40:47.519
And of course, they balance.

40:47.518 --> 40:54.158
Notice that once the loop is
fully inside the magnetic field,

40:54.157 --> 40:58.027
the light bulb will stop
glowing.

40:58.030 --> 40:59.400
Do you know why?

40:59.400 --> 41:00.180
Can you see why?

41:00.179 --> 41:00.759
Yes?

41:00.760 --> 41:02.720
Student:  Because the
flux in the loop isn't changing

41:02.715 --> 41:03.075
any more.

41:03.079 --> 41:03.889
Prof: No, no.

41:03.889 --> 41:07.039
You're pulling new laws,
in terms of what I've told you.

41:07.039 --> 41:10.709
Student:  The magnetic
field, there's no new field

41:10.713 --> 41:12.423
coming through the wire.

41:12.420 --> 41:15.630
The field is constant that's
coming through the wire.

41:15.630 --> 41:17.370
Prof: Just use what
we've used so far.

41:17.369 --> 41:19.189
Yes
Student:  The field is

41:19.193 --> 41:20.033
acting on the left segment.

41:20.030 --> 41:20.680
Prof: Right.

41:20.679 --> 41:23.349
In other words,
without invoking any new ideas,

41:23.351 --> 41:26.781
once this piece gets into the
loop, it's going to have a

41:26.780 --> 41:29.800
v x B there and
a v x B there.

41:29.800 --> 41:31.490
Which way should the current go?

41:31.489 --> 41:34.039
In fact, it won't go anywhere,
because the line integral of

41:34.041 --> 41:34.791
this will be 0.

41:34.789 --> 41:38.769
You will get contribution from
there, and - the contribution

41:38.771 --> 41:39.651
from there.

41:39.650 --> 41:42.490
So we can demolish this problem.

41:42.489 --> 41:43.919
You understand it completely.

41:43.920 --> 41:46.610
It doesn't pose any
difficulties.

41:46.610 --> 41:51.830
And it does not give any
grounds to monkey with those

41:51.826 --> 41:53.026
equations.

41:53.030 --> 41:56.820
And now comes the bad news.

41:56.820 --> 42:03.970
Suppose I go to a frame of
reference moving with the loop.

42:03.969 --> 42:06.879
It may well be that the loop
was just there,

42:06.880 --> 42:11.080
and I hired a couple of guys to
carry a magnet and run under my

42:11.079 --> 42:12.229
house, right?

42:12.230 --> 42:15.340
And also, my light bulb will
glow, and I don't have to go

42:15.338 --> 42:15.948
anywhere.

42:15.949 --> 42:18.089
They will carry the magnet.

42:18.090 --> 42:22.370
Now you realize that you're
certainly entitled to see it

42:22.367 --> 42:24.777
from that frame of reference.

42:24.780 --> 42:27.090
Now if you believe in
relativity, that means a person

42:27.094 --> 42:29.594
should be able to say from the
point of view of the loop,

42:29.586 --> 42:30.696
"I'm not moving.

42:30.699 --> 42:33.019
Someone's carrying the magnet
and running with it."

42:33.019 --> 42:35.649
But the light bulb must glow.

42:35.650 --> 42:38.960
Lots of things are relative,
but whether a light bulb is

42:38.956 --> 42:40.936
glowing or not is not relative.

42:40.940 --> 42:42.760
You can go to any frame of
reference.

42:42.760 --> 42:44.810
A glowing light bulb is a
glowing light bulb.

42:44.809 --> 42:47.949
We don't know how much power it
consumes and so on,

42:47.949 --> 42:51.339
but the fact that it's on and
glowing is undeniable.

42:51.340 --> 42:56.750
So how is the person in that
fixed loop supposed to

42:56.753 --> 43:01.413
understand the glowing of the
light bulb?

43:01.409 --> 43:04.269
What do you do,
what do you say,

43:04.273 --> 43:06.403
in terms of anything?

43:06.400 --> 43:07.660
Why are charges going around?

43:07.659 --> 43:09.479
I'm a static loop.

43:09.480 --> 43:12.540
Okay, someone's moving with the
magnet, but there's no v

43:12.536 --> 43:15.196
x B force because
nobody's moving in my loop.

43:15.199 --> 43:19.549
The magnet's irrelevant.

43:19.550 --> 43:21.210
Of course, there's one
difference.

43:21.210 --> 43:23.560
This magnet is not like the
usual problem,

43:23.556 --> 43:25.726
because it's a changing
magnetic field,

43:25.731 --> 43:28.881
whereas previously you had a
static magnetic field.

43:28.880 --> 43:31.280
So you might say, "Well,
maybe the force on a charge due

43:31.275 --> 43:34.515
to moving magnetic field is
different from its force due to

43:34.521 --> 43:36.371
changing magnetic field."

43:36.369 --> 43:39.029
But that's not acceptable,
because if you believe in the

43:39.034 --> 43:41.464
reality of the field,
what it does at a given point

43:41.458 --> 43:42.668
depends on its value.

43:42.670 --> 43:45.790
It doesn't depend on anything
else.

43:45.789 --> 43:48.989
But I still haven't explained
why charges like to go around a

43:48.992 --> 43:49.582
wire now.

43:49.579 --> 43:55.239
So what force could be pushing
them?

43:55.239 --> 43:58.879
What force do you think will
push them around a loop?

43:58.880 --> 43:59.760
Yes?

43:59.760 --> 44:01.230
Would you like to guess?

44:01.230 --> 44:02.270
Yes, you.

44:02.268 --> 44:08.288
Student:
>

44:08.289 --> 44:10.899
Prof: The magnetic field
is changing with time.

44:10.900 --> 44:14.410
You agree that the loop is
going to be fixed,

44:14.407 --> 44:17.757
and a little later,
it will look like this,

44:17.755 --> 44:19.425
and so on, right?

44:19.429 --> 44:22.039
But that's not going to move
the current around the wire,

44:22.038 --> 44:25.018
because the magnetic field has
no cross product with anything.

44:25.018 --> 44:30.368
So what's the only thing that
can make the charges move around

44:30.373 --> 44:31.693
the wire now?

44:31.690 --> 44:34.150
What made them move in the DC
circuit?

44:34.150 --> 44:34.870
Student:  Electric field.

44:34.869 --> 44:35.659
Prof: Electric field.

44:35.659 --> 44:39.459
Look, I want you to understand
that these equations carry a

44:39.460 --> 44:42.740
large amount of information,
and they in fact carry

44:42.737 --> 44:45.487
everything as far as you need to
know.

44:45.489 --> 44:48.779
The only force on a charge is
either E or v x

44:48.775 --> 44:49.395
B.

44:49.400 --> 44:51.030
That's it.

44:51.030 --> 44:55.130
If v x B is out,
it's E.

44:55.130 --> 44:56.790
I mean, there may be brand new
phenomena.

44:56.789 --> 44:58.439
It turns out that we don't need
anything else.

44:58.440 --> 45:01.160
E and B will do.

45:01.159 --> 45:03.529
In which case,
what are we saying?

45:03.530 --> 45:06.550
We are saying that in the other
frame of reference,

45:06.548 --> 45:09.448
where the loop is fixed and the
magnet is moving,

45:09.447 --> 45:11.317
there is an electric field.

45:11.320 --> 45:16.590
Not only that,
it's an electric field whose

45:16.594 --> 45:21.024
line integral is not 0,
because it is still true that

45:21.023 --> 45:23.553
charges in the loop are going
round and round and doing work,

45:23.550 --> 45:25.860
and someone's providing work
for every charge that goes

45:25.860 --> 45:26.630
around the loop.

45:26.630 --> 45:30.810
And the emf is the work done on
a unit charge.

45:30.809 --> 45:36.119
So this makes us believe now
that whenever there's a changing

45:36.117 --> 45:40.097
magnetic field,
there is an electric field.

45:40.099 --> 45:44.019
This is an electric field of
brand new origin.

45:44.018 --> 45:49.068
All the electric fields you
studied before were produced by

45:49.065 --> 45:51.585
charges, by coulombs force.

45:51.590 --> 45:54.350
This says, without the help of
any charge.

45:54.349 --> 45:55.929
Charges are nowhere to be found.

45:55.929 --> 45:57.499
They could be infinitely far.

45:57.500 --> 46:01.120
If I change the magnetic field,
I'm going to get an electric

46:01.119 --> 46:01.609
field.

46:01.610 --> 46:04.620
Of course, we have to find out
what is the connection.

46:04.619 --> 46:06.129
How much electric field do you
get?

46:06.130 --> 46:08.320
How is it connected to the
magnetic field?

46:08.320 --> 46:11.450
These are things you have to
deduce and you cannot deduce

46:11.452 --> 46:12.462
them from logic.

46:12.460 --> 46:15.450
You can deduce them only from
experiment.

46:15.449 --> 46:16.739
Certain things come from logic.

46:16.739 --> 46:19.099
If you have an equation,
you solve it using mathematics.

46:19.099 --> 46:20.869
If you like,
that's a kind of logic.

46:20.869 --> 46:24.359
Or you use symmetry and say,
"If I turn the wire and it

46:24.358 --> 46:26.778
looks the same,
the answer should look the

46:26.782 --> 46:28.262
same" and so on.

46:28.260 --> 46:32.490
But what electric field should
be produced by changing magnetic

46:32.492 --> 46:35.362
field you cannot derive from
pure logic.

46:35.360 --> 46:38.400
If you combine it with other
principles like relativity,

46:38.400 --> 46:40.340
you could try to get them,
but right now,

46:40.340 --> 46:42.930
with what we have,
we just know that in the moving

46:42.932 --> 46:44.632
frame--
look, you don't need Einstein

46:44.630 --> 46:45.390
to tell you this.

46:45.389 --> 46:48.889
Even before Einstein,
you knew that you're certainly

46:48.894 --> 46:52.814
free to imagine a loop being
fixed and somebody moving the

46:52.813 --> 46:53.573
magnet.

46:53.570 --> 46:57.420
And everything in you tells you
the answer's got to be the same,

46:57.420 --> 46:59.960
because in the end,
it's the relative motion

46:59.956 --> 47:02.726
between the loop and the magnet
that decides.

47:02.730 --> 47:05.010
And if you provide the same
relative motion,

47:05.005 --> 47:06.905
you've got to get the same
answer.

47:06.909 --> 47:09.339
And the only way to get that is
to say when there's a changing

47:09.342 --> 47:10.992
magnetic field,
I have an electric field,

47:10.989 --> 47:13.609
and furthermore,
electric field which is no

47:13.606 --> 47:17.796
longer conservative because its
line integral is not 0.

47:17.800 --> 47:19.700
So we've got to find out what
it is.

47:19.699 --> 47:23.879
So I'm going to tell you now an
equation that really answers

47:23.882 --> 47:26.792
that question,
and we cannot derive it.

47:26.789 --> 47:29.009
It's a summary of a lot of
experiment.

47:29.010 --> 47:38.920
And that says that the
electromotive force--

47:38.920 --> 47:42.760
in fact, due to electric and
magnetic origin,

47:42.760 --> 47:49.830
em = -d
Φ/dt,

47:49.829 --> 47:58.699
and I'll tell you what this
equation means.

47:58.699 --> 48:08.229
So you take a loop,
some contour c,

48:08.230 --> 48:14.480
and you define Φ
to be equal to the integral of

48:14.480 --> 48:22.230
the magnetic field on a surface
whose boundary is this loop.

48:22.230 --> 48:27.700
That's called the magnetic flux.

48:27.699 --> 48:30.159
So you know operationally how
to calculate it.

48:30.159 --> 48:32.719
Take any surface you want.

48:32.719 --> 48:35.049
It may be a flat one,
a bulging one,

48:35.054 --> 48:36.194
doesn't matter.

48:36.190 --> 48:40.230
Take the flux going through any
surface with that as the

48:40.228 --> 48:44.628
boundary, and find its rate of
change, and that will be equal

48:44.632 --> 48:46.912
to the electromotive force.

48:46.909 --> 48:52.749
In fact, you just write
E electromotive force,

48:52.751 --> 48:56.911
= -d
Φ/dt.

48:56.909 --> 48:59.639
So you know operationally how
to find this thing,

48:59.643 --> 49:00.103
right?

49:00.099 --> 49:03.859
Now again, you may ask,
"Which surface do you want

49:03.855 --> 49:06.145
to take for a given loop?"

49:06.150 --> 49:08.560
This flat one or the curvy one
or the one with more bumps in

49:08.556 --> 49:08.756
it?

49:08.760 --> 49:11.410
The answer is,
again, it does not matter.

49:11.409 --> 49:12.419
The reason is the same.

49:12.420 --> 49:16.010
If you've got some magnetic
lines coming through this guy,

49:16.005 --> 49:19.465
they also have to come out
through that guy because they

49:19.465 --> 49:20.405
don't stop.

49:20.409 --> 49:23.509
Magnetic field lines that enter
one surface have to also enter

49:23.510 --> 49:26.610
another one with the same edge,
because they cannot escape.

49:26.610 --> 49:28.210
So you can take any surface you
like.

49:28.210 --> 49:31.420
So whenever you have these laws
where a surface is invoked,

49:31.420 --> 49:34.040
and it's ambiguous and not
defined by the perimeter,

49:34.039 --> 49:36.689
the answer is,
it doesn't matter which

49:36.692 --> 49:37.412
surface.

49:37.409 --> 49:40.309
The reason is that field lines
don't stop and end,

49:40.309 --> 49:43.799
therefore whatever enters this
surface has to leave this.

49:43.800 --> 49:47.400
If you had magnetic monopoles,
the lines can enter this and

49:47.402 --> 49:49.642
terminate on a negative
monopole,

49:49.639 --> 49:55.079
then of course it won't be
true, but you don't have that.

49:55.079 --> 49:59.709
Now this - sign has got the
name Lenz attached to it,

49:59.706 --> 50:02.816
and this thing is called
Faraday.

50:02.820 --> 50:03.790
Faraday and Lenz.

50:03.789 --> 50:08.069
And I will tell you what the -
sign due to Lenz means.

50:08.070 --> 50:13.510
Lenz's law tells you that if
you change the magnetic flux in

50:13.505 --> 50:18.385
a system, there's going to be a
very long sentence.

50:18.389 --> 50:21.669
If you take the magnetic flux
in a system and it changes,

50:21.670 --> 50:26.420
it will induce an emf which,
if it could drive a current,

50:26.420 --> 50:31.160
will do so in order to fight
that change.

50:31.159 --> 50:33.689
That's why it's a long
statement.

50:33.690 --> 50:36.370
In other words,
if you had a real circuit and

50:36.367 --> 50:38.927
you shove it into region of
changing field,

50:38.925 --> 50:41.355
an emf will start driving a
current.

50:41.360 --> 50:44.440
That current itself will
produce its own magnetic field.

50:44.440 --> 50:45.550
Which way will it point?

50:45.550 --> 50:48.410
The answer is,
it will point it such a way as

50:48.413 --> 50:50.173
to neutralize the change.

50:50.170 --> 50:54.450
For example,
if you took a loop and it had

50:54.449 --> 50:58.819
some flux going through it,
if you tried to increase the

50:58.818 --> 51:01.498
flux going through it,
it will try to decrease it,

51:01.500 --> 51:03.660
so the current will flow like
that.

51:03.659 --> 51:05.949
If you try to decrease the flux
through it, it will try to prop

51:05.954 --> 51:06.884
it up to its old value.

51:06.880 --> 51:08.050
The current will go like that.

51:08.050 --> 51:11.060
That helps you find the
direction of the current.

51:11.059 --> 51:15.729
The emf will be such as to
neutralize the change.

51:15.730 --> 51:18.530
It does not necessarily oppose
the external field,

51:18.530 --> 51:19.730
the magnetic field.

51:19.730 --> 51:21.710
It opposes the change in the
magnetic field.

51:21.710 --> 51:23.300
That's what I want you to
understand.

51:23.300 --> 51:26.600
For example,
suppose I've got lines coming

51:26.603 --> 51:27.493
out here.

51:27.489 --> 51:31.369
And they get weaker,
namely some source of magnetism

51:31.367 --> 51:33.267
makes the lines weaker.

51:33.268 --> 51:37.318
Then the current will start
going this way to prop up the

51:37.320 --> 51:39.490
thing back to its old value.

51:39.489 --> 51:41.589
That's what you have to
understand.

51:41.590 --> 51:44.740
That's the meaning of the -
sign and that's going to guide

51:44.735 --> 51:47.215
us much better than all the
cross products.

51:47.219 --> 51:50.699
Now I will show you that with
this one law,

51:50.697 --> 51:53.097
you can explain everything.

51:53.099 --> 51:55.699
In other words,
you can explain the wire,

51:55.699 --> 51:58.259
the current loop moving in the
magnetic field,

51:58.260 --> 52:01.610
in the laboratory frame,
in which the magnet was at rest

52:01.612 --> 52:04.352
and the loop was moving,
or in the loop frame,

52:04.346 --> 52:07.136
where the loop was at rest and
the magnet was moving.

52:07.139 --> 52:10.829
You can explain everything with
this one great law.

52:10.829 --> 52:12.619
It's a remarkable law.

52:12.619 --> 52:16.019
And you just have to say,
we got it from experiment.

52:16.018 --> 52:18.628
I'm going to tell you how it
explains everything.

52:18.630 --> 52:22.370
So let's go back to this loop.

52:22.369 --> 52:25.339
By the way, some of you gave
that answer, so it was not the

52:25.340 --> 52:26.110
wrong answer.

52:26.110 --> 52:27.920
I mean, I think it's perfectly
valid,

52:27.920 --> 52:31.130
except that I did not want you
to use something we had not done

52:31.126 --> 52:32.676
yet,
but from this moment on,

52:32.684 --> 52:33.734
it's a valid answer.

52:33.730 --> 52:36.690
So I'm going to give you the
other explanation.

52:36.690 --> 52:38.390
So here is my field.

52:38.389 --> 52:45.929
I think I had everybody going
in, right?

52:45.929 --> 52:51.989
Here is the thing-- so I want
you to know what I'm trying to

52:51.985 --> 52:52.905
do now.

52:52.909 --> 52:57.049
I'm trying to tell you that
this one law will explain the

52:57.054 --> 53:01.424
loop going through the field in
both frames of reference.

53:01.420 --> 53:03.010
So first I'm going to do the
easy part.

53:03.010 --> 53:11.370
When there's a fixed magnetic
field and I'm dragging the loop.

53:11.369 --> 53:14.419
I'm not going to worry about
and - signs, because I will get

53:14.418 --> 53:15.348
that in the end.

53:15.349 --> 53:24.929
Let us ask, what is the flux
penetrating this loop?

53:24.929 --> 53:30.139
Can you see that it's just the
magnetic field times the width

53:30.141 --> 53:35.181
of the loop times the length of
the loop that is inside?

53:35.179 --> 53:39.539
Not the whole length,
but the length of the loop that

53:39.543 --> 53:42.653
is still inside the magnetic
field.

53:42.650 --> 53:44.210
That's a very simple result.

53:44.210 --> 53:46.440
B times A is the
flux.

53:46.440 --> 53:48.740
Now let's take the time
derivative of this,

53:48.740 --> 53:50.550
dΦ/dt.

53:50.550 --> 53:53.770
B is not changing,
w is not changing,

53:53.768 --> 53:58.908
but l is changing,
because when I move the loop,

53:58.909 --> 54:03.829
the rate of change of l
is just the velocity of the

54:03.826 --> 54:04.426
loop.

54:04.429 --> 54:11.459
But that's exactly the emf we
got before.

54:11.460 --> 54:15.170
Now the only thing is with the
- sign, let's figure out if the

54:15.173 --> 54:16.273
sign is correct.

54:16.268 --> 54:21.868
This has got magnetic flux
going into the board.

54:21.869 --> 54:27.789
When you pull the loop to the
right, you've got less magnetic

54:27.789 --> 54:30.649
field going into the board.

54:30.650 --> 54:35.080
Therefore the current should
flow in such a way as to

54:35.077 --> 54:36.777
continue the flux.

54:36.780 --> 54:39.080
So which way should the current
flow?

54:39.079 --> 54:41.389
So if I'm pulling the loop--I'm
sorry.

54:41.389 --> 54:46.099
I've got the loop drawn
backwards.

54:46.099 --> 54:47.079
This is okay too.

54:47.079 --> 54:50.119
Let's take this loop and pull
it to the right with a velocity

54:50.121 --> 54:50.681
v.

54:50.679 --> 54:56.589
So the flux into the board is
increasing, right?

54:56.590 --> 55:01.080
So it should produce flux
coming out of the board now to

55:01.079 --> 55:02.549
compensate that.

55:02.550 --> 55:08.660
That means the current has to
flow like this.

55:08.659 --> 55:09.129
See that?

55:09.130 --> 55:11.860
When the current flows like
this, you'll start producing

55:11.864 --> 55:12.564
upward flux.

55:12.559 --> 55:14.569
In other words,
you're getting more and more

55:14.568 --> 55:16.528
flux into the loop going into
the board.

55:16.530 --> 55:18.300
To cancel it,
I will produce flux coming out

55:18.300 --> 55:18.920
of the board.

55:18.920 --> 55:21.840
That's what the loop tells you.

55:21.840 --> 55:23.630
That way you don't have to
worry about all the cross

55:23.632 --> 55:24.022
products.

55:24.018 --> 55:26.498
At the very end,
you ask what's happening to the

55:26.500 --> 55:26.870
flux?

55:26.869 --> 55:29.049
How do I keep it from changing?

55:29.050 --> 55:31.590
And the answer is,
you keep it,

55:31.592 --> 55:35.322
in this case,
from changing by producing flux

55:35.322 --> 55:38.122
coming out of the blackboard.

55:38.119 --> 55:40.839
And finally,
remember when the loop is

55:40.836 --> 55:43.996
entirely inside,
there's going to be no more

55:43.996 --> 55:47.886
emf, because the flux through it
is not changing.

55:47.889 --> 55:50.719
Because in the beginning,
as it moves to the right,

55:50.719 --> 55:53.129
it's gaining more and more flux
from the back end of it,

55:53.130 --> 55:56.040
but once it's fully in,
it's not getting any more flux,

55:56.039 --> 55:57.429
so the current will stop
flowing.

55:57.429 --> 56:02.679
That comes from this argument,
from this way of thinking,

56:02.679 --> 56:05.869
that
dΦ/dt

56:05.865 --> 56:06.705
is 0.

56:06.710 --> 56:11.590
So in other words,
I've given you something that I

56:11.588 --> 56:13.478
say is a new law.

56:13.480 --> 56:17.450
The law says the rate of change
of flux = emf,

56:17.449 --> 56:19.919
but so far, it doesn't look
very new,

56:19.920 --> 56:24.340
because everything it gave you,
you were able to get without

56:24.340 --> 56:25.680
this law,
right?

56:25.679 --> 56:28.229
We managed to understand the
loop completely.

56:28.230 --> 56:29.950
So is there any new information
in this?

56:29.949 --> 56:32.189
And that's what I'm going to
talk about.

56:32.190 --> 56:35.470
The new information's going to
come in when I eventually use it

56:35.471 --> 56:38.701
to understand the answer in the
moving frame of reference,

56:38.699 --> 56:40.999
namely, what's the electric
field produced by changing

56:41.001 --> 56:41.741
magnetic field?

56:41.739 --> 56:43.549
That's also contained in this
one.

56:43.550 --> 56:46.630
But I'm just telling you,
this part of it is a letdown,

56:46.632 --> 56:48.632
because it's not very
impressive.

56:48.630 --> 56:51.120
It gave you an emf you can get
from the v x B.

56:51.119 --> 56:54.619
It is just v x B
all over again.

56:54.619 --> 56:57.159
But let's see what is new.

56:57.159 --> 57:02.339
To really find what is new,
we have to take this problem

57:02.344 --> 57:07.064
and apply it to the very general
case of a loop.

57:07.059 --> 57:10.259
Here is some loop.

57:10.260 --> 57:15.390
I'm going to have the loop
actually move in time.

57:15.389 --> 57:17.379
Take a loop in one shape.

57:17.380 --> 57:20.820
A little later,
it's got another shape.

57:20.820 --> 57:23.200
And the magnetic field is free
to change.

57:23.199 --> 57:25.189
I'm going to take a very
general situation.

57:25.190 --> 57:28.560
The loop is moving,
field is changing.

57:28.559 --> 57:32.889
So let's say this surface is
S_1 and a

57:32.885 --> 57:37.205
little later there's a surface
S_2.

57:37.210 --> 57:40.560
You should imagine this is
stacked on top of this.

57:40.559 --> 57:47.249
Maybe I should hide that part.

57:47.250 --> 57:52.390
Here is this surface,
and it's moved to a new

57:52.393 --> 57:53.683
location.

57:53.679 --> 57:56.839
Now this surface,
let me call S and let me

57:56.844 --> 58:00.674
call S Δs and you
will see what I mean by S

58:00.668 --> 58:01.788
ΔS.

58:01.789 --> 58:09.789
You can build up that surface
by taking this surface and

58:09.791 --> 58:13.431
gluing to it this edge.

58:13.429 --> 58:15.349
Is that clear or not clear?

58:15.349 --> 58:16.099
Not clear.

58:16.099 --> 58:19.009
Let me take a simpler surface
than this one.

58:19.010 --> 58:26.020
Here is a surface,
here is another surface.

58:26.018 --> 58:30.108
So one is a cylinder with the
round sides on the bottom.

58:30.110 --> 58:31.820
The other is the top.

58:31.820 --> 58:37.330
Do you agree,
they are both surfaces with the

58:37.327 --> 58:39.327
same boundary?

58:39.329 --> 58:41.419
Are you with me there?

58:41.420 --> 58:44.810
This top loop can either take
the top of the cylinder as its

58:44.809 --> 58:47.319
boundary,
or it's a hollow cylinder,

58:47.324 --> 58:49.924
the curvy side and the flat
bottom,

58:49.920 --> 58:53.610
are also another surface with
the same boundary.

58:53.610 --> 58:59.720
But I can think of the second
surface as essentially the old

58:59.719 --> 59:01.789
surface the sides.

59:01.789 --> 59:06.469
So you agree that S' = S
ΔS,

59:06.469 --> 59:11.429
where delta S is the
stuff you glue on the side to

59:11.425 --> 59:15.405
the old surface to make up the
new surface.

59:15.409 --> 59:17.879
Yes or no?

59:17.880 --> 59:19.340
I think the cylinder is the
easiest one.

59:19.340 --> 59:25.110
I tried more ambitiously to
draw this, but here is the

59:25.106 --> 59:26.736
simpler case.

59:26.739 --> 59:31.169
So now the question is,
what is the rate of change of

59:31.170 --> 59:33.130
flux in this problem?

59:33.130 --> 59:36.840
So the rate of change of flux
I'm going to find as follows.

59:36.840 --> 59:42.250
I'm going to find the flux at
time t Δt - phi at

59:42.246 --> 59:43.686
time t.

59:43.690 --> 59:50.530
That's what I want.

59:50.530 --> 59:53.740
Now at time t Δt,
my loop is here,

59:53.744 --> 59:58.414
but I'm going to use,
rather than that surface,

59:58.413 --> 1:00:03.423
I'm going to use the old
surface the patch job to make it

1:00:03.416 --> 1:00:05.556
into the new surface.

1:00:05.559 --> 1:00:09.589
So therefore,
I want the magnetic field,

1:00:09.590 --> 1:00:15.850
B at t Δt
on the surface S ΔS

1:00:15.853 --> 1:00:19.963
- the integral of the magnetic
field,

1:00:19.960 --> 1:00:25.820
dA on the old surface at
time t. This part is

1:00:25.824 --> 1:00:29.774
quite subtle and it took me a
while,

1:00:29.768 --> 1:00:33.698
when I first learned it,
to understand what the point

1:00:33.704 --> 1:00:34.164
was.

1:00:34.159 --> 1:00:37.289
See,the surface used to be the
bottom half of the cylinder,

1:00:37.293 --> 1:00:38.323
the bottom plate.

1:00:38.320 --> 1:00:40.430
The new surface is the top
plate.

1:00:40.429 --> 1:00:42.099
So natural instinct will be,
let's find

1:00:42.097 --> 1:00:44.197
B⋅v as
on the top face,

1:00:44.204 --> 1:00:45.964
but we don't want to do it that
way.

1:00:45.960 --> 1:00:48.390
We purposely want to take
another surface with the same

1:00:48.387 --> 1:00:50.047
boundary, because we're allowed
to.

1:00:50.050 --> 1:00:52.960
That is the old boundary the
vertical sides.

1:00:52.960 --> 1:00:54.520
That's the main point.

1:00:54.518 --> 1:00:56.948
You take the old boundary
because it's going to make it

1:00:56.947 --> 1:00:59.687
easy for you to cancel some of
the stuff with some of this.

1:00:59.690 --> 1:01:00.820
That's why.

1:01:00.820 --> 1:01:04.340
So B(t Δt)
⋅dA,

1:01:04.340 --> 1:01:12.740
you can write it as B(t
Δt) ⋅

1:01:12.744 --> 1:01:21.314
dA on S integral
B(t)⋅

1:01:21.306 --> 1:01:26.536
dA on ΔS.

1:01:26.539 --> 1:01:29.379
In other words,
the surface integral is

1:01:29.380 --> 1:01:31.400
changing for two reasons.

1:01:31.400 --> 1:01:33.940
The surface itself,
the loop itself around which

1:01:33.936 --> 1:01:36.796
you take the line integral is
changing, and the fields

1:01:36.797 --> 1:01:39.007
themselves may be changing with
time.

1:01:39.010 --> 1:01:40.830
So they can both contribute to
the change.

1:01:40.829 --> 1:01:43.129
And I've broken up into two
parts.

1:01:43.130 --> 1:01:47.430
One says take B at time
t Δt on the same

1:01:47.434 --> 1:01:49.494
old surface,
and then take

1:01:49.485 --> 1:01:53.895
B⋅dA
on the incremental surface,

1:01:53.900 --> 1:02:01.160
and subtract from it
B(t)⋅dA

1:02:01.157 --> 1:02:04.857
on the old surface.

1:02:04.860 --> 1:02:06.840
So do you guys see that?

1:02:06.840 --> 1:02:09.970
Phi of t Δt
I've written as two parts.

1:02:09.969 --> 1:02:14.529
One is the flux on the lower
face, but at a later time,

1:02:14.528 --> 1:02:17.398
and the flux on the curvy
sides.

1:02:17.400 --> 1:02:19.900
Now on the curvy side,
you don't have to worry about

1:02:19.896 --> 1:02:22.536
whether it's at a later time or
at an earlier time,

1:02:22.539 --> 1:02:25.729
because ΔS is
first ordered in time.

1:02:25.730 --> 1:02:27.370
It's proportional to
Δt,

1:02:27.369 --> 1:02:29.389
and this difference is going to
be proportional to

1:02:29.393 --> 1:02:31.883
Δt,
so you don't have to worry

1:02:31.876 --> 1:02:35.596
about the change in time over an
infinitesimal surface,

1:02:35.599 --> 1:02:39.799
because that's second ordered
in time.

1:02:39.800 --> 1:02:47.070
So now I have to bring all that
stuff over here.

1:02:47.070 --> 1:02:58.740
So I'm going to say the change
in Φ = dB/dt times

1:02:58.737 --> 1:03:04.977
dA,
Δt on the old

1:03:04.978 --> 1:03:07.928
surface,
here is the stuff we have to do

1:03:07.932 --> 1:03:08.632
that's extra.

1:03:08.630 --> 1:03:12.780
So do you understand how I
cancel B(t Δt) −

1:03:12.775 --> 1:03:14.975
B(t),
that's given by the rate of

1:03:14.981 --> 1:03:19.361
change of B with time,
times the change in time?

1:03:19.360 --> 1:03:22.050
You get their contribution only
because B is changing

1:03:22.047 --> 1:03:23.917
with time,
because you're taking the very

1:03:23.920 --> 1:03:26.560
same surface,
as you did the first time,

1:03:26.561 --> 1:03:28.181
but at a later time.

1:03:28.179 --> 1:03:30.279
Therefore the difference is due
to the rate of change.

1:03:30.280 --> 1:03:35.210
But now we have to do the
second part, which is,

1:03:35.206 --> 1:03:40.026
on the surface Δs,
we want to take

1:03:40.030 --> 1:03:43.700
B⋅dA.

1:03:43.699 --> 1:03:48.639
Now what is ΔS?

1:03:48.639 --> 1:03:49.219
Let me see.

1:03:49.219 --> 1:03:59.219


1:03:59.219 --> 1:04:00.959
So I will have to draw the
picture like this.

1:04:00.960 --> 1:04:08.810
So here's one surface,
here's another surface,

1:04:08.809 --> 1:04:14.739
and I've gone from here to
here.

1:04:14.739 --> 1:04:17.069
I've gone from this loop to
this loop in a time

1:04:17.074 --> 1:04:17.994
Δt.

1:04:17.989 --> 1:04:23.889
This is my segment of the loop,
and that's the distance,

1:04:23.891 --> 1:04:25.931
vΔt.

1:04:25.929 --> 1:04:30.499
So the area vector associated
with this section is just

1:04:30.501 --> 1:04:33.721
vΔt x
dl.

1:04:33.719 --> 1:04:38.459
That is the surface area.

1:04:38.460 --> 1:04:39.200
You see that?

1:04:39.199 --> 1:04:42.049
A parallelogram formed with the
loop size dl here

1:04:42.047 --> 1:04:43.927
and the distance traveled in
dt.

1:04:43.929 --> 1:04:45.789
Area of that is just the cross
product.

1:04:45.789 --> 1:04:55.659
So this = 1 v (I'm going
to put the Δv outside)

1:04:55.664 --> 1:05:03.034
v x
dl⋅B.

1:05:03.030 --> 1:05:04.610
So I'm not going to touch this
expression.

1:05:04.610 --> 1:05:07.310
I'm going to fiddle with this
for a while.

1:05:07.309 --> 1:05:11.509
Cancel dt everywhere if
you like, divide by dt.

1:05:11.510 --> 1:05:13.610
Forget that and forget that.

1:05:13.610 --> 1:05:17.480
Then I get
dΦ/dt =

1:05:17.483 --> 1:05:21.913
dB/dt
⋅dA on the

1:05:21.909 --> 1:05:27.149
surface,
the line integral of v x

1:05:27.148 --> 1:05:31.318
dl⋅B.

1:05:31.320 --> 1:05:33.970
Do you understand the last part
what I did?

1:05:33.969 --> 1:05:37.869
The extra surface that you got
here is made up of little

1:05:37.867 --> 1:05:42.117
rectangular tiles with the one
end being dl, other end

1:05:42.121 --> 1:05:43.611
being Bdt.

1:05:43.610 --> 1:05:46.430
The area vector comes like
that, the magnetic field may

1:05:46.429 --> 1:05:49.499
look like that,
and the dot product of the two

1:05:49.500 --> 1:05:53.970
is what I have as the extra flux
coming out of the curvy side of

1:05:53.974 --> 1:05:55.044
the surface.

1:05:55.039 --> 1:06:00.049
Now you should know enough
about cross products to know

1:06:00.052 --> 1:06:03.492
that this cross product is the
same as

1:06:03.487 --> 1:06:07.847
dl⋅B x
v,

1:06:07.849 --> 1:06:09.859
because you can rotate the
factors in an A x

1:06:09.864 --> 1:06:12.414
B⋅C is
B⋅C x

1:06:12.405 --> 1:06:13.695
A,
is C⋅A

1:06:13.695 --> 1:06:14.105
x B.

1:06:14.110 --> 1:06:16.390
I've done that once.

1:06:16.389 --> 1:06:21.799
But then this = - the integral
dl

1:06:21.802 --> 1:06:25.922
⋅v x
B.

1:06:25.920 --> 1:06:30.380
So I'm going to write down what
I have.

1:06:30.380 --> 1:06:32.640
What I have is that the rate of
change of flux,

1:06:32.639 --> 1:06:42.859
dΦ/dt
= -dB/dt times the area -

1:06:42.860 --> 1:06:51.600
integral v x
B⋅dl.

1:06:51.599 --> 1:06:58.559
I'm sorry, it's d/dt
that.

1:06:58.559 --> 1:07:01.469
Therefore −d
Φ/dt,

1:07:01.465 --> 1:07:04.065
we'll have a - sign here and a
sign here.

1:07:04.070 --> 1:07:05.600
So just write the formula for
−d

1:07:05.599 --> 1:07:06.469
Φ/dt.

1:07:06.469 --> 1:07:07.959
Put a - on everything here.

1:07:07.960 --> 1:07:10.350
This is what I want you to know.

1:07:10.349 --> 1:07:12.619
Now you've got to go home and
you have to think about this.

1:07:12.619 --> 1:07:14.929
I don't claim it's easy.

1:07:14.929 --> 1:07:16.739
I will also tell you something.

1:07:16.739 --> 1:07:18.779
If you really don't want to
know everything,

1:07:18.778 --> 1:07:21.148
you don't have to follow this
particular detail.

1:07:21.150 --> 1:07:22.670
I'm going to tell you what I
get out of this.

1:07:22.670 --> 1:07:24.340
That's what you have to know.

1:07:24.340 --> 1:07:27.050
I want to make sure that if you
want to follow everything,

1:07:27.048 --> 1:07:28.188
you're given a chance.

1:07:28.190 --> 1:07:35.800
So here's what I get from the
old Faraday and Lenz law.

1:07:35.800 --> 1:07:39.660
This says that rate of change
of flux has two parts.

1:07:39.659 --> 1:07:41.839
You take a contour and you find
the flux changing through it,

1:07:41.836 --> 1:07:42.596
it's got two parts.

1:07:42.599 --> 1:07:48.319
One, because the field itself
is changing over the region.

1:07:48.320 --> 1:07:52.390
Second, because the loop itself
is changing.

1:07:52.389 --> 1:07:53.999
And together,
they make these two

1:07:54.003 --> 1:07:54.813
contributions.

1:07:54.809 --> 1:08:02.909
But emf = - integral of all the
electric forces and the magnetic

1:08:02.911 --> 1:08:04.071
forces.

1:08:04.070 --> 1:08:09.740
That's =
-(dB/dt)dA

1:08:09.735 --> 1:08:19.115
integral v x
B⋅dl.

1:08:19.118 --> 1:08:21.318
Now here's something very
beautiful to look at.

1:08:21.319 --> 1:08:25.579
It says the rate of change of
flux by miracle manages to

1:08:25.582 --> 1:08:30.472
measure the line integral of the
electric field and the magnetic

1:08:30.465 --> 1:08:31.235
field.

1:08:31.238 --> 1:08:33.438
And the rate of change of flux
contains two parts.

1:08:33.439 --> 1:08:37.419
The second part is really the
v x B force that

1:08:37.416 --> 1:08:41.526
comes because the contour itself
is moving, the wire itself is

1:08:41.528 --> 1:08:42.268
moving.

1:08:42.270 --> 1:08:46.050
So you balance this term with
this term.

1:08:46.050 --> 1:08:48.850
If this is all you were looking
at, there's nothing new in the

1:08:48.845 --> 1:08:49.345
equation.

1:08:49.350 --> 1:08:52.420
That's why I said we're not
impressed, because it's just the

1:08:52.422 --> 1:08:53.572
v x B.

1:08:53.569 --> 1:08:57.159
But the real beauty is,
when you cancel these terms and

1:08:57.157 --> 1:09:01.667
you look at what it says here,
that says line integral of

1:09:01.671 --> 1:09:06.221
E⋅dl =
- rate of change of

1:09:06.224 --> 1:09:09.264
B⋅dA.

1:09:09.260 --> 1:09:13.300
The meaning of the partial
derivative is that B can

1:09:13.297 --> 1:09:16.267
depend on x,
y, z, and t, but

1:09:16.273 --> 1:09:19.323
this is only due to change in
t.

1:09:19.319 --> 1:09:22.789
Only due to change in t
because I did the integral on

1:09:22.793 --> 1:09:25.153
the same surface,
only a little later.

1:09:25.149 --> 1:09:33.629
This is called Faraday's law.

1:09:33.630 --> 1:09:36.330
You can see that because of
Faraday's law,

1:09:36.328 --> 1:09:41.998
one of the four equations I
wrote hidden somewhere here,

1:09:42.000 --> 1:09:46.290
in which the line integral of
E is 0 is no longer 0.

1:09:46.288 --> 1:09:50.018
So this finally tells you that
in general,

1:09:50.020 --> 1:09:53.010
if you take the line integral
of the electric field around a

1:09:53.009 --> 1:09:55.979
closed loop,
you will not get 0 if the

1:09:55.976 --> 1:09:59.946
magnetic flux through that loop
is changing.

1:09:59.948 --> 1:10:03.368
But this has nothing to do
whatsoever with a real

1:10:03.372 --> 1:10:05.302
conducting loop any more.

1:10:05.300 --> 1:10:08.290
See, this formula has a lot to
do with the conducting loop,

1:10:08.288 --> 1:10:10.468
because v is the
velocity of the portion

1:10:10.470 --> 1:10:13.770
dl,
as the loop moves through space.

1:10:13.770 --> 1:10:15.750
But the loop has been banished
from the two sides.

1:10:15.750 --> 1:10:18.840
These equations have nothing to
do with any loop.

1:10:18.840 --> 1:10:22.140
They just say take a fixed
contour in space and the line

1:10:22.135 --> 1:10:25.725
integral of E on that
contour is the rate of change of

1:10:25.729 --> 1:10:26.269
flux.

1:10:26.270 --> 1:10:29.340
It's an intrinsic relation
between electric and magnetic

1:10:29.337 --> 1:10:32.237
fields, not having anything to
do with conductors.

1:10:32.238 --> 1:10:35.548
It's a property of E and
B, namely that a changing

1:10:35.551 --> 1:10:37.971
B can produce a
circulating E.

1:10:37.970 --> 1:10:45.000
That's the content of this
equation.