WEBVTT

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Prof: Okay,
well I want to start by

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cleaning up a little bit of
stuff done near the end of last

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time.

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So I told you the whole thing
with magnetism is going to

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contain two parts,
just like with electricity,

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namely how do things react to a
magnetic field and how do things

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produce a magnetic field?

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Just like the electric charges,
electric charges produce

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electric fields and electric
charges react to electric

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fields.

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So the first part is very easy
for magnetism.

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The force of magnetism on a
charge q depends on its

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velocity, because if it doesn't
move, it doesn't feel it.

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And it depends on the magnetic
field, which I will denote by

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some symbol B.

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Then you can ask yourself,
"How am I going to get a

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vector out of two vectors?"

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namely the velocity and
B, and a natural

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candidate is the cross product.

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Now that doesn't mean that's
the right answer.

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For example,
I can multiply this by the

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absolute value of velocity
squared and that will still be a

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vector, because that's just a
scalar.

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Anyway, ignore that,
because that's not what

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happens.

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Nature is very kind and it's
just

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v x B.

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And you got to know how to do
the cross products,

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okay?

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There are two rules you're
going to use all the time.

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One is the cross product,
which is for me the screwdriver

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rule,
that if you've got a vector

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v and you've got a vector
B,

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you turn the screwdriver from
v to B and the way

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it comes out is the direction of
the cross product.

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So everyone's doing that.

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You can also do it with your
hand.

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Have you seen all the rap guys?

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They're all computing cross
products, because there turns

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out to be a right hand rule and
a left hand rule.

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So you can do that,
or you can do this.

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Another one is whenever there's
a current or something,

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you don't know which way the
magnetic field is going,

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you're supposed to wrap your
hands around the current.

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Now don't do this at home.

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Wrap it around the current,
and the thumb points the

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direction of the field.

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So whenever you have doubts
about which way something is

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going,
if you see something looping

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around,
wrap your four fingers around

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that loop,
and thumb will point in the

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direction of the answer.

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I've been using it a lot
without spelling it out for you,

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but you should know at least
the technical definition of

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cross product.

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Another thing is,
I didn't tell you anything

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about units.

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So B is measured in
Tesla, and the meaning of a

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Tesla is that if you have a 1
coulomb charge moving at 1 meter

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per second,
perpendicular to the magnetic

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field B of 1 tesla,
it will experience a force of 1

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Newton.

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That's it.

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So Tesla is cooked up so that
in the mks units that we use,

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if you put in the numbers,
you will get the force in

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Newtons.

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Then I said from this
microscopic description of what

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happens to a single charge,
you can go to a piece of wire

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carrying some current and take a
segment dl,

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and the current is I
going through the loop.

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The force, which I write as
dF, meaning an

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infinitesimal force on an
infinitesimal segment is

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dl x B.

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So if you stick a wire in a
magnetic field,

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different parts of the wire
have current going in different

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directions.

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I'm taking a tiny section,
dl, and in that tiny

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section, I'm computing the cross
product.

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That's going to be the force.

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For example,
if at this location the

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magnetic field points that way,
then the force will be coming

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out of the board.

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So this is the first half,
and I tried to do a problem

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near the end.

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Let me repeat that,
because I sort of switched

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coordinates in the middle.

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So I wanted to do this problem.

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There's a wire.

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This is a semicircle.

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I'm not worried about this
portion and this portion.

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I just want to know,
what is the force on this

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segment here?

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Which way is it going to act?

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So B is like this.

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First of all,
you can tell,

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the force is going to be in or
out of the blackboard,

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because if B is in the
plane of the blackboard,

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I mean, velocity of the current
is in the plane of the

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blackboard,
B is in the plane,

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cross product will be
perpendicular to two of them,

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will come out.

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You've just got to figure out,
is it coming out or is it going

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in?

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So take a segment here,
if you like.

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That's a segment dl,
and here is the B.

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Turn a screwdriver from
dl to B,

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it's coming out of the
blackboard.

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B is assumed to be
uniform and in the plane of the

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blackboard.

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So let's now compute the
contribution from this segment.

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The contribution from this
segment is going to be--I'm

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going to ignore the vector
nature, because it's coming out

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of the board.

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I already know that.

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So there is the current,
then there is the dl,

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then there is the B,
and there's the sine of the

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angle, which means these two
guys.

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I claim that that is the same
as this angle here.

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This is an ancient trick.

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The ancient trick says if the
angle between two lines is

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θ,
then the angle between the

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perpendicular to the two lines
is also θ.

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So the two lines are this and
this and the perpendiculars to

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them,
this guy's perpendicular to

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this one and the tangent vector
is perpendicular to the radius.

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So that's the same angle
θ.

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So that sinθ is
this one.

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So the total force will be
IB--and what is

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dl?

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The segment length dl is
just R times

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dθ.

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So you've got sinθ
dθ integral from 0

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to Π.

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The reason I repeated this is
that normally we measure theta

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like that, but there's nothing
sacred.

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You can measure it any way you
like.

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If you measure it this way,
it's increasing in this

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direction, you should go from 0
here to Π there.

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So integral of sine is -cosine.

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You do all the limits,
you get 2IBR.

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By the way, it's very useful to
know a quick rule,

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is that the force on a segment,
where everybody is

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perpendicular to everybody,
is BLl.

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Segment of length L
carrying current I in a

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magnetic field B.

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If B and I are
all perpendicular,

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then the force is just
BIL.

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So if you look at it that way,
it looks like B times

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I times 2R,
2R being this length.

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In other words,
if the current,

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instead of going around like
this, had gone straight from

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here to here,
it would have a length

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2R.

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B is this way,
perpendicular.

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Take the cross product,
you will get the same force.

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In other words,
the force on the curved segment

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between two points is the same
as the force on a straight

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segment joining the two points.

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This happens to be universally
true in a uniform field.

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In other words,
even if the wire did this,

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don't bother to find the force.

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It's just simply the force on
that segment.

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Now you will prove that in your
homework.

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It's not obvious,
but I don't want to tell you

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exactly why, but that's
something you can prove.

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So that was one line segment.

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But then I started taking a
loop.

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So here is the loop,
and let's say the magnetic

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field is going up like this and
the loop is oriented like this.

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It's carrying a current
I in the sense shown

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here.

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So what are the forces on the
four sides.

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B is this way.

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Dl or I is this
way, and B is up.

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If you turn a screwdriver from
there to B,

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you will find this force here
is like that,

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and this force is like that.

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Both are horizontal forces.

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There's a force on this side
and a force on that side,

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which are equal and opposite.

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Yes?

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Student:  What are the
forces on _________ B

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inside and outside the force?

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Prof: They are
perpendicular.

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I will show you a side view
where you will see exactly how

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they are pointing.

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At the moment,
this is limited by my artistic

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skills, but I will show you in a
moment a side view that will

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make it very clear.

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But it should be clear right
now, the total force on this

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loop is 0,
because it's a uniform magnetic

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field,
if there's a current going this

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way,
there's the same current going

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the opposite way for the same
length in the same field,

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so those forces will cancel.

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So the loop doesn't feel a net
force.

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But it feels a net torque,
which I think you can imagine,

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is trying to straighten this
loop out and make it horizontal.

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And if you want to see the
torque, look at the loop from

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this end, from this edge.

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It looks like this.

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That's the loop and that's the
B field.

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And that's the normal to the
loop or the area vector of the

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loop looks like that.

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So the force looks like this
here and looks like that in the

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top section.

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So let us say this length is
l and this width

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is w.

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Then the force on this segment
is BlI,

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I told you that,
and this force is also

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BlI,
and that is w and this

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angle and this angle are both
theta,

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then the torque =
BlIwsinθ.

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I hope you understand why it's
wsinθ,

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because
wsinθ is

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that distance.

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By the way, when you compute
torques,

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you can find the torque around
this point or torque around that

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point,
it doesn't matter around which

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point,
as long as the forces add up to

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0, and finding it around this
point.

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If you like,
you can find the torque around

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the midpoint,
then this will give a

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contribution,
that will give an equal

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contribution.

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You will add them,
but no matter what you do,

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you will get
wsinθ

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times either force.

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So that's the torque.

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But whenever we see a sine
theta and a lot of vectors,

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we know we're looking at a
cross product.

11:20.009 --> 11:23.099
And the cross product,
the vector formula for tau,

11:23.096 --> 11:27.406
is going to be written as
μ x B,

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and μ is a quantity
which is area vector times the

11:33.957 --> 11:39.757
current,
because l times w

11:39.755 --> 11:41.625
is the area.

11:41.629 --> 11:42.299
Right?

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The area of the loop which
points perpendicular to that has

11:45.855 --> 11:48.915
a value equal to l
times w.

11:48.918 --> 11:50.738
so l times
w is area,

11:50.740 --> 11:53.410
so I times area is
called the magnetic moment,

11:53.408 --> 12:00.278
and that times B times
sinθ is the cross

12:00.275 --> 12:01.435
product.

12:01.440 --> 12:06.680
So what you notice is that the
torque will turn it till

12:06.684 --> 12:09.604
μ x B is 0.

12:09.600 --> 12:12.350
That will occur when
μ and B are

12:12.349 --> 12:14.009
parallel, or anti-parallel.

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There are the only two options.

12:15.798 --> 12:18.078
When they are the parallel,
the area vector's along

12:18.077 --> 12:18.577
B.

12:18.580 --> 12:21.460
When they're anti-parallel,
the area vector's exactly

12:21.456 --> 12:22.726
opposite to B.

12:22.730 --> 12:26.320
And they are the two
equilibrium positions.

12:26.320 --> 12:28.510
But one is the point of maximum
energy;

12:28.509 --> 12:32.099
one is the point of minimum
energy, just like with the

12:32.100 --> 12:33.320
electric dipole.

12:33.320 --> 12:37.490
Once you have a torque like
this, you can find the potential

12:37.494 --> 12:41.104
energy and you can show the
potential energy is just

12:41.101 --> 12:45.011
-μBcosθ,
which is

12:45.011 --> 12:49.781
-μ⋅B.

12:49.779 --> 12:50.109
Right?

12:50.105 --> 12:52.005
The integral of
sinθ is

12:52.009 --> 12:56.439
-cosθ,
so basically the point I'm

12:56.441 --> 13:00.111
making is,
this loop would like to have

13:00.113 --> 13:03.893
its area vector parallel to the
magnetic field.

13:03.889 --> 13:05.819
That's what you've got to
understand.

13:05.820 --> 13:09.420
It will align itself so that
the most amount of magnetic

13:09.423 --> 13:11.263
field lines go through it.

13:11.259 --> 13:14.139
So one way is to be parallel to
it, the other is to be

13:14.144 --> 13:16.434
anti-parallel,
and they don't have the same

13:16.431 --> 13:17.031
energy.

13:17.028 --> 13:20.758
The energy looks like this,
- cosine theta means it looks

13:20.763 --> 13:23.233
like that as the function of
theta.

13:23.230 --> 13:27.010
Theta = 0 is the best and theta
= pi is the worst.

13:27.009 --> 13:30.789
If you keep it here,
it's a stable equilibrium.

13:30.788 --> 13:32.718
If the loop is perturbed a
little, little bit,

13:32.717 --> 13:34.087
it will rattle back and forth.

13:34.090 --> 13:37.760
If you keep it there,
it's like being on the top of a

13:37.764 --> 13:38.264
hill.

13:38.259 --> 13:40.239
If you perturb it,
it will come all the way down

13:40.235 --> 13:44.435
the other side;
it will flip over.

13:44.440 --> 13:50.070
So this loop is really like a
tiny compass needle.

13:50.070 --> 13:52.450
If this B field were due
to the earth,

13:52.450 --> 13:54.280
then the normal to the area
vector,

13:54.279 --> 13:57.139
if you can put a little--you
could take a plane and you paint

13:57.140 --> 14:01.570
a little arrow on top of it,
the arrow will line up with the

14:01.573 --> 14:03.103
magnetic field.

14:03.100 --> 14:08.500
All right, so you should
remember then that this is

14:08.501 --> 14:11.421
called a magnetic dipole.

14:11.419 --> 14:12.349
Why?

14:12.350 --> 14:15.390
Because we studied the electric
dipole in which if you got a

14:15.394 --> 14:19.644
plus charge and a minus charge,
and you got an electric field

14:19.636 --> 14:24.436
E and the torque on it
was p x E,

14:24.440 --> 14:31.560
where the magnitude of p
is the charge times the distance

14:31.562 --> 14:33.262
between them.

14:33.259 --> 14:36.529
And a dipole would also align
itself with the electric field,

14:36.533 --> 14:39.813
with the charge along E
and the - charge down here.

14:39.808 --> 14:43.038
The other configuration was
exactly anti-parallel,

14:43.038 --> 14:45.508
but that's the configuration of
highest energy,

14:45.509 --> 14:51.599
and the energy again was
-p⋅E.

14:51.600 --> 14:54.850
So a current loop is like an
electric dipole,

14:54.849 --> 14:57.359
except it's the magnetic
dipole.

14:57.360 --> 15:00.960
It is to the magnetic field
what the electric dipole is to

15:00.961 --> 15:03.871
the electric field,
with one big difference.

15:03.870 --> 15:07.790
If you look at an electric
dipole, it really has a charge

15:07.788 --> 15:10.098
and a - charge at the two ends.

15:10.100 --> 15:15.780
A magnetic dipole,
like a current loop here,

15:15.777 --> 15:21.717
doesn't have a and - magnetic
charge there.

15:21.720 --> 15:24.520
It's as if there is a and -
magnetic charge,

15:24.523 --> 15:26.873
but there are no magnetic
charges.

15:26.870 --> 15:30.260
There are no isolated magnetic
charges that produce magnetic

15:30.263 --> 15:33.603
fields the way electric charges
produce electric fields.

15:33.600 --> 15:37.330
If you had a magnetic monopole,
it will produce a radially

15:37.330 --> 15:39.620
outgoing field,
like 1/r^(2),

15:39.620 --> 15:41.650
but there is no such thing.

15:41.649 --> 15:44.269
But there is something that
looks like a magnetic dipole.

15:44.269 --> 15:46.539
In other words,
if this was a magnetic charge,

15:46.538 --> 15:49.598
that was a - magnetic charge,
then they would feel a magnetic

15:49.596 --> 15:52.906
field and they would obviously
line up with the magnetic field.

15:52.908 --> 15:57.508
The dipole will line up with
the field.

15:57.509 --> 16:01.379
Now it turns out you can get
some money out of this.

16:01.379 --> 16:03.819
You can build a device.

16:03.820 --> 16:11.710
The device I'm going to build
is an electric motor,

16:11.711 --> 16:20.081
because you can take a bar
magnet or any other magnet,

16:20.077 --> 16:26.547
with some magnetic field coming
here.

16:26.548 --> 16:29.008
This is the other end of the
magnet.

16:29.009 --> 16:35.659
You put a current loop here.

16:35.658 --> 16:39.778
Here are some wires coming out
of the current loop.

16:39.779 --> 16:43.609
If you drive a current through
it, then let's see,

16:43.610 --> 16:46.890
if I drive the current like
this, the magnetic moment looks

16:46.893 --> 16:48.873
like that,
and the magnetic field,

16:48.866 --> 16:50.316
B, looks like this.

16:50.320 --> 16:54.060
So it will flip till μ
aligns with B.

16:54.059 --> 16:55.159
Do you understand?

16:55.158 --> 16:59.998
It will flip so that the plane
of the loop is perpendicular to

16:59.998 --> 17:03.408
the field lines,
or parallel to this or that

17:03.408 --> 17:05.628
face of the two magnets.

17:05.630 --> 17:07.170
But to do that,
of course, you have to drive

17:07.172 --> 17:07.712
some current.

17:07.710 --> 17:11.540
So I take this current,
a source of current,

17:11.538 --> 17:14.118
and I connect it like that.

17:14.118 --> 17:19.588
Now, do you have any problem
with this invention?

17:19.589 --> 17:20.139
Yes?

17:20.140 --> 17:22.020
What's your problem?

17:22.019 --> 17:24.169
Student:  It will only
go until new aligned with

17:24.170 --> 17:26.770
B, and then it will stop.

17:26.769 --> 17:27.629
Prof: That's right.

17:27.630 --> 17:29.860
It's not going to run very long.

17:29.858 --> 17:32.908
It's going to flip till it
lines up and that's the end.

17:32.910 --> 17:35.480
And if it's already lined up,
it won't even do that.

17:35.480 --> 17:37.620
So this is not going to sell.

17:37.619 --> 17:39.129
So what do you have to do?

17:39.130 --> 17:41.630
You guys might know what you
have to do.

17:41.630 --> 17:42.920
Student:
>

17:42.920 --> 17:45.980
AC?

17:45.980 --> 17:47.600
Prof: AC?

17:47.598 --> 17:53.818
Suppose you only have a DC
voltage, what are you going to

17:53.815 --> 17:54.365
do?

17:54.368 --> 18:01.078
Suppose you're very quick with
your hands, what will you do?

18:01.079 --> 18:02.009
Pardon me?

18:02.009 --> 18:03.479
Student:  Switch the
magnets.

18:03.480 --> 18:04.930
Prof: Okay, two things.

18:04.930 --> 18:07.490
One is, switch the magnet.

18:07.490 --> 18:09.530
You can also switch the wires
of the terminal,

18:09.527 --> 18:09.887
right?

18:09.890 --> 18:12.810
You can keep doing this and you
can keep reversing the current.

18:12.808 --> 18:15.538
Every time this guy thinks it's
found happiness,

18:15.540 --> 18:17.460
you say no, and turn it around.

18:17.460 --> 18:19.790
And you go the other end,
you do it again.

18:19.788 --> 18:22.188
But that means you can never
leave this motor and go

18:22.191 --> 18:22.711
anywhere.

18:22.710 --> 18:24.580
It's high maintenance.

18:24.579 --> 18:26.849
So here is a very clever device.

18:26.848 --> 18:29.868
If you haven't thought about it
or if you don't know the answer,

18:29.866 --> 18:31.876
you'll be very impressed the
way--I was.

18:31.880 --> 18:34.970
Because that's the gap between
pure science and applied

18:34.968 --> 18:35.538
science.

18:35.539 --> 18:37.539
It makes all the difference.

18:37.539 --> 18:39.759
Here's what they did.

18:39.759 --> 18:47.859
These two wires feed into two
semicircular pieces of metal.

18:47.858 --> 18:50.588
Imagine that comes there,
that comes there.

18:50.588 --> 18:52.668
I've just blown it up for your
view.

18:52.670 --> 18:57.370
Then the battery is connected
like this, not with a hard wire,

18:57.374 --> 18:59.924
but with a spring loaded brush.

18:59.920 --> 19:02.390
There's a brush here,
there's a spring holding it.

19:02.390 --> 19:06.360
So these brushes make contact
with these two metallic

19:06.364 --> 19:09.424
hemispheres, if you like,
semi cylinders,

19:09.421 --> 19:12.251
with them connected to this
wire.

19:12.250 --> 19:13.480
Now you can see what happens.

19:13.480 --> 19:15.780
As the coil spins,
as shown here,

19:15.775 --> 19:20.075
the terminal is connected to
the top guy, the - to the bottom

19:20.077 --> 19:23.877
guy, so the current goes like
this and comes out.

19:23.880 --> 19:27.710
But half a revolution later,
this guy will be touching this

19:27.708 --> 19:31.668
one, that guy will be touching
this one, and the current will

19:31.666 --> 19:32.456
reverse.

19:32.460 --> 19:35.820
So the current reverses
automatically because you have

19:35.821 --> 19:39.501
this split circle and you have
the top half and bottom half

19:39.501 --> 19:41.471
split,
and then the current enters

19:41.472 --> 19:43.362
this metallic part and goes like
this,

19:43.358 --> 19:45.548
and then later on goes the
other way.

19:45.549 --> 19:47.709
That will keep running.

19:47.710 --> 19:49.920
Also, in real life what happens
is,

19:49.920 --> 19:56.640
once it flips over,
in a real system,

19:56.640 --> 19:59.050
made up of real coils,
once it comes to equilibrium,

19:59.048 --> 20:03.518
it won't stop there,
even without the reversal.

20:03.519 --> 20:07.639
It will go oscillating,
because it's got some moment of

20:07.644 --> 20:08.414
inertia.

20:08.410 --> 20:10.480
You just cannot lose your
potential energy and have

20:10.476 --> 20:11.506
nothing to show for it.

20:11.509 --> 20:13.699
So what will happen is,
it will overshoot its goal a

20:13.701 --> 20:16.541
little bit, and then it will get
carried away to the other side.

20:16.538 --> 20:20.568
Then it will keep going round
and round.

20:20.568 --> 20:24.288
Okay, so that is an application
of the torque.

20:24.288 --> 20:26.278
That is for a DC motor and
you're quite right.

20:26.278 --> 20:28.728
If you have an AC motor,
then you put an alternating

20:28.732 --> 20:31.652
current supply,
but then you'll have to make

20:31.647 --> 20:35.737
sure that the frequency of the
current supply will be the

20:35.743 --> 20:37.723
frequency of this motor.

20:37.720 --> 20:41.010
Okay, so let's leave this board
here, because this is the end of

20:41.009 --> 20:41.739
one section.

20:41.740 --> 20:44.560
You have to know where you
stand.

20:44.558 --> 20:46.778
We have done the first part of
magnetism,

20:46.779 --> 20:51.269
which is, what are the forces
and what are the torques on

20:51.269 --> 20:54.959
either moving charges or
electric currents,

20:54.960 --> 20:57.480
in some simple situations?

20:57.480 --> 21:01.170
Now we are going to come to the
more interesting part,

21:01.165 --> 21:04.915
also the more difficult part,
which is, how do currents

21:04.922 --> 21:06.872
produce magnetic fields?

21:06.868 --> 21:09.718
Now we're going to the cause of
the magnetic field.

21:09.720 --> 21:13.600
At the end, everything is
produced by charges,

21:13.598 --> 21:16.618
microscopic charges,
but the formula for that is

21:16.618 --> 21:20.338
quite difficult to calculate
because if you look at all the

21:20.344 --> 21:22.334
charges,
and they're moving around

21:22.333 --> 21:25.653
producing magnetic field,
the field at any one location,

21:25.648 --> 21:28.268
because of the delay of
interaction,

21:28.269 --> 21:31.789
depends on what they were doing
at various times in the past.

21:31.788 --> 21:34.818
Just like electric fields are
also difficult to calculate if

21:34.816 --> 21:37.736
charges are moving because of
the relativistic theory that

21:37.740 --> 21:40.870
says you cannot communicate
instantly with another place.

21:40.868 --> 21:44.418
You can only do it at the speed
of light, so the field here

21:44.419 --> 21:47.789
depends on what was going on
earlier for the more remote

21:47.785 --> 21:48.515
charges.

21:48.519 --> 21:51.549
But in electrostatics,
we beat the problem by saying,

21:51.545 --> 21:54.275
"Look, none of these
charges ever moved.

21:54.279 --> 21:56.069
They've been there
forever."

21:56.068 --> 21:59.198
Then where they are now is
where they were before and we

21:59.200 --> 22:00.680
can calculate the field.

22:00.680 --> 22:05.240
The analogous thing for
magnetism is to say that if you

22:05.244 --> 22:09.054
have currents and the currents
are steady,

22:09.048 --> 22:14.068
then it's a case where the
magnetic field is also going to

22:14.067 --> 22:15.297
be constant.

22:15.298 --> 22:18.148
Remember, a steady current
doesn't mean there's no motion.

22:18.150 --> 22:21.330
It means the motion of these
charges round and round the loop

22:21.327 --> 22:22.067
is constant.

22:22.069 --> 22:24.419
It's not changing with time.

22:24.420 --> 22:29.540
Do not confuse a steady current
with a single particle moving at

22:29.542 --> 22:31.172
a steady velocity.

22:31.170 --> 22:33.870
That is not a steady current.

22:33.869 --> 22:36.709
You see the difference?

22:36.710 --> 22:39.900
Because if you sit at any one
point in a wire in a closed

22:39.904 --> 22:42.134
loop, the current is always the
same.

22:42.130 --> 22:44.170
If you have the instrument,
called the ammeter,

22:44.167 --> 22:46.517
that reads the current,
you can put it where you like,

22:46.516 --> 22:47.886
you'll get the same answer.

22:47.890 --> 22:50.240
But if a single charge is
carrying the current,

22:50.243 --> 22:53.113
there's a current here,
there's nothing anywhere else.

22:53.108 --> 22:57.328
There's a current only where
there is a charge.

22:57.328 --> 23:00.238
So it's like saying that when I
go on the freeway at 40 miles an

23:00.240 --> 23:02.660
hour,
I do not myself constitute

23:02.659 --> 23:05.399
steady traffic,
because there is no traffic

23:05.400 --> 23:06.230
where I am not.

23:06.230 --> 23:07.670
It's the traffic with me now.

23:07.670 --> 23:09.410
When I'm over there,
there's traffic there.

23:09.410 --> 23:13.460
What we mean by steady traffic
is cars are flowing constantly

23:13.464 --> 23:16.714
so at any instant,
every place looks the same.

23:16.710 --> 23:20.090
So a steady current is what
we're talking about,

23:20.086 --> 23:22.956
not a charge going at steady
velocity.

23:22.960 --> 23:27.820
So the question is,
what is the magnetic field

23:27.816 --> 23:33.856
produced by a tiny piece of
current sitting somewhere?

23:33.858 --> 23:36.648
This is a piece of current,
I times dl

23:36.651 --> 23:37.711
, do you understand?

23:37.710 --> 23:40.030
You cannot have an isolated
piece of wire.

23:40.029 --> 23:43.499
It's part of a bigger loop,
but I'm focusing on a tiny

23:43.496 --> 23:44.146
section.

23:44.150 --> 23:52.380
And let us say it is located at
r'.

23:52.380 --> 23:58.320
Let's draw a picture so you
guys can see anything.

23:58.318 --> 24:03.518
I go to the point r',
and there I catch a piece of

24:03.522 --> 24:05.942
wire carrying a current.

24:05.940 --> 24:08.450
The current is I,
the segment is called

24:08.450 --> 24:09.510
dl.

24:09.509 --> 24:12.949
I want the field at the point
r.

24:12.950 --> 24:13.720
There's nothing here.

24:13.720 --> 24:21.690
I just want the field here,
and this vector separating them

24:21.692 --> 24:25.132
is r −
r'.

24:25.130 --> 24:33.010
So every segment of wire will
produce a little magnetic field,

24:33.009 --> 24:34.559
dB.

24:34.558 --> 24:37.808
And it's going to depend on the
orientation of the segment,

24:37.809 --> 24:41.279
the current it carries and the
separation between where you are

24:41.284 --> 24:42.914
and where the current is.

24:42.910 --> 24:49.130
And this is called the law of
Biot and Savart.

24:49.130 --> 24:52.770
It says the answer is
μ_0 over

24:52.772 --> 24:55.452
4Π times I
times

24:55.450 --> 25:00.580
dl x
e_rr`,

25:00.584 --> 25:04.974
divided by r −
r' squared.

25:04.970 --> 25:09.150
So let's take a minute to
understand this.

25:09.150 --> 25:10.710
μ_0/4Π,
like

25:10.707 --> 25:16.397
1/4Πε_0 is
a number whose value happens to

25:16.403 --> 25:20.083
be 10^(-7) in the units we are
using.

25:20.078 --> 25:23.058
That number is cooked up so
that if you've got 1 ampere here

25:23.057 --> 25:25.527
and 1 meter here and 10 meters
here and so on,

25:25.528 --> 25:30.368
you crank out all the numbers,
the field comes out in Tesla.

25:30.368 --> 25:32.418
So whenever you pick a system
of units,

25:32.420 --> 25:35.010
this has all the factors that
control the problem,

25:35.009 --> 25:37.669
you have to put a number in
front so that with all these

25:37.674 --> 25:40.894
measured in standard units,
this comes out in standard

25:40.886 --> 25:44.196
units and that number is
μ_0/4Π.

25:44.200 --> 25:46.330
That happens to be 10^(-7).

25:46.328 --> 25:51.328
r −r' is the separation
vector between where the current

25:51.326 --> 25:53.206
is and where you are.

25:53.210 --> 25:56.680
I is the current flowing in
that circuit.

25:56.680 --> 26:03.030
And e_rr` is
the unit vector from r'

26:03.031 --> 26:04.541
to r.

26:04.538 --> 26:06.798
You can call it
e_rr` or

26:06.801 --> 26:09.861
e_r'r ,
as long as you know it is from

26:09.855 --> 26:11.435
the cause to the effect.

26:11.440 --> 26:13.470
That's the formula.

26:13.470 --> 26:17.280
That is a nasty formula,
because unlike Coulomb's

26:17.279 --> 26:19.899
law--see, why is it more nasty?

26:19.900 --> 26:21.060
Have you thought about it?

26:21.058 --> 26:25.618
Why is it so much harder now
than for Coulomb?

26:25.618 --> 26:29.518
Remember, Coulomb's law was
just the unit vector divided by

26:29.520 --> 26:30.530
r^(2).

26:30.528 --> 26:34.748
Why are we having all of these
nasty cross products here?

26:34.750 --> 26:35.550
Yes?

26:35.548 --> 26:38.258
Student:
>

26:38.259 --> 26:40.599
Prof: What's making it?

26:40.598 --> 26:42.378
What's bringing in all the
extra vectors?

26:42.380 --> 26:46.190
Student:  The magnetic
field can never be worked

26:46.185 --> 26:48.085
>

26:48.088 --> 26:50.328
Prof: Right,
but what's the cause of the

26:50.325 --> 26:53.285
magnetic field compared to the
cause of the electric field?

26:53.289 --> 26:54.139
Yes?

26:54.140 --> 26:55.030
Student:  Moving charges.

26:55.029 --> 26:57.539
Prof: Right,
but the electric field is a

26:57.540 --> 26:58.360
point charge.

26:58.358 --> 27:00.908
There is no vector associated
with that point charge.

27:00.910 --> 27:01.670
It's just sitting there.

27:01.670 --> 27:05.190
The only vector you've got is
from the charge to the location.

27:05.190 --> 27:06.840
The current,
on the other hand,

27:06.838 --> 27:10.138
has got its own direction,
nothing to do with where it is.

27:10.140 --> 27:13.270
It's the way the little wire is
going at that point.

27:13.269 --> 27:16.539
It's the presence of the extra
vector and separation vector

27:16.541 --> 27:20.041
that gives you a chance to form
yet another vector by combining

27:20.040 --> 27:21.170
the two of them.

27:21.170 --> 27:22.920
That's where you get these
cross products.

27:22.920 --> 27:24.840
But anyway, that's how nature
is.

27:24.838 --> 27:27.028
That's what happens,
so we've just got to say what

27:27.031 --> 27:27.391
it is.

27:27.390 --> 27:30.900
So you've got to take this and
learn how to use it,

27:30.895 --> 27:32.855
so that's what we will do.

27:32.858 --> 27:39.698
So the first application of
this is going to be the field of

27:39.702 --> 27:44.692
a little circular loop of some
radius r,

27:44.690 --> 27:49.130
and I'm going to find the field
at a distance z on the

27:49.134 --> 27:50.324
z axis.

27:50.318 --> 27:52.408
Say the loop is in the
xy plane.

27:52.410 --> 27:58.370
I want the field here.

27:58.368 --> 28:01.768
So what you will have to
do--think about what you have to

28:01.772 --> 28:02.322
do now.

28:02.318 --> 28:04.518
You've got this formula,
but what do you think you will

28:04.516 --> 28:04.716
do?

28:04.720 --> 28:10.750
The current here is I.

28:10.750 --> 28:14.350
You guys have a game plan,
at least in principle?

28:14.348 --> 28:21.108
What will it take to calculate
the B field?

28:21.108 --> 28:23.028
Anybody from this wing has an
idea?

28:23.029 --> 28:23.819
Yes?

28:23.818 --> 28:27.788
Student:  Find the field
due to a point charge on the

28:27.785 --> 28:30.995
circle and then you integrate--
Prof: He said find a

28:30.998 --> 28:32.648
field due to a point charge and
then integrate.

28:32.650 --> 28:34.430
Except for one thing,
you are right.

28:34.430 --> 28:35.620
Student:
>

28:35.619 --> 28:36.269
Prof: That's right.

28:36.269 --> 28:41.949
Take a tiny line segment,
dl, here.

28:41.950 --> 28:44.350
Join at that point,
take the cross product.

28:44.348 --> 28:47.168
That's a vector you've got to
add to all the vectors coming

28:47.174 --> 28:48.934
from all the points in the
circle.

28:48.930 --> 28:52.040
That's your answer.

28:52.039 --> 28:53.679
So I'm going to show you.

28:53.680 --> 28:55.840
First of all,
is it clear to you that by the

28:55.837 --> 28:58.547
fact that the problem is
symmetric under rotations,

28:58.548 --> 29:02.288
with respect to this axis,
I can pick any one point and

29:02.285 --> 29:06.425
whatever it does can be used to
deduce what other points will

29:06.434 --> 29:06.924
do.

29:06.920 --> 29:08.250
They're all in the same footing.

29:08.250 --> 29:12.310
I want you to take this point
and this point for a minute.

29:12.308 --> 29:15.528
Here the current is coming out
of the blackboard,

29:15.529 --> 29:19.019
and here the current is going
into the blackboard.

29:19.019 --> 29:20.549
Are you with me now?

29:20.548 --> 29:23.418
The loop is half behind the
blackboard, half in front of the

29:23.416 --> 29:25.066
blackboard, and I slice it here.

29:25.068 --> 29:27.628
I just look at this guy and see
what it does.

29:27.630 --> 29:30.020
There's an element dl,
if you want,

29:30.019 --> 29:32.119
coming out of the blackboard
here.

29:32.118 --> 29:36.178
And the separation vector is
this one.

29:36.180 --> 29:40.090
If you take the cross product
of that vector with that vector,

29:40.088 --> 29:41.868
you've got to think about where
it will be,

29:41.868 --> 29:45.118
and it's got to be
perpendicular to this vector,

29:45.118 --> 29:47.248
therefore in the plane of the
blackboard.

29:47.250 --> 29:48.820
And it's got to be
perpendicular to this vector.

29:48.818 --> 29:53.968
And if you do your signs right,
it will give you magnetic

29:53.974 --> 29:58.214
contribution dB that
looks like that.

29:58.210 --> 30:01.730
But you know that the part of
dB that's going to

30:01.730 --> 30:05.710
survive is the component here,
because this guy will produce a

30:05.705 --> 30:07.005
field like this.

30:07.009 --> 30:09.829
And by symmetry,
they will have the same angle,

30:09.828 --> 30:12.708
so you only have to keep the
z component,

30:12.709 --> 30:15.099
not the one in the xy
plane.

30:15.098 --> 30:18.278
Because for every element I
find here, I can find a

30:18.278 --> 30:21.518
compensating one on the other
side of the circle.

30:21.519 --> 30:26.169
So therefore,
dB along the z

30:26.170 --> 30:30.140
direction is
μ_0I/

30:30.143 --> 30:36.283
4Π times the
segment coming out of the

30:36.282 --> 30:38.212
blackboard.

30:38.210 --> 30:39.370
What about the
sinθ?

30:39.368 --> 30:42.028
Sine theta, here's where you've
got to be very careful.

30:42.029 --> 30:48.879
What do you think the
sinθ is between the

30:48.884 --> 30:55.614
segment coming out of the
blackboard and this guy?

30:55.608 --> 30:58.098
What's the angle between a line
that lies in the plane of the

30:58.096 --> 31:00.416
blackboard, and the line coming
out of the blackboard?

31:00.420 --> 31:02.090
Student:  Ninety.

31:02.089 --> 31:02.769
Prof: Ninety.

31:02.769 --> 31:04.699
Don't be fooled by anything
else.

31:04.700 --> 31:07.370
Anything coming out of the
blackboard is orthogonal to

31:07.365 --> 31:09.475
anything in the plane of the
blackboard.

31:09.480 --> 31:11.820
The wire is coming out of the
blackboard, this vector's in the

31:11.821 --> 31:12.131
plane.

31:12.130 --> 31:14.950
There's no sine thetas to worry
about.

31:14.950 --> 31:19.350
But you've got to divide by the
square of the distance,

31:19.346 --> 31:23.576
and the square of the distance
happens to be z^(2)

31:23.579 --> 31:24.719
R^(2).

31:24.720 --> 31:26.640
That's the square of the
distance.

31:26.640 --> 31:31.940
That will give you this part,
but I want that part.

31:31.940 --> 31:35.360
So that's cosine of this angle
α.

31:35.358 --> 31:38.438
Now let's see where I can get
the same cosine.

31:38.440 --> 31:42.110
The angle between two lines is
the same as the angle between

31:42.109 --> 31:45.219
the perpendiculars,
so that guy is perpendicular to

31:45.221 --> 31:46.031
this guy.

31:46.029 --> 31:49.469
This guy is perpendicular to
that guy, so that will also be

31:49.465 --> 31:49.935
alpha.

31:49.940 --> 32:01.460
And cosine α
will be R divided by z^(2)

32:06.858 --> 32:08.918
So the sinθ didn't
come because of a

32:08.920 --> 32:10.680
sinθ in the cross
product.

32:10.680 --> 32:13.960
This factor,
it came because you want the

32:13.955 --> 32:17.965
projection of this vector up the
z axis.

32:17.970 --> 32:19.980
We are not done,
because this is a contribution

32:19.978 --> 32:22.248
from a tiny link dl coming out
of the blackboard,

32:22.250 --> 32:25.380
but notice the contribution is
independent of where you are on

32:25.382 --> 32:26.052
the circle.

32:26.048 --> 32:29.218
Everything makes a contribution
up the z axis.

32:29.220 --> 32:32.210
Each one makes a contribution
dl, so if you add up all

32:32.213 --> 32:34.513
of these, there's nothing to
integrate here.

32:34.509 --> 32:35.749
That's what I want you to
notice.

32:35.750 --> 32:37.160
z and R are fixed.

32:37.160 --> 32:40.650
You've just got to integrate
dl, which will be

32:40.654 --> 32:41.734
2ΠR.

32:41.730 --> 32:47.340
So I get μ_0
I/2ΠR squared,

32:47.338 --> 32:57.698
divided by 4Π,
divided by z^(2) R^(2)

32:57.700 --> 33:01.980
to the power 3/2.

33:01.980 --> 33:05.740
That is the magnetic field here.

33:05.740 --> 33:12.880
So along the axis,
it will go like this.

33:12.880 --> 33:17.360
And what is it at the center of
the loop?

33:17.358 --> 33:18.838
That's a useful formula to
remember.

33:18.838 --> 33:23.538
At the center,
you put z = 0,

33:23.538 --> 33:26.958
you get
μ_0I times

33:26.964 --> 33:33.294
2ΠR^(2),
divided by 4ΠR^(3).

33:33.288 --> 33:41.968
That gives me
μ_0I/2R

33:41.973 --> 33:44.873
Tesla.

33:44.868 --> 33:47.358
And it's going to be pointing
straight up.

33:47.358 --> 33:52.198
And if you go down here,
you will find it looks like

33:52.195 --> 33:52.855
this.

33:52.858 --> 33:55.358
Now if you want to find the
field somewhere else,

33:55.361 --> 33:57.711
it's a lot of work,
because you don't have the

33:57.705 --> 34:00.255
symmetries, but I will tell you
what you get.

34:00.259 --> 34:05.939
You will get these lines that
look exactly like--I'm not

34:05.935 --> 34:11.505
showing you the other half,
but that looks the same.

34:11.510 --> 34:16.780
Now that is supposed to remind
you again of a dipole.

34:16.780 --> 34:20.660
In fact, if you go to very,
very long distances,

34:20.655 --> 34:25.845
when z goes to infinity,
what does B look like?

34:25.849 --> 34:32.609
You go to the formula and let
z go to infinity,

34:32.608 --> 34:38.218
you will get--maybe I'll do it
again here.

34:38.219 --> 34:45.159
z goes to infinity,
B goes to

34:45.155 --> 34:51.425
μ_0
IΠR^(2),

34:51.429 --> 34:59.939
divided by 2Πz^(3).

34:59.940 --> 35:01.930
I'll tell you why I don't
cancel the pi's.

35:01.929 --> 35:06.469
I know you guys are dying to do
that, but ΠR^(2) times

35:06.474 --> 35:09.654
I is the dipole moment of
the loop.

35:09.650 --> 35:13.140
Remember, it's the area times
the current.

35:13.139 --> 35:17.249
So ignoring all the factors,
it looks like the dipole moment

35:17.250 --> 35:21.290
divided by the distance cubed,
that's exactly how the field

35:21.289 --> 35:23.519
went for an electric dipole.

35:23.518 --> 35:26.758
So the magnetic dipole and
electric dipole are,

35:26.760 --> 35:27.960
again, similar.

35:27.960 --> 35:30.680
So they're similar in two ways.

35:30.679 --> 35:33.829
Just like the electric dipole
aligns with the electric field,

35:33.827 --> 35:36.607
the magnetic dipole aligns with
the magnetic field.

35:36.610 --> 35:38.360
That's the first thing.

35:38.360 --> 35:42.030
The field of an electric dipole
has got these little circles

35:42.030 --> 35:45.390
going out and the field of a
magnetic dipole also looks

35:45.391 --> 35:46.761
exactly like that.

35:46.760 --> 35:49.280
The only difference is,
if you go into an electric

35:49.277 --> 35:52.307
dipole, you'll find two guys,
and -, producing the field.

35:52.309 --> 35:54.559
If you go into a magnetic
dipole, you'll find nothing at

35:54.557 --> 35:55.087
the center.

35:55.090 --> 35:58.000
You'll just find a loop.

35:58.000 --> 36:03.110
So nature gives us magnetic
dipoles, but not magnetic

36:03.110 --> 36:04.290
monopoles.

36:04.289 --> 36:05.819
That doesn't mean they're not
there.

36:05.820 --> 36:07.410
People have been looking for
them.

36:07.409 --> 36:11.129
They've been spotted once,
but it's not been spotted

36:11.126 --> 36:11.706
again.

36:11.710 --> 36:13.700
It could be because they're
very rare, and they're very

36:13.702 --> 36:15.882
heavy and they're very hard to
measure, or maybe they're not

36:15.880 --> 36:16.250
there.

36:16.250 --> 36:17.140
We don't know.

36:17.139 --> 36:21.129
But we don't see them the way
we see electric charges to be

36:21.130 --> 36:22.370
sure they exist.

36:22.369 --> 36:29.489
Okay, now suppose I take one
coil and put under it a second

36:29.494 --> 36:37.364
coil and a third coil and I join
them all up and wrap them around

36:37.358 --> 36:44.528
some little cardboard tube,
you can imagine then these

36:44.527 --> 36:51.377
fields will all add up and
you'll get a field that looks

36:51.380 --> 36:53.000
like this.

36:53.000 --> 37:03.660
That looks exactly like a
magnetic field around a bar

37:03.664 --> 37:05.514
magnet.

37:05.510 --> 37:06.230
Can you see that?

37:06.230 --> 37:07.730
You saw the field due to one.

37:07.730 --> 37:11.240
You can stack them up and then
you can compare the field.

37:11.239 --> 37:17.869
As far as a compass needle is
concerned, if you put it here or

37:17.871 --> 37:21.461
there, it behaves the same way.

37:21.460 --> 37:24.010
So now we have a question.

37:24.010 --> 37:27.770
Here is a magnetic effect due
to current carrying loop,

37:27.768 --> 37:29.228
or due to currents.

37:29.230 --> 37:31.950
Here is a magnetic effect due
to no currents.

37:31.949 --> 37:34.959
Your magnet's not connected to
anything, so you have an option.

37:34.960 --> 37:38.570
Either you can say that's a new
kind of magnetism produced by

37:38.574 --> 37:40.404
god knows what,
or you can say,

37:40.396 --> 37:43.046
"I believe that everything
is coming from electric

37:43.052 --> 37:45.022
currents,"
and the question is,

37:45.018 --> 37:48.908
where are the currents in this
guy?

37:48.909 --> 37:58.579
Do you know where the currents
are in a bar magnet?

37:58.579 --> 38:02.139
No?

38:02.139 --> 38:04.349
Okay, let me give you a hint.

38:04.349 --> 38:08.519
It comes from the atomic theory
of materials.

38:08.518 --> 38:09.918
Don't think of this as a lump
of something.

38:09.920 --> 38:10.590
Yes?

38:10.590 --> 38:11.900
Student:  The electrons.

38:11.900 --> 38:13.030
Prof: That's right.

38:13.030 --> 38:16.310
You know the atom has got
electrons going around,

38:16.307 --> 38:18.627
and every electron is a
current.

38:18.630 --> 38:25.280
Now you can ask,
let's put one electron here

38:25.284 --> 38:32.564
going like that,
then put another electron going

38:32.559 --> 38:39.989
like that, another electron
going like that.

38:39.989 --> 38:41.649
What do you think they all do
together?

38:41.650 --> 38:44.510
Suppose I've got nine electrons
in the plane of the blackboard

38:44.514 --> 38:45.974
all going around their atoms.

38:45.969 --> 38:48.509
You can see that in the region
between the atoms,

38:48.512 --> 38:50.262
they go in opposite directions.

38:50.260 --> 38:52.140
This guy's going like that,
this guy's coming like this.

38:52.139 --> 38:54.609
They cancel.

38:54.610 --> 38:57.900
The only thing that doesn't
cancel is along the perimeter.

38:57.900 --> 39:02.590
Everybody--what are these guys
doing?

39:02.590 --> 39:03.230
Here we go.

39:03.230 --> 39:04.950
Sorry.

39:04.949 --> 39:09.419
Yes, they do this.

39:09.420 --> 39:13.340
It's a physical fact that when
many orbits are put next to each

39:13.336 --> 39:16.876
other, they cancel on the
interior and give you something

39:16.876 --> 39:18.136
on the boundary.

39:18.139 --> 39:21.199
So even though each atom is a
very tiny loop,

39:21.195 --> 39:24.525
if you've got 10^(23) atoms,
they can look like a

39:24.530 --> 39:25.990
microscopic loop.

39:25.989 --> 39:28.419
Now that is just a two
dimensional loop,

39:28.418 --> 39:31.218
but you can take a solid,
a magnetic solid.

39:31.219 --> 39:34.009
It's go these little orbits
here.

39:34.010 --> 39:38.720
In every layer it's got orbits,
and each one is like a current.

39:38.719 --> 39:43.389
So magnetic material is like
having a sheet of current,

39:43.393 --> 39:47.813
and of course that will produce
a magnetic field.

39:47.809 --> 39:50.479
There are a lot of subtleties,
but this is roughly what

39:50.476 --> 39:50.966
happens.

39:50.969 --> 39:53.749
This is how you get magnetism.

39:53.750 --> 39:54.800
There are some questions.

39:54.800 --> 39:58.120
One of them is,
why isn't everything magnetic?

39:58.119 --> 40:00.059
Why not this guy?

40:00.059 --> 40:03.689
It's got electrons, right?

40:03.690 --> 40:06.950
Why aren't they lined up,
and which way should they line

40:06.952 --> 40:07.252
up?

40:07.250 --> 40:09.790
Which way should the magnet
point?

40:09.789 --> 40:12.429
So the answer is,
many materials may contain

40:12.429 --> 40:15.989
equal number of electrons going
in opposite directions,

40:15.989 --> 40:18.169
so that some are going one way,
some are going the other way,

40:18.170 --> 40:21.990
and for that reason,
each atom may cancel itself

40:21.992 --> 40:22.482
out.

40:22.480 --> 40:25.700
Another possibility is,
every atom may have one net

40:25.699 --> 40:29.049
uncancelled electrons,
but the plane of the orbit for

40:29.050 --> 40:30.790
each atom is different.

40:30.789 --> 40:33.159
Some are rotating in this
plane, some are rotating in that

40:33.155 --> 40:35.305
plane,
so there are little magnetic

40:35.307 --> 40:38.927
moment vectors pointing in
random directions and not lined

40:38.927 --> 40:39.307
up.

40:39.309 --> 40:42.459
And that happens because of
thermal agitation.

40:42.460 --> 40:44.560
Things like to jiggle when you
heat them.

40:44.559 --> 40:48.249
But when you cool them,
they can all line up and you

40:48.246 --> 40:51.206
suddenly find the material is
magnetic.

40:51.210 --> 40:55.030
So if you take a regular bar
magnet in your fridge and you

40:55.032 --> 40:58.452
put it on a hotplate for a
while, and you remove it,

40:58.452 --> 41:00.802
you'll find it's less magnetic.

41:00.800 --> 41:03.740
So it's magnetic because you
caused these guys to jiggle

41:03.735 --> 41:04.105
more.

41:04.110 --> 41:06.700
And beyond a temperature called
curie temperature,

41:06.699 --> 41:10.009
which depends on the material,
the jiggling will be so intense

41:10.007 --> 41:13.087
that magnetism is destroyed,
not atom by atom,

41:13.085 --> 41:18.095
but because they're no longer
able to act together and produce

41:18.096 --> 41:20.146
a net magnetic moment.

41:20.150 --> 41:23.600
Then you can also take
materials in which every atom

41:23.599 --> 41:27.729
has an electron that's willing
to line up, but they don't know

41:27.728 --> 41:29.418
which way to line up.

41:29.420 --> 41:31.430
They can line up in any way
they like,

41:31.429 --> 41:34.079
but then if you put them in a
strong magnetic field,

41:34.079 --> 41:36.819
then because every dipole likes
to line up with a magnetic

41:36.817 --> 41:39.127
field,
atomic moments line up with the

41:39.128 --> 41:40.098
magnetic field.

41:40.099 --> 41:43.009
And even if you take off the
magnetic field,

41:43.007 --> 41:44.967
the dipoles remain aligned.

41:44.969 --> 41:48.119
Do you know why they remain
aligned?

41:48.119 --> 41:51.509
Because they are producing
their own magnetic field which

41:51.514 --> 41:52.854
keeps them in place.

41:52.849 --> 41:55.669
That's called a self consistent
solution.

41:55.670 --> 41:57.210
So the magnets,
if they all agree,

41:57.211 --> 41:58.951
"Hey,
let's line up,"

41:58.952 --> 42:01.332
can actually make it a stable
situation,

42:01.329 --> 42:05.139
because when they do line up,
they produce their own magnetic

42:05.143 --> 42:07.753
field which encourages them to
line up.

42:07.750 --> 42:10.090
These are all interesting
things in magnetism,

42:10.090 --> 42:15.660
but the starting point always
is to find out how to take these

42:15.655 --> 42:19.665
microscopic moments and make
them line up.

42:19.670 --> 42:25.240
So now I'm going to do--so this
is at least an explanation of

42:25.239 --> 42:29.389
why bar magnets are magnetic,
because they are really

42:29.393 --> 42:33.123
circulating currents,
and it's no different from this

42:33.121 --> 42:33.861
current.

42:33.860 --> 42:42.010
Okay, now I'm going to find the
magnetic field for a very

42:42.012 --> 42:48.132
classic problem,
and that's the field of an

42:48.128 --> 42:50.748
infinite wire.

42:50.750 --> 42:55.310
So here is the infinite wire
and we want to find the field it

42:55.313 --> 42:56.153
produces.

42:56.150 --> 43:05.280
You take a point here,
and let's take some segment of

43:05.275 --> 43:08.605
length dx.

43:08.610 --> 43:15.400
And this is the separation
vector, and we want to find the

43:15.398 --> 43:19.448
field due to--dl is here.

43:19.449 --> 43:21.289
Now do the cross product.

43:21.289 --> 43:26.229
Turn screwdriver from dl
to R, it comes out of the

43:26.230 --> 43:27.250
blackboard.

43:27.250 --> 43:31.220
So this little guy makes a
contribution that comes out of

43:31.215 --> 43:32.415
the blackboard.

43:32.420 --> 43:35.610
In fact, you can check that,
as you go along the line,

43:35.606 --> 43:39.336
everything acts in the same way
and makes a contribution coming

43:39.335 --> 43:40.895
out of the blackboard.

43:40.900 --> 43:44.530
So what you can imagine is that
I just happened to slice it in

43:44.529 --> 43:48.039
this plane, but the real picture
is, the field lines will go

43:48.039 --> 43:49.229
around the wire.

43:49.230 --> 43:51.250
Right now they're coming out of
the blackboard.

43:51.250 --> 43:53.620
Down here they're going into
the blackboard,

43:53.617 --> 43:55.817
and they are orbiting around
the wire.

43:55.820 --> 43:59.800
If you look at the wire from
the end, then the current is

43:59.804 --> 44:01.304
coming towards you.

44:01.300 --> 44:03.080
The field lines will look like
this.

44:03.079 --> 44:07.429
And what we're trying to
calculate is how strong is that

44:07.427 --> 44:08.057
field?

44:08.059 --> 44:12.879
So you go back to the famous
law, dB =

44:12.876 --> 44:18.346
μ_0I/4Π
times dl,

44:18.349 --> 44:28.149
which is just dx,
divided by x^(2) a^(2).

44:28.150 --> 44:34.280
Let's say the distance is
a.

44:34.280 --> 44:38.200
Then you need the sine of the
angle between the separation

44:38.199 --> 44:42.759
vector of the segment,
and the sine of the angle,

44:42.760 --> 44:48.990
sinθ, is a divided
by square root of x^(2)

44:48.989 --> 44:50.409
a^(2).

44:50.409 --> 44:58.589
Then the integral of dB
is the integral of all these

44:58.592 --> 44:59.582
guys.

44:59.579 --> 45:04.069
Now you can guess what the
final answer is going to look

45:04.067 --> 45:07.247
like, just from dimensional
analysis.

45:07.250 --> 45:09.310
dx is a length,
a is a length.

45:09.309 --> 45:11.339
There are two lengths on the
top.

45:11.340 --> 45:12.840
The bottom there are three
powers of length,

45:12.840 --> 45:15.240
an x^(2) or a^(2)
from this one,

45:15.239 --> 45:16.969
and the square root of a length
squared,

45:16.969 --> 45:18.459
which is another length.

45:18.460 --> 45:19.410
You see that?

45:19.409 --> 45:22.039
Two powers of length at the
top, three in the bottom.

45:22.039 --> 45:24.909
The answer has to,
in the end, have dimensions of

45:24.914 --> 45:27.554
1 over length,
and the only length in town is

45:27.550 --> 45:27.910
a.

45:27.909 --> 45:32.589
So you know it's going to look
like μ_0I/a.

45:32.590 --> 45:35.690
But then it could be half of
a, or 10 times a.

45:35.690 --> 45:40.680
If you do that integral,
you'll find it's over

45:40.675 --> 45:42.445
2Πa.

45:42.449 --> 45:45.119
And I'm not doing the integral,
because we have seen this

45:45.121 --> 45:45.981
integral before.

45:45.980 --> 45:48.640
dx over x^(2)
a^(2), to the 3/2.

45:48.639 --> 45:51.879
You make the tangent
substitution and then get secant

45:51.876 --> 45:53.306
this and secant that.

45:53.309 --> 45:55.989
So I don't want to do that
integral again,

45:55.990 --> 45:59.130
but the answer is
μ_0I/2a.

45:59.130 --> 46:03.790
so the magnetic field from an
infinite wire falls from a

46:03.789 --> 46:06.499
distance like 1 over a.

46:06.500 --> 46:08.270
Very much like the electric
problem.

46:08.268 --> 46:11.038
Even though single electric
charge produces a field that

46:11.036 --> 46:12.896
goes like 1 over r
squared,

46:12.900 --> 46:15.780
if you remember,
infinite line produces a field

46:15.782 --> 46:17.852
that goes like 1 over r.

46:17.849 --> 46:20.059
Similarly, even though
elemental wire has a

46:20.061 --> 46:22.961
contribution that goes like 1
over distance squared,

46:22.960 --> 46:25.200
when integrated over an
infinite line,

46:25.199 --> 46:27.529
the final field goes like
a,

46:27.530 --> 46:33.320
1 over a,
where a is the distance

46:33.317 --> 46:36.837
from the axis of the wire.

46:36.840 --> 46:40.120
For a homework problem,
I want you to notice that in

46:40.121 --> 46:42.441
this integral from - to
infinity,

46:42.440 --> 46:47.170
the segment for the axis,
the segment for the - axis make

46:47.170 --> 46:49.030
equal contributions.

46:49.030 --> 46:51.440
In other words,
if you take this to be origin,

46:51.440 --> 46:54.760
that segment and that segment
contribute equally to the field

46:54.762 --> 46:57.802
at any point,
because it's an even function

46:57.804 --> 46:58.774
of x.

46:58.768 --> 47:02.048
You will find it useful when
you do the problem set.

47:02.050 --> 47:06.290
Okay, so here is an interesting
result, μ_0

47:06.291 --> 47:07.391
I/2Πa.

47:07.389 --> 47:10.199
Now that's going to explain a
few things.

47:10.199 --> 47:14.139
So here is a wire and I told
you long back when we started

47:14.141 --> 47:16.601
magnetism,
I said, if you put a test

47:16.599 --> 47:18.909
charge here and it moves like
this,

47:18.909 --> 47:22.459
it's attracted to the wire.

47:22.460 --> 47:26.070
Now we can understand their
attraction.

47:26.070 --> 47:29.450
We understand the attraction as
follows - look at the thing from

47:29.452 --> 47:31.682
end on,
the current is coming--what do

47:31.677 --> 47:34.767
you think the direction of the
magnetic field here is?

47:34.768 --> 47:37.928
Again, you put your thumbs
around this one,

47:37.934 --> 47:41.104
it's curling,
therefore it's going into the

47:41.099 --> 47:45.469
blackboard here and coming out
of the blackboard here.

47:45.469 --> 47:50.309
That is the azimuthal magnetic
field seen from the plane of the

47:50.313 --> 47:51.333
blackboard.

47:51.329 --> 47:54.399
You've got this guy going to
velocity v here.

47:54.400 --> 47:58.560
Take the cross product of
v with somebody going

47:58.556 --> 48:02.236
into the blackboard,
you'll find you get a force

48:02.242 --> 48:03.892
towards the wire.

48:03.889 --> 48:06.869
That's why the charged particle
going along the current is

48:06.867 --> 48:09.737
attracted to the current,
if it's positively charged.

48:09.739 --> 48:13.269
If it's going in the opposite
direction, if you reverse the

48:13.266 --> 48:15.756
velocity, you will get the
opposite force,

48:15.760 --> 48:18.010
so it will go away from the
wire.

48:18.010 --> 48:20.920
Finally, I want to mention the
experiment that people did

48:20.922 --> 48:23.992
before any of this which is to
notice that if you've got two

48:23.990 --> 48:25.600
currents,
l_1 and

48:25.599 --> 48:28.829
l_2,
they attract each other if they

48:28.831 --> 48:32.981
are parallel and they repel if
they're anti-parallel.

48:32.980 --> 48:36.040
So if you went and asked
yourself, "How am I going

48:36.041 --> 48:39.221
to find the force that's
attractive when they're parallel

48:39.215 --> 48:41.195
or they're anti-parallel?"

48:41.199 --> 48:43.449
you can think of all kinds of
simple rules,

48:43.452 --> 48:46.782
maybe involving this current
vector and that current vector.

48:46.780 --> 48:50.600
But the correct answer is a
pretty complicated series of

48:50.603 --> 48:53.943
cross products on why these two
wires attract.

48:53.940 --> 48:56.010
Let me tell you why they
attract.

48:56.010 --> 48:57.810
I think you can see yourself.

48:57.809 --> 49:01.389
Take the force on this guy due
to I_1.

49:01.389 --> 49:06.359
It produces a field that looks
like this there and looks like

49:06.364 --> 49:07.364
that here.

49:07.360 --> 49:12.030
So just like the single charge
here, this experiences a force

49:12.025 --> 49:15.365
towards the other wire,
every part of it.

49:15.369 --> 49:20.059
So the force that current 2
feels due to current 1 is

49:20.059 --> 49:24.389
μ_0I
_1/2Πa,

49:24.389 --> 49:26.489
where a is the distance between
them.

49:26.489 --> 49:30.769
That is my B.

49:30.768 --> 49:32.958
I, and that's my
l.

49:32.960 --> 49:35.340
If l_2
is the length of this wire.

49:35.340 --> 49:39.300
I'm just using the formula
BIl I told you

49:39.302 --> 49:39.892
about.

49:39.889 --> 49:42.579
So the B is the B
due to the first wire,

49:42.581 --> 49:45.531
I is the current of the
second wire, l is the

49:45.525 --> 49:47.045
length of the second wire.

49:47.050 --> 49:49.450
So if you took an infinite
second wire, you're going to get

49:49.454 --> 49:51.824
nonsense, because you'll get
infinity as the force between

49:51.815 --> 49:52.765
two infinite wires.

49:52.768 --> 49:55.378
So it's more common to take the
force.

49:55.380 --> 49:58.540
Let me drop the arrow on this,
because I'm looking at the

49:58.539 --> 50:00.119
magnitude per unit length.

50:00.119 --> 50:05.459
Then it is just this,
μ_0I

50:05.463 --> 50:12.113
_1I_2
over 2Πa.

50:12.110 --> 50:13.680
And the current,
you know, is measured in

50:13.677 --> 50:16.067
amperes,
which means if you took 1

50:16.072 --> 50:19.842
ampere in this wire and 1 ampere
in that wire,

50:19.840 --> 50:23.100
and they were 1 meter apart,
the force between them will be

50:23.096 --> 50:27.756
μ_0/2Π,
which will be 2 times 10^(-7)

50:27.764 --> 50:30.194
newtons per unit length.

50:30.190 --> 50:32.200
In other words,
if you're all holding this wire

50:32.204 --> 50:34.904
from running off to that wire,
and you are given 1 meter of

50:34.896 --> 50:39.146
the wire to take care of,
you will have to exert that

50:39.148 --> 50:39.938
force.

50:39.940 --> 50:43.680
So attraction between wires in
the end has an explanation,

50:43.675 --> 50:45.505
but it's very complicated.

50:45.510 --> 50:49.500
It does not simply involve the
two current directions.

50:49.500 --> 50:51.850
The first current,
through a series of cross

50:51.853 --> 50:54.233
products,
produces a magnetic field

50:54.226 --> 50:57.836
perpendicular to this wire,
and another cross product of

50:57.835 --> 51:01.405
that perpendicular field and the
current here gives you another

51:01.413 --> 51:03.783
vector which is the attraction
vector.

51:03.780 --> 51:05.840
So the force of attraction,
whereas force of attraction

51:05.840 --> 51:07.600
between two charges is simply
q_1

51:07.597 --> 51:09.427
q_2 over r
squared,

51:09.429 --> 51:11.289
force of attraction between two
wires,

51:11.289 --> 51:13.159
even though it has a simple
formula,

51:13.159 --> 51:16.449
comes in the end from a whole
bunch of cross products.

51:16.449 --> 51:28.049
That's what you have to know.

51:28.050 --> 51:45.170
All right, so now I'm going to
do a very basic result called

51:45.166 --> 51:49.516
Ampere's law.

51:49.518 --> 51:53.898
Ampere's law says the following
- look at the current coming

51:53.900 --> 51:54.940
towards you.

51:54.940 --> 52:00.470
The field lines go like this.

52:00.469 --> 52:04.289
Let me call that angle Φ.

52:04.289 --> 52:08.659
Then the magnetic field is a
unit vector in the Φ

52:08.655 --> 52:12.595
direction times
μ_0I divided

52:12.601 --> 52:16.711
by 2Πr,
where r is the distance.

52:16.710 --> 52:18.640
This is the wire, okay?

52:18.639 --> 52:21.649
And I am looking at various
circles.

52:21.650 --> 52:25.640
This is the circle of radius
r. Here is what B

52:25.639 --> 52:26.519
looks like.

52:26.518 --> 52:30.238
So B is circulating
around the wire,

52:30.235 --> 52:32.975
very different from E.

52:32.980 --> 52:37.990
E emanates from charges,
but has no circulation.

52:37.989 --> 52:41.389
Its line integral on a closed
loop is 0, electrostatically.

52:41.389 --> 52:43.419
The magnetic field is just the
opposite.

52:43.420 --> 52:46.430
It's always going around in
circles and therefore has a line

52:46.427 --> 52:48.387
integral,
and I will mention later on

52:48.389 --> 52:51.809
that it has no surface integral,
because Bs don't start

52:51.811 --> 52:52.881
and end anywhere.

52:52.880 --> 52:55.590
The lines of B,
therefore if you took any

52:55.588 --> 52:58.178
surface, whatever line comes in
has to go out,

52:58.182 --> 52:59.742
because it cannot stop.

52:59.739 --> 53:03.489
I'll come to that in a minute,
but let's look at the line

53:03.494 --> 53:06.114
integral of B around a
circle.

53:06.110 --> 53:11.800
B⋅dr
around a circle.

53:11.800 --> 53:17.500
So on a circle,
take a section of a circle.

53:17.500 --> 53:22.040
So here is my portion of a
circle.

53:22.039 --> 53:26.459
dR =
rdΦ times unit

53:26.463 --> 53:29.553
vector in the phi direction.

53:29.550 --> 53:31.140
Do you agree with that?

53:31.139 --> 53:33.809
The tangent vector to the
circle, a tiny little guy,

53:33.811 --> 53:36.381
has length r times
dΦ.

53:36.380 --> 53:39.370
r is this one,
and points in the direction of

53:39.371 --> 53:41.661
the unit vector in the phi
direction.

53:41.659 --> 53:44.519
So
e_Φ

53:44.518 --> 53:46.858
is unit vector like that.

53:46.860 --> 53:50.580
B is already this.

53:50.579 --> 54:00.439
Therefore B⋅e
_rdr looks

54:00.438 --> 54:06.118
like
μ_0I/2Π

54:06.119 --> 54:09.629
times integral...

54:09.630 --> 54:12.900
because the dot product of the
two vectors, because they're

54:12.896 --> 54:15.366
both azimuthal,
is simply the product of this

54:15.373 --> 54:16.673
guy times that guy.

54:16.670 --> 54:18.520
I'm sorry.

54:18.518 --> 54:27.218
Please change this R to
little r.

54:27.219 --> 54:30.849
In other words,
I want that distance to be

54:30.847 --> 54:32.437
little r.

54:32.440 --> 54:33.410
You can also use big R.

54:33.409 --> 54:35.979
It's going to cancel out,
but I'm not taking a fixed

54:35.981 --> 54:36.891
length r.

54:36.889 --> 54:38.919
I'm taking r to be
variable, so I want to use

54:38.916 --> 54:39.906
little r for it.

54:39.909 --> 54:41.719
So rdΦ is the
length.

54:41.719 --> 54:45.599
The point to notice is that the
field goes like 1 over r,

54:45.601 --> 54:47.821
the arc length goes like
r.

54:47.820 --> 54:49.320
So they cancel.

54:49.320 --> 54:51.950
Integral of
dΦ is

54:51.949 --> 54:55.819
2Π and I get
μ_0I.

54:55.820 --> 55:00.040
So the line integral of the
magnetic field are on a closed

55:00.038 --> 55:02.858
loop,
penetrated by a current,

55:02.860 --> 55:08.530
= μ_0 times
the current coming out of that

55:08.527 --> 55:09.547
surface.

55:09.550 --> 55:12.030
Independent of the radius of
the circle.

55:12.030 --> 55:14.890
You can take a small circle,
you can take a big circle,

55:14.885 --> 55:16.625
you always get the same answer.

55:16.630 --> 55:19.310
Because if you take a big
circle, you've got a big

55:19.313 --> 55:21.453
circumference growing like
r,

55:21.449 --> 55:25.579
but the field goes like 1 over
r and the effects cancel.

55:25.579 --> 55:30.659
This remind you of something
you've seen before?

55:30.659 --> 55:31.489
Which one?

55:31.489 --> 55:32.819
Student:  The plane.

55:32.820 --> 55:35.720
Prof: No,
where these things cancel.

55:35.719 --> 55:36.549
Pardon me?

55:36.550 --> 55:42.530
Student:  Gauss's law
_______________.

55:42.530 --> 55:44.080
Prof: Like Gauss's law,
you mean?

55:44.079 --> 55:44.829
Student:  Yes.

55:44.829 --> 55:45.859
Prof: Yes,
Gauss's law,

55:45.856 --> 55:48.866
where you took a single charge,
the surface integral of E

55:48.867 --> 55:51.967
was independent of the size
of the sphere,

55:51.969 --> 55:54.979
because the field went like 1
over r^(2).

55:54.980 --> 55:57.100
Area of the sphere went like
r^(2),

55:57.097 --> 55:59.987
and surface integral just said
it's equal to the q

55:59.989 --> 56:00.919
that's inside.

56:00.920 --> 56:03.770
Then of course we found it's
true, even if the shape is not

56:03.769 --> 56:04.359
spherical.

56:04.360 --> 56:07.480
Again, we find here,
first you take a circle and you

56:07.478 --> 56:11.208
find the answer is independent
of the radius of the circle.

56:11.210 --> 56:15.650
Then I'm going to show you that
the answer is independent,

56:15.648 --> 56:18.138
even of the shape of the loop.

56:18.139 --> 56:22.109
So let the current be here.

56:22.110 --> 56:28.050
So take a loop like that.

56:28.050 --> 56:35.900
So take a portion of that loop,
this portion here.

56:35.900 --> 56:41.850
It's got a radial part and a
tangential part.

56:41.849 --> 56:46.989
So you can see the dr of
the loop is dr times

56:46.987 --> 56:51.147
e_r
rdΦ times

56:51.152 --> 56:54.432
e_
Φ.

56:54.429 --> 56:56.019
Let me blow this up for you if
you like.

56:56.018 --> 57:01.088
So that is that segment there,
dr, and that's got an

57:01.092 --> 57:03.982
angular part and a radial part.

57:03.980 --> 57:08.060
The radial part is length
dr, the angular part is

57:08.059 --> 57:10.359
length rdΦ.

57:10.360 --> 57:13.660
So integrating along that
curve, which I can approximate

57:13.659 --> 57:16.929
by a straight line,
I can break up that elemental

57:16.931 --> 57:20.501
vector dr into the tiny portion
in the angular one,

57:20.500 --> 57:22.340
tiny portion in the radial one.

57:22.340 --> 57:27.530
But B =
μ_0I/2Πr

57:27.534 --> 57:32.274
times e_
Φ.

57:32.268 --> 57:35.048
So when you take
B⋅dr,

57:35.050 --> 57:36.190
e_r
⋅

57:36.188 --> 57:37.358
e_Φ is
0,

57:37.360 --> 57:39.560
because they are perpendicular
vectors.

57:39.559 --> 57:46.849
All you get then is
B⋅dr

57:46.846 --> 57:53.536
will be μ_0I
over 2Π

57:53.538 --> 57:56.808
dΦ.

57:56.809 --> 58:01.829
So the line integral of B
receives the contribution

58:01.827 --> 58:07.367
equal to simply the angle of the
two rays that bound this little

58:07.373 --> 58:08.433
segment.

58:08.429 --> 58:10.889
Then you can see if you go
around the whole loop,

58:10.889 --> 58:14.349
you will just add up all the
dΦ's and you

58:14.346 --> 58:16.546
will get 2Π and you
will get

58:16.550 --> 58:18.280
μ_0I.

58:18.280 --> 58:22.030
So the answer is true,
even if it is not a circle,

58:22.025 --> 58:25.845
as long as it is a closed loop,
enclosing the wire,

58:25.846 --> 58:29.206
the answer is equal to current
enclosed.

58:29.210 --> 58:41.110
What if you took a loop here
and the current's actually here?

58:41.110 --> 58:44.730
There's no current going out of
this guy, so we'd better get 0

58:44.731 --> 58:46.811
for that and we want to know
why.

58:46.809 --> 58:49.469
You'll get 0 because if you do
the same formula,

58:49.469 --> 58:50.909
it will give you integral of
dΦ,

58:50.909 --> 58:54.779
but notice if you go round here
and come back all the way here,

58:54.780 --> 58:56.190
the change in Φ is 0.

58:56.190 --> 58:58.270
Can you see that?

58:58.268 --> 59:02.158
In any loop bounded by some
lines, the range of Φ

59:02.163 --> 59:04.713
as you follow it goes back to
0.

59:04.710 --> 59:08.970
Whereas if you encircle the
origin, then Φ

59:08.969 --> 59:10.769
changes by 2Π.

59:10.768 --> 59:13.868
So if you take a point here and
you go around in a loop,

59:13.871 --> 59:15.791
you come back to the same
Φ.

59:15.789 --> 59:18.389
If you take a point here and
you go around,

59:18.393 --> 59:19.823
phi goes up by 2Π.

59:19.820 --> 59:23.200
That's why a loop not enclosing
a current will get 0

59:23.204 --> 59:24.204
contribution.

59:24.199 --> 59:28.639
A loop enclosing a current will
get μ_0I.

59:28.639 --> 59:32.839
And finally,
by superposition,

59:32.835 --> 59:40.355
if you've got many,
many wires coming out of this.

59:40.360 --> 59:41.870
Some are going in,
some are coming out,

59:41.869 --> 59:43.979
some are going in,
some are coming out,

59:43.980 --> 59:49.650
then integral of
B⋅dr =

59:49.646 --> 59:55.546
μ_0 times sum of
all the currents,

59:55.550 --> 1:00:16.130
I_j penetrates
the surface bounded by the loop.

1:00:16.130 --> 1:00:19.250
That's because B is made
up of B due to this guy,

1:00:19.250 --> 1:00:22.300
that guy, that guy,
but the integral of the total

1:00:22.300 --> 1:00:24.780
B is the individual
integrals.

1:00:24.780 --> 1:00:27.120
Each one gives you its own
current.

1:00:27.119 --> 1:00:31.029
So superposition is a great and
powerful result.

1:00:31.030 --> 1:00:33.650
If you can do anything for one
charge or one current,

1:00:33.652 --> 1:00:35.722
you're able to add everything
together.

1:00:35.719 --> 1:00:50.959
So this is called the law of
Ampere.

1:00:50.960 --> 1:00:57.090
Now I might as well tell you,
the surface integral of

1:00:57.094 --> 1:01:02.644
B on any closed surface
is actually 0.

1:01:02.639 --> 1:01:07.289
Because if you took electric
charges, they've got lines

1:01:07.286 --> 1:01:09.866
coming out like this,
right?

1:01:09.869 --> 1:01:13.459
And if you took a surface,
you get some contribution from

1:01:13.458 --> 1:01:17.238
the dot product of the field
vector with the area vector.

1:01:17.239 --> 1:01:19.919
Magnetic lines never end
anywhere.

1:01:19.920 --> 1:01:21.830
They always close.

1:01:21.829 --> 1:01:26.389
So take whatever surface you
want, any line that enters it

1:01:26.389 --> 1:01:30.629
will also leave it,
because there's nowhere to end.

1:01:30.630 --> 1:01:34.240
Had there been magnetic
charges, magnetic monopoles,

1:01:34.244 --> 1:01:38.364
lines will either come out of
the monopole or collapse into

1:01:38.355 --> 1:01:39.555
the monopole.

1:01:39.559 --> 1:01:42.689
Then the surface integral of
B will count the magnetic

1:01:42.688 --> 1:01:43.208
charges.

1:01:43.210 --> 1:01:44.680
But we have not found any.

1:01:44.679 --> 1:01:46.989
So as of now,
the surface integral of

1:01:46.994 --> 1:01:49.764
B is 0,
the line integral of B

1:01:49.757 --> 1:01:52.327
is the current trapped by that
loop.

1:01:52.329 --> 1:01:52.859
Do you understand?

1:01:52.860 --> 1:01:57.780
You take that loop and you draw
a plane with that loop as a

1:01:57.784 --> 1:01:58.724
boundary.

1:01:58.719 --> 1:02:02.479
Then any current coming out of
that is going to be equal to the

1:02:02.476 --> 1:02:04.716
line integral up to this factor
mu.

1:02:04.719 --> 1:02:07.089
For the electric field,
just for comparison,

1:02:07.090 --> 1:02:12.030
the line integral of the
electric field is 0 and the

1:02:12.027 --> 1:02:17.347
surface integral of the electric
field = the charge over

1:02:17.351 --> 1:02:19.871
ε_0.

1:02:19.869 --> 1:02:21.829
So here is this one integral.

1:02:21.829 --> 1:02:25.489
Let me write it here,
B⋅dr

1:02:25.485 --> 1:02:29.495
= μ_0 times all the
currents crossing it.

1:02:29.500 --> 1:02:34.860
So it's very useful for you to
write these 4 mathematical

1:02:34.860 --> 1:02:36.010
equations.

1:02:36.010 --> 1:02:47.620
These are called the integral
form of the Maxwell equations,

1:02:47.619 --> 1:02:50.849
namely, it's the relation
between electric and magnetic

1:02:50.851 --> 1:02:54.141
fields written in terms of the
integrals of electric and

1:02:54.143 --> 1:02:55.343
magnetic fields.

1:02:55.340 --> 1:02:57.210
There's something called the
differential form of the

1:02:57.208 --> 1:02:58.608
magnetic field which is more
useful,

1:02:58.610 --> 1:03:06.560
which requires more mathematics
than we have done in this

1:03:06.561 --> 1:03:07.841
course.

1:03:07.840 --> 1:03:11.510
But this is all for the case of
electrostatics and

1:03:11.514 --> 1:03:14.444
magnetostatics,
very, very important.

1:03:14.440 --> 1:03:17.060
All the currents have to be
constant;

1:03:17.059 --> 1:03:19.259
all the charges have to be
fixed.

1:03:19.260 --> 1:03:25.060
Only then is this true.

1:03:25.059 --> 1:03:28.099
It turns out this is all you
really need to determine

1:03:28.103 --> 1:03:29.923
electric and magnetic fields.

1:03:29.920 --> 1:03:32.950
Suppose someone comes to you
and says, "I'm going to

1:03:32.952 --> 1:03:34.472
give you all the currents.

1:03:34.469 --> 1:03:37.139
I'm going to give you all the
charges.

1:03:37.139 --> 1:03:39.619
Can you find E and
B?"

1:03:39.619 --> 1:03:41.229
It's a mathematical problem.

1:03:41.230 --> 1:03:44.510
The mathematical statement is
you know the line integral of

1:03:44.512 --> 1:03:47.972
either field around any loop and
you know the surface integral

1:03:47.967 --> 1:03:49.267
around any surface.

1:03:49.268 --> 1:03:52.468
It turns out that's all you
need.

1:03:52.469 --> 1:03:55.419
The line and surface integrals
are known for every possible

1:03:55.422 --> 1:03:58.632
loop and every possible surface,
then the solution is unique.

1:03:58.630 --> 1:04:03.450
That's why this is all you
really need to do any problem in

1:04:03.447 --> 1:04:04.857
magnetostatics.

1:04:04.860 --> 1:04:07.130
Because you might think,
maybe this guy's going to write

1:04:07.132 --> 1:04:08.002
two more equations.

1:04:08.000 --> 1:04:11.180
I'm saying mathematically,
this is complete.

1:04:11.179 --> 1:04:13.859
On the other hand,
if I stopped at one of them,

1:04:13.860 --> 1:04:16.820
except for the electric field,
if all I told you was this one,

1:04:16.820 --> 1:04:23.220
then mathematically it's not
enough to solve the problem.

1:04:23.219 --> 1:04:29.309
Okay, so I'm going to use
Ampere's law the way I used

1:04:29.313 --> 1:04:35.293
Gauss's law to show you some
simple applications.

1:04:35.289 --> 1:04:39.739
Suppose someone gives you an
infinite wire and says,

1:04:39.737 --> 1:04:43.137
"Find the field around
it."

1:04:43.139 --> 1:04:46.739
Now you can do that nasty
integral which I did on the top

1:04:46.744 --> 1:04:49.774
right corner,
but now we don't have to do it.

1:04:49.768 --> 1:04:55.078
Because you will argue by
symmetry that the field here

1:04:55.079 --> 1:05:00.989
cannot be pointing to the left
or pointing to the right,

1:05:00.989 --> 1:05:03.899
should lie in the plane,
because there's no reason to

1:05:03.902 --> 1:05:05.472
bias one way or the other.

1:05:05.469 --> 1:05:08.639
And in the plane perpendicular
to it, the field should be

1:05:08.639 --> 1:05:12.089
invariant when you rotate the
wire, because the wire looks the

1:05:12.094 --> 1:05:13.684
same when you rotate it.

1:05:13.679 --> 1:05:16.679
So the only possible things I
can think of are,

1:05:16.681 --> 1:05:20.271
the field can look like this,
or the field can look like

1:05:20.271 --> 1:05:20.861
this.

1:05:20.860 --> 1:05:24.600
They both have the property
that when you rotate the wire,

1:05:24.601 --> 1:05:25.981
they look the same.

1:05:25.980 --> 1:05:28.710
But you can rule this one out.

1:05:28.710 --> 1:05:31.870
One way to rule it out is to
say that if all the magnetic

1:05:31.867 --> 1:05:35.137
field lines end on the wire,
there must be magnetic charges

1:05:35.137 --> 1:05:36.657
and we don't have them.

1:05:36.659 --> 1:05:39.099
Another way which is more
subtle is to say,

1:05:39.099 --> 1:05:42.749
if this was the field
configuration for this current,

1:05:42.750 --> 1:05:46.860
if I reverse the current,
everything should reverse and

1:05:46.862 --> 1:05:49.532
the lines should go outwards
now.

1:05:49.530 --> 1:05:52.080
On the other hand,
if I took this infinite wire

1:05:52.081 --> 1:05:54.691
and I rotated it like that
around this axis,

1:05:54.690 --> 1:05:56.790
the current will point the
other way,

1:05:56.789 --> 1:06:00.689
but the field lines would still
keep going radially in.

1:06:00.690 --> 1:06:04.240
That means if I took that
current and I flipped it around,

1:06:04.235 --> 1:06:07.775
the field lines still go the
center, but I has become - I,

1:06:07.780 --> 1:06:09.710
so that rules this one out.

1:06:09.710 --> 1:06:13.410
But you can easily satisfy
yourself with this configuration

1:06:13.409 --> 1:06:16.919
of circling things will satisfy
all the requirements.

1:06:16.920 --> 1:06:20.760
So I know the field is
azimuthal, goes around the wire.

1:06:20.760 --> 1:06:23.470
I know it depends only on the
distance from the wire and not

1:06:23.472 --> 1:06:26.742
where I am in the circle,
therefore I look at the wire

1:06:26.737 --> 1:06:30.757
end on and I know that there's a
field here in the angular

1:06:30.762 --> 1:06:31.682
direction.

1:06:31.679 --> 1:06:35.919
I also know the line integral
of B⋅dr

1:06:35.923 --> 1:06:38.343
= μ_0I.

1:06:38.340 --> 1:06:39.970
This is like Gauss's law.

1:06:39.969 --> 1:06:42.629
You cannot find from it
B separately,

1:06:42.630 --> 1:06:45.980
except in the lucky case where
B⋅dr

1:06:45.978 --> 1:06:48.608
is a constant over the
entire circle,

1:06:48.610 --> 1:06:52.180
which it is in this case,
because it's simply B

1:06:52.177 --> 1:06:53.657
times 2Πr.

1:06:53.659 --> 1:06:56.439
If B is azimuthal,
which I've argued for,

1:06:56.438 --> 1:06:59.388
the line integral is B
times 2Πr =

1:06:59.394 --> 1:07:01.114
μ_0I.

1:07:01.110 --> 1:07:08.020
That means B = μ_0
I/2Πr.

1:07:08.018 --> 1:07:11.408
So what I'm telling you is just
like Gauss's law,

1:07:11.414 --> 1:07:13.824
will only give you one
equation.

1:07:13.820 --> 1:07:19.000
If you told me to find the
field due to an arbitrary

1:07:18.996 --> 1:07:23.356
distribution of currents,
I can still write this true

1:07:23.362 --> 1:07:26.512
statement,
but I cannot deduce from it the

1:07:26.507 --> 1:07:28.017
value of B.

1:07:28.018 --> 1:07:30.168
But if it's due to this
infinite wire and you've got

1:07:30.170 --> 1:07:32.570
enough symmetries and the only
thing you don't know is the

1:07:32.574 --> 1:07:34.604
magnitude of B as you
vary r,

1:07:34.599 --> 1:07:36.429
then on a contour of fixed
r,

1:07:36.429 --> 1:07:37.699
there's only one unknown,
which is,

1:07:37.699 --> 1:07:40.649
how big is B as a
function of r in the

1:07:40.648 --> 1:07:41.918
azimuthal direction.

1:07:41.920 --> 1:07:46.530
Then the one number you get
from this one equation.

1:07:46.530 --> 1:07:52.560
Now here's another variation of
that problem.

1:07:52.559 --> 1:07:56.969
They all have analogs in
electrostatics.

1:07:56.969 --> 1:08:03.579
Imagine that I have a solid
cable carrying current I,

1:08:03.583 --> 1:08:07.063
coming out of the blackboard.

1:08:07.059 --> 1:08:11.779
And let's say this cable has
radius R.

1:08:11.780 --> 1:08:17.310
Then I want the magnetic field
here.

1:08:17.310 --> 1:08:20.090
Now the real problem is much
more difficult than before,

1:08:20.091 --> 1:08:23.381
because there you had a single
line, you could do the integral.

1:08:23.380 --> 1:08:25.540
If you have a blob,
a cylindrical blob,

1:08:25.538 --> 1:08:28.838
you do all the dl cross
r's for everything is

1:08:28.836 --> 1:08:30.936
going to be very,
very difficult.

1:08:30.939 --> 1:08:35.219
But now we're going to argue by
symmetry that this one should

1:08:35.220 --> 1:08:39.360
also have necessarily fields
that go around in circles,

1:08:39.359 --> 1:08:42.219
with a uniform strength on a
circle of given radius.

1:08:42.220 --> 1:08:46.150
Then you conclude then that
2Πr times B =

1:08:46.154 --> 1:08:48.024
μ_0I.

1:08:48.020 --> 1:08:54.160
Therefore B =
μ_0I divided

1:08:54.158 --> 1:08:59.178
by 2Πr,
which is exactly the field you

1:08:59.176 --> 1:09:03.956
would get from a point wire
carrying all the current.

1:09:03.960 --> 1:09:06.440
Again, this should remind you
of Gauss's law,

1:09:06.439 --> 1:09:10.099
but if you have a ball of
charge, the field due to it

1:09:10.101 --> 1:09:14.541
looks like the field of a point
charge sitting at its center.

1:09:14.538 --> 1:09:17.328
Likewise the field of a uniform
distribution,

1:09:17.328 --> 1:09:19.778
a tube of current coming
towards you,

1:09:19.779 --> 1:09:23.729
outside this tube looks like
it's all coming from the center,

1:09:23.729 --> 1:09:27.229
because that's what it will
look like if the entire current

1:09:27.228 --> 1:09:29.278
were concentrated on the center.

1:09:29.279 --> 1:09:31.429
But you will see the difference
now.

1:09:31.430 --> 1:09:36.270
If you go inside this and ask,
what's the magnetic field on a

1:09:36.270 --> 1:09:40.150
circle lying inside this,
then again you can say

1:09:40.149 --> 1:09:44.269
2Πr times B =
μ_0 times the

1:09:44.270 --> 1:09:47.290
current that's enclosed by that
loop,

1:09:47.289 --> 1:09:49.389
not the entire current.

1:09:49.390 --> 1:09:54.220
And I think by now you guys
should know, that will be this

1:09:54.217 --> 1:09:56.927
fraction of the total current.

1:09:56.930 --> 1:10:00.040
Or if you want to see the
details, you can put a pi there

1:10:00.036 --> 1:10:01.586
and you can put a pi here.

1:10:01.590 --> 1:10:04.000
I/ΠR^(2) is
the current density and

1:10:04.003 --> 1:10:06.043
ΠR^(2) is the area of
this guy.

1:10:06.038 --> 1:10:09.248
But by now you guys should know
that things scale like distance

1:10:09.246 --> 1:10:11.156
squared if you're looking at
areas.

1:10:11.159 --> 1:10:12.469
So what does that tell me now?

1:10:12.470 --> 1:10:15.640
It's
μ_0Ir^(2)

1:10:15.641 --> 1:10:17.951
over big R^(2).

1:10:17.948 --> 1:10:21.478
So for B,
it looks like

1:10:21.479 --> 1:10:28.539
μ_0Ir
divided by 2ΠR^(2),

1:10:28.538 --> 1:10:31.268
with r less than or
equal to R.

1:10:31.270 --> 1:10:34.530
And this is the formula for
r bigger than or equal to

1:10:34.530 --> 1:10:35.140
R.

1:10:35.140 --> 1:10:41.630
And you can test the formula
when little r = R.

1:10:41.630 --> 1:10:42.520
That's correct.

1:10:42.520 --> 1:10:45.970
Little r = R, they agree.

1:10:45.970 --> 1:10:51.660
Therefore the magnetic field as
a function of r grows

1:10:51.662 --> 1:10:56.682
inside the cylinder and then
decays as 1/r.

1:10:56.680 --> 1:10:59.540
All this should be familiar to
you if you still remember

1:10:59.537 --> 1:11:01.407
similar problems from Gauss's
law.

1:11:01.408 --> 1:11:03.988
If you've got a ball of charge,
the electric charge grows

1:11:03.988 --> 1:11:05.828
linearly when you're inside the
ball,

1:11:05.828 --> 1:11:08.318
because as you grow,
you're enclosing more and more

1:11:08.320 --> 1:11:10.100
charge,
which is growing like

1:11:10.103 --> 1:11:12.553
r^(3),
but the sphere is growing like

1:11:12.554 --> 1:11:13.274
r^(2).

1:11:13.270 --> 1:11:16.200
So it grows like r,
but once you cross the sphere,

1:11:16.198 --> 1:11:18.188
you go to bigger and bigger
spheres,

1:11:18.189 --> 1:11:20.429
you're not enclosing more
charge, but the sphere for

1:11:20.430 --> 1:11:23.690
Gauss's law is getting bigger,
so it falls like 1/r^(2).

1:11:23.689 --> 1:11:27.289
Similarly here,
it falls like 1/r when

1:11:27.291 --> 1:11:31.471
you're outside the cylinder,
and grows linearly with

1:11:31.466 --> 1:11:35.146
r when you're inside the
cylinder.

1:11:35.149 --> 1:11:39.559
Now there are a couple of other
miscellaneous applications which

1:11:39.560 --> 1:11:41.100
I will do next time.

1:11:41.100 --> 1:11:46.000