WEBVTT

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Prof: Okay, let's start.

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I'm going to do some leftover
stuff from circuits,

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because I didn't finish it last
time.

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So I want you to remember the
following things about a

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circuit.

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I'm always thinking of
different ways to explain it.

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So the things we normally learn
is,

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if you've got some battery
here, some number of volts,

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let's give it the symbol
E, and you have a

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resistor,
we just say there's a 1.5 volt

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difference.

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So you can say this is 0 this
is 1.5.

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Current goes around and comes
back.

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And we say the work is done,
because the battery is somehow

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making the current go around.

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But we look under the hood and
found it's somewhat more

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complex.

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It is not done entirely with
the electrostatic force.

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The electrostatic force is due
to these charges here,

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sitting on the plate,
but they cannot do work for a

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charge that goes round and
round.

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Because if you go round in a
full loop in an electrostatic

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field, the net work done by the
field is 0.

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That's because the net work
done is the integral of the

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electrostatic field and because
it's conservative,

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you'll get 0.

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Yet when charges go round and
round the circuit,

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they are doing some work,
because the resistor is heating

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up, delivering heat.

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And the hidden secret of course
is not the electrostatic force,

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but another force inside the
battery that's doing the job.

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So I gave an analogy which I
think is worth repeating.

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So here is a ski slope and you
start here and you come down.

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Let's imagine on the way down
you bump into a lot of trees,

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that you pretty much come down
at 0 speed.

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Then you stagger back to the
foot of this ski lift.

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Now that's the end of your
skiing.

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There's nothing else,
because gravity is not going to

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help you anymore,
because gravity is not going to

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help you go from here to there.

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So it's not obviously run by
just gravity.

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That's when you have the ski
lift, which I'm going to portray

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as a vertical thing,
where you've got a lot of

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chairs, so you can sit on one of
them and you can ride to the

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top.

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So that ski lift exerts a force.

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Gravity, which is acting down
here,

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is also acting down here,
and the ski lift,

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F_ski and
F_gravity, ski

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lift cancels gravity here and
takes you to the top.

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So if you define the analog of
an emf, there is no

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"electro"
about this one.

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Let's just call it emf.

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As the integral of the forces
acting on this one skier,

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as he or she goes around the
loop, you can write that as the

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force due to gravity ⋅
dr and the force due to the

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lift--
or what did I call it?

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Hey guys, you know what this
was called?

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Student:  Ski.

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Prof: Ski.

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Okay, ski dot dr.

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this is no good.

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Gravity does something to you
when you come down and takes it

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all back when you go up.

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This guy, the force inside the
ski, has a non-zero line

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integral, because it's a
non-zero force only in here.

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The line integral's basically
the ski force times the

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distance.

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The ski force is just
mg, the magnitude.

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That times the distance is
mgh.

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That's why you can say this
point is a potential mgh

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that lifts up to that.

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But inside this region,
gravity is pointing down and

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the machine takes you against
gravity and puts you on the top.

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And that's the non-conservative
force that does the work.

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So the main point was,
electrostatics is conservative,

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and yet when a charge goes
around and around the loop,

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the line integral of the total
force is not 0,

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so it's not entirely driven by
electrostatics,

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but electrostatic forces are
important.

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Just like gravity's important
here, because gravity's what

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helps you come down here.

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The ski lift is all hidden
inside, and then it takes you

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back up.

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So I'm going to draw the
circuit in a way that really

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indicates the gravitational
analogy.

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So I'm going to draw it not
this way but this way.

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Here's the top plate and here's
the bottom plate and you may

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directly visualize the height
here to be the potential.

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Then there is a wire that has
no resistance.

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It's going to be flat.

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Potential doesn't have to be up
or down for current to flow down

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this.

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Then here is a resistor.

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Then there's another little
wire, which is pretty much flat

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and joins at the bottom.

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There are charges piled up
here, there are charged piled up

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here,
and they will produce an

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electrostatic field that will do
something there,

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but inside, they're bad.

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Electrostatic field goes down
here.

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So if this is the whole story,
you cannot do anything,

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because maybe if you charged it
once,

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it will go round and come back
here and it's stuck,

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because it cannot go up.

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That's when the tiny little
hands of the chemical forces

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will come and drag the charge,
against this field.

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That's the F_chemical
force.

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And it's the line integral of
the chemical force around a

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closed loop that's called emf.

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If you write the chemical force
per unit charge,

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integrate over a closed loop,
that's what we call the emf.

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But that's very easily computed.

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That's what I didn't realize,
a much easier way to do the

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computation.

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This thing is just a product of
that force times the distance,

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but in magnitude,
that force precisely cancels

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the downward electric force,
just like the lift cancels the

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force of gravity.

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So the line integral in
magnitude is just the electric

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field times the distance between
the two plates,

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and that is just the voltage.

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Therefore the voltage
difference between this point

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and this point will be the emf.

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Now if you don't want to know
the whole details,

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you can go back to drawing
pictures this way,

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and if you have a circuit like
this,

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and maybe there's a switch
here, you can be assured that

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between this point and this
point,

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if you go around and measure
the voltage difference,

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that will be the emf of the
battery.

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That's all you really need to
know.

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The way it comes out is that
this battery is like a pump that

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lifts you up by an electrical
height equal to 1.5 volts.

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The emf is literally the
voltage difference.

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Yes?

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Student:  What's the
symbol under

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F_c in the
first?

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Prof: Charge.

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Because electric field is the
force per unit charge.

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We should take whatever force
there is and write it as a force

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per unit charge.

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All right, so I wanted to do
some simple problems.

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This stuff you've done already
in high school,

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but I want to go over it in
view of everything we have

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learned.

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So here's the simplest circuit
in the world.

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You have an emf.

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There is a resistor and it
comes right back.

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You want to know what current
is flowing in the circuit.

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So the logic,
there are two equations you

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write now.

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One equation is,
if you start at any point on

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the circuit and you go around a
path and you come back to the

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same point,
the change in voltage should be

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zero,
because voltage is like height.

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You can go anywhere you want
and you can come back,

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then the change in height
should add up to zero.

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If you want,
sum of all the change in

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voltage should be zero around a
closed loop.

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Second thing is,
current is conserved.

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These are the only equations
you need.

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Current doesn't disappear.

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It's uniform here.

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So current I comes here,
drops through the resistor and

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comes out of the other side.

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So I'm going to write the
following equation.

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I'm going to start at this
point and keep track of all the

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changes in voltage.

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First when I go around this
battery, I told you many times,

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the change in voltage will be
just E.

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When I come here,
current flows downhill.

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I know the voltage drop here is
RI, but if I follow it,

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I'm dropping in potential.

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It's a skier;
it's going downhill.

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It's a ski lift where it's
going up.

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And then you come back here to
where you started,

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the total change must be 0.

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Now this is pretty trivial.

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You probably wrote this without
any further thought,

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but we then just say we get
from that E = IR.

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Then we're going to do a more
complicated problem,

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one resistor,
second resistor,

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R_1,
R_2.

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Again, if you apply an emf
here, that current is going to

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go through this guy and go
through that guy,

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and it's the same current
because it has nowhere to go.

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So the equation now will be
E, which is increased

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when you go up here.

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Then you drop
R_1 times

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I R_2
times I and that equals

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0,
so current is

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E/(R_1
R_2).

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That means if you put two
resistors in series,

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it's equal to a single resistor
equal to their sum.

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In other words,
if these two were enclosed in a

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black box,
you didn't know what's inside,

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you've got two leads coming out
and someone says,

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"Hey, what's inside the
black box?

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What's the resistor
inside?"

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you will apply voltage,
you will measure the current

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and that ratio you will call
R.

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E/I = R,
but here, you can see

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E/I =
R_1

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R_2.

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So that's the value you will
ascribe to the two guys in

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series.

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Now if you put them in
parallel--I think again I'm

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repeating it,
but maybe it's harmless right

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now--so here is one resistor,
here's the second resistor.

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This is R_1,
this is R_2.

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This is some E.

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If you put this inside a black
box and someone says,

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"Hey, tell me what's
inside,"

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you don't know.

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It may not be this,
but there'll be some effective

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resistance.

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And we compute that by saying
let a current I leave this

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battery here,
do whatever it wants and come

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out of the other side.

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Inside the box,
this current will split into an

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I_1.

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I can call it
I_2,

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but I'm just going to call it
I - I_1,

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because that's what
I_2 should be.

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And it comes out of this side.

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So let me say the total current
= I_1--I

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changed my mind,
guys.

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Sorry.

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Let me call this
I_2.

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The total current is
I_1

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I_2.

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But I_1 is
V/R_1 because

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the voltage V is acting
across these two ends.

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The voltage V is also
acting across those two ends,

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so it will be
V/R_2,

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and that's going to be equal to
V divided by whatever the

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resistance inside the box is,
by definition.

11:47.038 --> 11:50.538
If you compare the two,
you find 1/R is

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1/R_1
1/R_2.

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I mention this just to tell you
that it comes from current

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conservation,
where I used I,

11:58.797 --> 12:00.857
is I_1
I_2.

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We might as well deal with the
two others, one other circuit

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element, which is the capacitor.

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So suppose someone takes two
capacitors.

12:14.210 --> 12:15.620
This guy is
C_1,

12:15.620 --> 12:17.210
this guy is
C_2,

12:17.210 --> 12:21.360
puts them in a black box and
says, "Tell me what's

12:21.356 --> 12:22.506
inside."

12:22.509 --> 12:28.879
So you will take your battery,
apply voltage E,

12:28.875 --> 12:34.395
then see how much charge leaves
the battery.

12:34.399 --> 12:39.179
Then the capacitance will be
the charge that leaves the

12:39.177 --> 12:43.067
battery divided by the emf or
the voltage.

12:43.070 --> 12:45.950
If a charge comes from here,
a charge Q_1

12:45.947 --> 12:48.517
will go from there and
-Q_1 will go

12:48.524 --> 12:49.034
there.

12:49.029 --> 12:52.619
Q_2 will go
there and -Q_2

12:52.621 --> 12:53.581
will go there.

12:53.580 --> 12:59.610
So the Q that's going in
will be Q_1

12:59.607 --> 13:02.017
Q_2...

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over E.

13:04.149 --> 13:09.629
But Q_1 = C_1
times E.

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Q_2 =
C_2 times

13:11.544 --> 13:12.064
E.

13:12.058 --> 13:17.498
So divided by E, you'll
get C_1

13:17.496 --> 13:19.626
C_2.

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This will tell you that the two
capacitors in parallel will act

13:23.705 --> 13:27.645
like a single capacitor whose
value is the sum of the two.

13:27.649 --> 13:34.119
That's actually very easy to
understand because here's one

13:34.118 --> 13:35.478
capacitor.

13:35.480 --> 13:37.450
Here's another guy.

13:37.450 --> 13:40.020
Let me just glue those plates
together.

13:40.019 --> 13:43.009
It doesn't matter because they
are the same voltage.

13:43.009 --> 13:44.339
And you don't need these two
wires.

13:44.340 --> 13:45.860
You can just dump that and dump
that.

13:45.860 --> 13:47.340
Current can just come in this
way.

13:47.340 --> 13:50.840
You've got a single capacitor
with the area equal to the sum

13:50.841 --> 13:51.971
of the two areas.

13:51.970 --> 13:54.030
Capacitance was
ε_0

13:54.027 --> 13:55.167
A/d.

13:55.168 --> 13:59.128
They have the same d,
but the areas add up so

13:59.130 --> 14:00.840
capacitances add up.

14:00.840 --> 14:03.630
And the final thing which I
don't want to do,

14:03.629 --> 14:05.519
because I don't want to move
onto more interesting things,

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is when you put two capacitors
in series,

14:08.769 --> 14:11.809
what happens is,
you can take it as a homework

14:11.807 --> 14:13.727
problem,
that if you put a charge

14:13.729 --> 14:16.379
Q on this guy,
and a -Q comes out of

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the other terminal,
these two guys have no choice

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but to take -Q and
Q,

14:21.658 --> 14:24.588
because these two plates are
isolated from the world.

14:24.590 --> 14:27.210
The charge they can have has to
be equal and opposite,

14:27.214 --> 14:29.844
but -Q has to balance
this, and Q has to

14:29.837 --> 14:30.677
balance that.

14:30.678 --> 14:34.538
Now if you use the fact that
the voltage drop on this one the

14:34.537 --> 14:37.427
voltage drop on that one is the
total emf,

14:37.428 --> 14:44.428
you will find 1/C =
1/C_1_

14:44.432 --> 14:48.662
1/C_2.

14:48.658 --> 14:52.178
So capacitors and resistors
combine in exactly the opposite

14:52.184 --> 14:52.554
way.

14:52.548 --> 14:54.458
This is just in series,
they're additive;

14:54.460 --> 14:56.770
capacitors in parallel are
additive.

14:56.769 --> 14:58.749
Whereas for resistance in
parallel,

14:58.750 --> 15:02.270
you add their inverse to get
the inverse of the total,

15:02.269 --> 15:07.149
and capacitors in series,
you add the inverses to get the

15:07.145 --> 15:09.145
inverse of the total.

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So now I want to do one
slightly more interesting

15:12.120 --> 15:12.740
problem.

15:12.740 --> 15:14.470
It looks like this.

15:14.470 --> 15:18.340
You have a voltage E
here.

15:18.340 --> 15:25.090
Then you have a capacitor C
and a resistor R,

15:25.086 --> 15:31.366
and there's a little switch
which we will close at some

15:31.365 --> 15:32.525
point.

15:32.528 --> 15:35.148
So we want to know what will
happen here.

15:35.149 --> 15:38.339
Initially you are told the
capacitor's empty,

15:38.342 --> 15:40.232
there's no charge on it.

15:40.230 --> 15:44.570
Once you close the switch,
you should sort of imagine what

15:44.572 --> 15:45.642
will happen.

15:45.639 --> 15:47.549
This plate is full of positive
charges.

15:47.549 --> 15:51.639
They don't want to be there.

15:51.639 --> 15:52.679
And this is full of negative
charges.

15:52.679 --> 15:53.829
They don't want to be there.

15:53.830 --> 15:56.250
So positive charges would like
to go there, negative would like

15:56.250 --> 15:57.930
to come around,
but the switch did not allow

15:57.928 --> 15:58.278
that.

15:58.279 --> 16:03.079
But now that it's closed,
some charges will leave this

16:03.080 --> 16:08.600
and some negative charges will
be formed on the other side and

16:08.604 --> 16:11.054
they'll come back here.

16:11.048 --> 16:14.668
It's very interesting to note
that current doesn't really flow

16:14.666 --> 16:16.086
through the capacitor.

16:16.090 --> 16:19.100
Instead what happens is,
positive charges come here,

16:19.104 --> 16:22.714
and positive charges leave that
plate, leaving behind negative

16:22.708 --> 16:23.358
charge.

16:23.360 --> 16:26.950
And that current will come
here, that's the current

16:26.945 --> 16:29.235
I, and goes back there.

16:29.240 --> 16:32.580
But once the capacitor begins
to charge, it bites the hand

16:32.581 --> 16:34.991
that feeds it,
because it's then trying to

16:34.985 --> 16:36.505
drive its own current.

16:36.509 --> 16:37.879
If you ask this guy,
"What do you want to

16:37.884 --> 16:38.224
do?"

16:38.220 --> 16:40.370
he'll say, "I want to
drive the current that

16:40.370 --> 16:40.910
way."

16:40.908 --> 16:44.868
The capacitor will start
opposing the external voltage,

16:44.870 --> 16:46.720
and as it charges up more and
more and more,

16:46.720 --> 16:51.150
eventually, its voltage will
equal the applied voltage and

16:51.154 --> 16:53.414
then the current will stop.

16:53.408 --> 16:57.298
And when that's happened,
the charge on this capacitor at

16:57.303 --> 16:58.643
the end,
I'm going to call it

16:58.638 --> 17:01.428
Q_infinity,
you'll see why,

17:01.432 --> 17:05.712
will be C times
E.

17:05.710 --> 17:11.460
Then the current would stop.

17:11.460 --> 17:14.750
So let's understand in detail
what happens to current from the

17:14.749 --> 17:16.259
time you close the switch.

17:16.259 --> 17:19.819
So you write down the basic
equation which says start here,

17:19.816 --> 17:23.616
go around and keep track of the
change in voltage and equate it

17:23.617 --> 17:24.167
to 0.

17:24.170 --> 17:27.690
So I get a E when I go
through this battery.

17:27.690 --> 17:32.390
When I come from here to here,
I drop by an amount

17:32.391 --> 17:34.311
Q/C.

17:34.308 --> 17:37.248
Then I drop by an amount
RI.

17:37.250 --> 17:41.780
Then I'm back to where I
started, which was my 0.

17:41.779 --> 17:44.179
That's the equation.

17:44.180 --> 17:48.730
So now you can write this
equation as--now what's the

17:48.731 --> 17:52.761
relation between I and
Q here?

17:52.759 --> 17:55.569
You should think about it.

17:55.568 --> 17:58.908
When we drain the capacitor,
remember, I did a problem

17:58.913 --> 18:02.513
earlier on where I only had a
capacitor and a resistor.

18:02.509 --> 18:05.089
There the current was
−dQ/dt

18:05.087 --> 18:07.937
because as the current flows,
the capacitor's getting

18:07.941 --> 18:08.601
drained.

18:08.598 --> 18:10.738
Here actually the current is
dQ/dt,

18:10.744 --> 18:12.984
so if you're not careful,
you'll get in trouble.

18:12.980 --> 18:15.210
It's dQ/dt,
because any charge that piles

18:15.210 --> 18:17.960
up on the capacitor leaves the
other plate and goes through

18:17.961 --> 18:18.391
this.

18:18.390 --> 18:22.610
So the current flow is the
charging of the capacitor so

18:22.612 --> 18:25.352
I is in fact
dQ/dt.

18:25.348 --> 18:28.278
Otherwise if I is
positive and the current is

18:28.281 --> 18:30.761
going this way,
charge is building up on the

18:30.755 --> 18:31.555
capacitor.

18:31.558 --> 18:37.808
So if you remember that,
you get the equation E

18:37.810 --> 18:41.350
= Q/C R dQ/dt.

18:41.348 --> 18:49.288
So that's the equation you want
to solve.

18:49.288 --> 18:51.188
So how do we think of this
equation?

18:51.190 --> 18:53.750
We tell ourselves if we wait a
long,

18:53.750 --> 18:58.140
long time, until the capacitor
is completely loaded,

18:58.140 --> 19:00.740
so that it doesn't allow any
more current to flow,

19:00.740 --> 19:03.390
its voltage will equal this one
and at that point,

19:03.390 --> 19:07.010
dQ/dt will vanish.

19:07.009 --> 19:08.719
And let me call that final
charge

19:08.717 --> 19:11.697
Q_infinity,
because you will find out it

19:11.704 --> 19:13.684
happens only after infinite
time.

19:13.680 --> 19:20.360
That
Q_infinity/C =

19:20.364 --> 19:21.514
emf.

19:21.509 --> 19:23.839
So the capacitor is empty to
begin with.

19:23.838 --> 19:25.378
After infinite time,
it's fully charged,

19:25.380 --> 19:28.980
meaning this battery cannot
charge it to any higher voltage,

19:28.980 --> 19:32.990
because its voltage is
precisely balancing that of

19:32.987 --> 19:33.557
this.

19:33.558 --> 19:36.278
So let me solve--our goal is
still to find Q as the

19:36.284 --> 19:38.774
function of t,
starting with this equation.

19:38.769 --> 19:41.789
So we are going to write
Q as a function of

19:41.787 --> 19:44.247
t, as
Q_infinity,

19:44.250 --> 19:47.700
which is the asymptotic value
some Q twiddle.

19:47.700 --> 19:50.810
You can always write your
answer as 96 something else,

19:50.813 --> 19:54.223
and something else will adjust
itself to make this true.

19:54.220 --> 19:58.260
But let's find the equation for
Q twiddle.

19:58.259 --> 20:01.259
So take this Q,
put that into the equation and

20:01.255 --> 20:02.345
see what you get.

20:02.349 --> 20:04.569
Left hand side is E.

20:04.568 --> 20:07.468
Right hand side is
Q/C,

20:07.473 --> 20:11.113
Q_infinity
/C Q

20:11.105 --> 20:12.795
twiddle/C.

20:12.798 --> 20:16.918
And when you take dQ/dt,
this is a constant.

20:16.920 --> 20:20.980
So this is just dQ
twiddle/dt. But

20:20.980 --> 20:25.040
Q_inf
inity/C is exactly

20:25.041 --> 20:29.491
equal to E,
so these terms cancel,

20:29.492 --> 20:34.872
and the equation I get is this
combination = 0.

20:34.869 --> 20:35.439
Right?

20:35.440 --> 20:44.880
This says dQ/dt,
dQ twiddle/dt =

20:44.880 --> 20:51.780
-RC--I'm sorry,
-Q/RC.

20:51.779 --> 20:53.459
Q twiddle/RC.

20:53.460 --> 20:54.610
And we know how to integrate
this.

20:54.609 --> 20:56.159
We've done it many times.

20:56.160 --> 21:02.870
With Q twiddle =
Q twiddle at time 0 times

21:02.865 --> 21:08.225
e to the
−t/RC.

21:08.230 --> 21:14.030
Now Q twiddle at time 0
has to be chosen to satisfy the

21:14.027 --> 21:18.777
following condition:
what is Q at time 0?

21:18.778 --> 21:22.378
You guys know what charge is on
the capacitor at time 0?

21:22.380 --> 21:24.010
Do you know?

21:24.009 --> 21:26.739
Student:  Sorry,
my mind is elsewhere.

21:26.740 --> 21:27.330
Prof: Okay.

21:27.329 --> 21:28.749
Welcome back here.

21:28.750 --> 21:31.680
So what is the charge on the
capacitor when I started

21:31.678 --> 21:32.408
everything?

21:32.410 --> 21:36.100
Student:  Charge on the
capacitor.

21:36.098 --> 21:37.918
Prof: At the beginning
of the experiment.

21:37.920 --> 21:43.270
Student:  The charge on
the capacitor should equal the

21:43.272 --> 21:43.632
0.

21:43.630 --> 21:44.880
Prof: No,
where I just started.

21:44.880 --> 21:47.230
Anybody know?

21:47.230 --> 21:48.120
Yes?

21:48.119 --> 21:48.959
Student:  0?

21:48.960 --> 21:51.370
Prof: 0,
because I told you the

21:51.365 --> 21:54.155
capacitor started out with
nothing on it.

21:54.160 --> 21:55.550
That's not a mathematical
result.

21:55.549 --> 21:56.879
It's a historical fact.

21:56.880 --> 21:59.730
You can't begin an experiment
with charge on the capacitor.

21:59.730 --> 22:02.750
I told you today that the
charge on the capacitor is 0.

22:02.750 --> 22:10.010
So I've got to get 0 at t =
0 and this guy is E/C Q

22:10.013 --> 22:16.193
twiddle, that's Q
twiddle of 0 over C.

22:16.190 --> 22:19.140
I'm sorry.

22:19.140 --> 22:20.770
Q_infinity is
EC.

22:20.769 --> 22:25.419
Is that right?

22:25.420 --> 22:31.390
Yes, Q_infinity
= EC Q twiddle

22:31.385 --> 22:34.365
of 0 and that's got to be 0.

22:34.368 --> 22:38.888
Therefore Q twiddle of 0
= -EC.

22:38.890 --> 22:42.760
That means Q(t) =
Q_infinity

22:42.761 --> 22:44.011
Q twiddle.

22:44.009 --> 22:54.179
That becomes EC times 1
- e^(-t/RC).

22:54.180 --> 22:58.150
So if you draw the graph of
this, it will satisfy all the

22:58.148 --> 22:59.778
expectations we have.

22:59.779 --> 23:02.829
If you draw Q as a
function of time,

23:02.827 --> 23:05.947
at t = 0,
e to the -0 is 1.

23:05.950 --> 23:10.090
1 cancels the 1,
you start with 0 charge.

23:10.088 --> 23:13.248
At Q = infinity,
this guy is gone.

23:13.250 --> 23:19.510
It is EC, which is just
that value.

23:19.509 --> 23:23.839
So the charge builds up to that
value, but it never quite

23:23.839 --> 23:26.159
reaches the value EC.

23:26.160 --> 23:27.610
In other words,
the voltage in the capacitor is

23:27.612 --> 23:28.972
never quite equal to that of the
battery.

23:28.970 --> 23:30.320
There's always some left over.

23:30.319 --> 23:31.379
Yes?

23:31.380 --> 23:33.730
Student:  Where does the
negative come from?

23:33.730 --> 23:34.600
Prof: Here?

23:34.599 --> 23:35.539
Student:  Yes.

23:35.539 --> 23:36.609
Prof: Okay.

23:36.608 --> 23:39.868
I said Q at time 0 is
this guy at times

23:39.873 --> 23:44.153
Q_infinity,
which is EC Q twiddle at

23:44.152 --> 23:44.952
time 0.

23:44.950 --> 23:46.750
But that had to be equal to 0.

23:46.750 --> 23:47.980
Student:  Oh, okay, yeah.

23:47.980 --> 23:51.300
Prof: So that's how I
got that.

23:51.298 --> 23:52.708
This is a trick to solve the
equation.

23:52.710 --> 23:56.000
See, we all sort of know how to
solve this equation.

23:56.000 --> 24:01.280
You may not know how to solve
an equation with an extra term

24:01.280 --> 24:05.220
here, so this is a trick for
solving that.

24:05.220 --> 24:08.950
So this is the part that's
interesting, in the sense that

24:08.953 --> 24:12.563
once you write the basic rules
of physics, you get some

24:12.556 --> 24:13.486
equations.

24:13.490 --> 24:16.420
You've got to solve the
equations and it's no longer up

24:16.416 --> 24:17.986
to you to see what happens.

24:17.990 --> 24:21.180
The mathematics rules after
that point.

24:21.180 --> 24:25.280
And whatever it tells you,
you rush out to the lab and see

24:25.279 --> 24:26.359
if it's true.

24:26.358 --> 24:28.578
And with the capacitor,
whose capacitance you have

24:28.582 --> 24:30.712
measured carefully,
and a resistor which you have

24:30.711 --> 24:32.851
measured carefully,
you put them together,

24:32.846 --> 24:36.256
here's a nice prediction on
what will happen as it charges

24:36.259 --> 24:36.619
up.

24:36.618 --> 24:38.648
For example,
you may want your capacitor to

24:38.645 --> 24:41.825
hold 80 percent of its maximum
charge and you may like to know,

24:41.828 --> 24:44.118
"How long should I
wait?"

24:44.118 --> 24:46.718
If you want it to hold 100
percent of the maximum charge,

24:46.720 --> 24:48.160
it's never going to get done.

24:48.160 --> 24:49.910
So pick some number.

24:49.910 --> 24:54.220
They'll tell you,
if you want 75 percent,

24:54.219 --> 24:57.989
that's how many seconds you
wait.

24:57.990 --> 25:01.020
So this is how you're supposed
to have interplay between

25:01.020 --> 25:03.850
mathematics and physics,
because what happened was,

25:03.845 --> 25:07.025
you got into a situation where
you have to solve an equation.

25:07.028 --> 25:09.908
So this is a differential
equation, but differential

25:09.913 --> 25:12.683
equations are just questions
which are opposite of

25:12.684 --> 25:13.594
derivatives.

25:13.588 --> 25:17.838
You're trying to guess a
function about whose derivative

25:17.842 --> 25:20.782
you know something,
namely this one.

25:20.779 --> 25:22.289
And it's all guess work.

25:22.288 --> 25:25.698
You keep on guessing and keep
on guessing and you make a table

25:25.699 --> 25:28.829
of integrals where people tell
you what they guessed,

25:28.828 --> 25:30.578
and that big fat table of
integrals,

25:30.579 --> 25:31.819
you can turn to.

25:31.818 --> 25:35.348
But in easy cases,
you can solve it yourself.

25:35.348 --> 25:39.238
Okay, now there's a homework
problem I'm going to give you

25:39.238 --> 25:42.648
guys, but I'll tell you what's
in store for you.

25:42.650 --> 25:47.770
We want to ask ourselves,
what happens to the energetics

25:47.765 --> 25:49.435
in this problem.

25:49.440 --> 25:57.360
When I started,
how much energy did I have,

25:57.357 --> 26:00.937
and where was it?

26:00.940 --> 26:07.340
Anybody know?

26:07.338 --> 26:11.088
Was there any charge in the
capacitor?

26:11.089 --> 26:11.759
Nothing.

26:11.759 --> 26:13.089
There's nothing to begin with.

26:13.087 --> 26:13.307
Yes.

26:13.308 --> 26:15.208
Student:  It's all in
your battery.

26:15.210 --> 26:16.720
Prof: Yes,
the battery had some internal

26:16.722 --> 26:17.022
energy.

26:17.019 --> 26:18.689
That's correct.

26:18.690 --> 26:21.920
But now, during this
experiment, once I close the

26:21.924 --> 26:25.834
switch, I think of the battery
as outside my universe so it

26:25.834 --> 26:27.524
gives me some energy.

26:27.519 --> 26:31.529
How much energy did the battery
give me during the whole

26:31.529 --> 26:32.259
process?

26:32.259 --> 26:33.459
Yes?

26:33.460 --> 26:38.370
Student:  Equal to the
potential energy stored in the

26:38.374 --> 26:42.544
capacitor the energy dissipated
by the resistor.

26:42.538 --> 26:43.148
Prof: Right.

26:43.151 --> 26:44.151
So let me repeat what he said.

26:44.150 --> 26:47.540
He said it's got to be = to the
energy stored in the capacitor

26:47.544 --> 26:49.774
the energy dissipated in the
resistor.

26:49.769 --> 26:52.269
And that's got to be equal to
the work done by the battery.

26:52.269 --> 26:55.409
It's like saying,
how much work did the ski lift

26:55.414 --> 26:55.754
do?

26:55.750 --> 26:58.850
Well, the work done by the ski
lift is the work it takes to

26:58.851 --> 27:01.201
carry each person from the
bottom to the top,

27:01.203 --> 27:03.293
multiplied by the number of
people.

27:03.289 --> 27:05.229
That's the work done.

27:05.230 --> 27:07.790
In electrical language,
that just means the charge

27:07.792 --> 27:10.882
transported from the bottom to
the top times the voltage.

27:10.880 --> 27:20.370
That means the total work done
by the battery = the emf times

27:20.365 --> 27:28.265
I(t) dt
from 0 to infinity.

27:28.269 --> 27:30.889
I(t)dt is
a charge that flows in a small

27:30.890 --> 27:33.030
time dt,
and I is not a constant.

27:33.029 --> 27:35.429
In fact, we can find
I(t) by the

27:35.425 --> 27:37.125
formula of Q(t).

27:37.130 --> 27:42.450
Remember, Q(t) =
EC times 1 - E to

27:42.451 --> 27:47.031
the −t/RC.
Then I, which is

27:47.032 --> 27:51.362
dQ/dt,
you can calculate is EC

27:51.357 --> 27:56.687
divided by RC times
e^(−t/RC).

27:56.690 --> 28:03.070
It becomes E/R times
e^(−t/RC).

28:03.068 --> 28:10.568
So the current falls
exponentially in this form.

28:10.568 --> 28:13.838
At t = 0 the current is
as if the capacitor were not

28:13.843 --> 28:16.893
even there, because the
capacitor's inert at t =

28:16.891 --> 28:17.231
0.

28:17.230 --> 28:18.240
It has got no charge.

28:18.240 --> 28:20.760
It's not opposing you.

28:20.759 --> 28:22.469
And eventually,
the current goes to 0.

28:22.470 --> 28:25.690
So take that formula for the
current, put it here and do some

28:25.689 --> 28:26.279
integral.

28:26.279 --> 28:28.259
You get some answer.

28:28.259 --> 28:32.249
That's the energy given by the
battery to us.

28:32.250 --> 28:33.890
And what have you done with it?

28:38.954 --> 28:41.924
where Q in the end
should be

28:41.924 --> 28:44.724
Q_infinity.

28:44.720 --> 28:48.760
The resistor burns power at the
rate VI,

28:48.761 --> 28:50.961
which is I^(2)R.

28:50.960 --> 28:54.460
So you take this I^(2),
take this I,

28:54.463 --> 28:57.683
square it, do the integral from
0 to infinity,

28:57.682 --> 28:59.472
multiply by R.

28:59.470 --> 29:04.410
That is the work done in the
resistor, or work dissipated in

29:04.413 --> 29:05.673
the resistor.

29:05.670 --> 29:09.210
You should make sure that this
energy delivered by the battery

29:09.209 --> 29:12.339
= what is stored by the
capacitor what is dissipated in

29:12.343 --> 29:13.333
the resistor.

29:13.329 --> 29:15.619
That's your homework.

29:15.618 --> 29:17.778
That just means doing these
integrals and making sure

29:17.776 --> 29:18.726
everything works out.

29:18.730 --> 29:19.930
Yes?

29:19.930 --> 29:23.920
Student:  Shouldn't it
be negative, negative

29:23.922 --> 29:25.042
t/RC?

29:25.039 --> 29:25.809
Prof: Here?

29:25.809 --> 29:27.659
Student:  Yes.

29:27.660 --> 29:31.400
Prof: Okay,
finally I have a chance.

29:31.400 --> 29:35.540
This negative sign is going to
cancel that.

29:35.539 --> 29:36.589
Student:  Oh, okay.

29:36.588 --> 29:38.458
Prof: Look,
don't give up,

29:38.460 --> 29:41.910
okay, because just like you
thought, I make mistakes too.

29:41.910 --> 29:45.470
So I never mind it when you
guys do that.

29:45.470 --> 29:48.430
In this case,
I'll tell you why I sort of

29:48.431 --> 29:50.431
knew what the answer was.

29:50.430 --> 29:53.100
I know the answer's positive
because the current's going to

29:53.096 --> 29:53.736
be positive.

29:53.740 --> 29:56.440
So if I'd kept the extra -
sign, it will mean the current's

29:56.438 --> 29:59.368
going against the battery and I
would know that doesn't work.

29:59.368 --> 30:02.108
Quite often,
we're allowed to use the sign

30:02.112 --> 30:04.792
of the answer,
we're allowed to guess the

30:04.788 --> 30:05.388
sign.

30:05.390 --> 30:07.630
For example,
what's the height of a person?

30:07.630 --> 30:11.600
You said the answer was 6 feet,
but I don't know if it's or -.

30:11.599 --> 30:13.049
Well, you should know it's .

30:13.048 --> 30:15.238
You're talking about the height
of a person.

30:15.240 --> 30:16.920
What's the depth to which he
sank?

30:16.920 --> 30:18.830
Well, there it can be negative.

30:18.828 --> 30:21.478
So you should know by context
whether the answer is positive

30:21.480 --> 30:22.110
or negative.

30:22.108 --> 30:25.618
That's a very useful check as
far as signs go.

30:25.618 --> 30:29.308
Now as far as overall formulas
go, you can always take extreme

30:29.307 --> 30:32.267
limits of every answer and see
if it's correct.

30:32.269 --> 30:35.169
In the beginning,
just when you close the switch,

30:35.174 --> 30:37.664
there's the battery,
there's the resistor,

30:37.656 --> 30:41.286
the capacitor doesn't oppose
you, so you get e/R.

30:41.288 --> 30:43.378
At the end of the day,
the capacitor's fully charged,

30:43.380 --> 30:46.470
it's neutralizing the battery
by driving the current in the

30:46.467 --> 30:49.537
opposite way with equal voltage,
so the current should vanish.

30:49.539 --> 30:55.129
These are some tests.

30:55.130 --> 31:00.420
All right, so let's continue
now to the last thing in

31:00.419 --> 31:02.709
circuits,
which I'm not going to do any

31:02.711 --> 31:04.741
more,
but I'll tell you the kind of

31:04.740 --> 31:06.390
stuff people throw at you.

31:06.390 --> 31:10.380
So here is some guy,
some resistor,

31:10.380 --> 31:16.480
branches into two other
branches, some other voltage,

31:16.483 --> 31:18.483
join it here.

31:18.480 --> 31:21.780
This is E,
this is R_1,

31:21.778 --> 31:24.528
R_2,
R_3.

31:24.528 --> 31:28.058
This is E_2,
let's say.

31:28.058 --> 31:30.558
You've done stuff like this
before, but let me remind you

31:30.555 --> 31:31.085
the trick.

31:31.089 --> 31:32.369
The trick is very ancient.

31:32.369 --> 31:34.229
We all know what the trick is.

31:34.230 --> 31:38.210
First is, you've got to know
how many unknowns there are.

31:38.210 --> 31:42.410
If you can say the current here
is I_1.

31:42.410 --> 31:43.850
I don't know what
I_1 is;

31:43.849 --> 31:45.439
that's an unknown.

31:45.440 --> 31:47.710
The current flowing through
this guy is

31:47.712 --> 31:49.092
I_2.

31:49.088 --> 31:51.148
The current flowing through
this guy is

31:51.145 --> 31:53.095
I_3,
but I know that

31:53.095 --> 31:56.175
I_3 = I_1
- I_2.

31:56.180 --> 31:56.850
Do you agree?

31:56.849 --> 31:58.399
I_1 comes here.

31:58.400 --> 32:00.010
If I_2 goes
there, I_1 -

32:00.008 --> 32:00.948
I_2 goes there.

32:00.950 --> 32:02.910
So I'm actually solving one of
the equations,

32:02.907 --> 32:04.907
which is that I_1
= I_2

32:04.910 --> 32:05.890
I_3.

32:05.890 --> 32:09.090
Let's put that into the
equation.

32:09.088 --> 32:11.268
Then you have to write,
so how many unknowns do I have?

32:11.269 --> 32:13.609
I have two unknowns,
I_1 and

32:13.612 --> 32:16.752
I_2. So I have
to get two equations for two

32:16.753 --> 32:18.883
unknowns,
and one equation will be--you

32:18.877 --> 32:19.827
can do many things.

32:19.828 --> 32:23.688
You can start here,
go around like this and say the

32:23.690 --> 32:25.620
change in voltage is 0.

32:25.618 --> 32:32.188
That would tell me
E_1 - R_1

32:32.192 --> 32:39.292
I_1 - R_2
I_2 = 0.

32:39.289 --> 32:40.979
Are you guys with me now?

32:40.980 --> 32:43.380
You drop here,
you drop here,

32:43.384 --> 32:45.194
then you come back.

32:45.190 --> 32:49.160
Then you can take another loop
that goes like this.

32:49.160 --> 32:53.130
Then you will say
E_1 - R_1

32:53.125 --> 32:57.735
I_1 - R_3
times I_1 -

32:57.739 --> 32:59.519
I_2.

32:59.519 --> 33:01.639
Now let's look here.

33:01.640 --> 33:06.200
You keep going the same way,
but here you drop by an amount

33:06.201 --> 33:08.011
E_2.

33:08.009 --> 33:16.739
Then you've come over here and
you come to the other end and

33:16.742 --> 33:18.522
you get 0.

33:18.519 --> 33:21.549
Now somebody can say,
"Hey, why don't I start

33:21.553 --> 33:25.213
here, go around that loop and
say that voltage difference is

33:25.207 --> 33:25.947
0?"

33:25.950 --> 33:30.060
I'm getting the third equation,
but there are only two

33:30.057 --> 33:30.907
unknowns.

33:30.910 --> 33:36.610
So you math-minded people
should sort of know what will

33:36.605 --> 33:40.505
happen if I write a third
equation.

33:40.509 --> 33:42.109
What do you think will happen?

33:42.109 --> 33:47.179
Student:  Nothing.

33:47.180 --> 33:50.290
Prof: What do you mean,
nothing will happen?

33:50.288 --> 33:52.288
Student:  It's not going
to help you get any more.

33:52.288 --> 33:55.498
Prof: In what manner
will it prove to be useless?

33:55.500 --> 33:58.950
Student:  It will be a
combination of the other two.

33:58.950 --> 34:00.010
Prof: Right.

34:00.009 --> 34:02.959
You can deduce the third
equation by fiddling with the

34:02.964 --> 34:03.694
other two.

34:03.690 --> 34:07.400
Maybe 9 times one equation - 6
times the second equation will

34:07.396 --> 34:08.876
be the third equation.

34:08.880 --> 34:11.680
Therefore it's not an
independent equation.

34:11.679 --> 34:14.749
It will always turn out that if
you've got two unknowns,

34:14.746 --> 34:16.806
you need two independent
equations.

34:16.809 --> 34:19.099
If you can get three and four
and five, you will find out

34:19.101 --> 34:21.721
they're not telling you anything
new, so that's when you stop.

34:21.719 --> 34:24.369
If you don't know that,
you may start writing all kinds

34:24.373 --> 34:25.113
of equations.

34:25.110 --> 34:27.060
You may think,
here's another loop.

34:27.059 --> 34:29.819
Here's another guy who wants to
do this.

34:29.820 --> 34:32.710
You can do all that,
but you'll keep getting the

34:32.710 --> 34:33.510
same stuff.

34:33.510 --> 34:48.680
Okay, now we will really start
new topic: magnetism.

34:48.679 --> 34:51.999
Well, you start this because
every time you think you're done

34:52.003 --> 34:54.783
with physics,
somebody does some experiment

34:54.784 --> 34:57.074
and it doesn't fit what you
know,

34:57.070 --> 35:00.050
so you've got to make up new
stuff.

35:00.050 --> 35:04.250
So magnetism you know was
discovered in Ancient Greece,

35:04.250 --> 35:08.110
when parents noticed kids are
sticking stuff on the

35:08.114 --> 35:11.674
refrigerator using some little
black things.

35:11.670 --> 35:13.280
So more experiments.

35:13.280 --> 35:15.960
I obviously don't know all the
details.

35:15.960 --> 35:18.150
But it was discovered in many
ways.

35:18.150 --> 35:21.680
One was in little compass
needles which told you which way

35:21.677 --> 35:22.417
was north.

35:22.420 --> 35:25.240
So I'm going to give you a
string of things that happened

35:25.235 --> 35:28.395
that tell you there is something
going on that is not covered by

35:28.403 --> 35:31.123
anything I've written down so
far in this course,

35:31.119 --> 35:33.259
new phenomena that don't make
sense.

35:33.260 --> 35:35.250
Here's the simplest one.

35:35.250 --> 35:38.720
There's a wire carrying some
current.

35:38.719 --> 35:41.259
There's a little charge sitting
here.

35:41.260 --> 35:42.630
Nothing happens.

35:42.630 --> 35:45.460
Because the wire is
electrically neutral,

35:45.463 --> 35:48.443
the charge Q doesn't do
anything.

35:48.440 --> 35:53.180
Now the charge begins to move
at velocity v.

35:53.179 --> 35:57.119
Then you suddenly find the
charge attracted to the wire.

35:57.119 --> 36:00.179
It starts bending in.

36:00.179 --> 36:03.029
That is not the electrical
force, because the electrical

36:03.027 --> 36:03.697
force is 0.

36:03.699 --> 36:05.769
The wire is electrically
neutral.

36:05.768 --> 36:09.558
It doesn't care if the charge
is moving or not,

36:09.556 --> 36:11.446
so it is a new force.

36:11.449 --> 36:15.989
And if the charge goes this
way, the force is repulsive.

36:15.989 --> 36:19.419
That's just one phenomenon you
notice.

36:19.420 --> 36:24.820
Another thing you notice is
there are little things called

36:24.822 --> 36:26.152
bar magnets.

36:26.150 --> 36:29.060
They seem to have a north and a
south.

36:29.059 --> 36:32.159
And if you bring it next to
another bar magnet,

36:32.157 --> 36:35.857
which has got a north and
south, you find they repel and

36:35.862 --> 36:37.752
south and north attract.

36:37.750 --> 36:41.230
Then you take this compass
needle that people use to find

36:41.228 --> 36:44.828
the north direction for the
earth, and you put it somewhere

36:44.829 --> 36:45.389
here.

36:45.389 --> 36:48.469
You find that it swings,
if it's free to pivot,

36:48.474 --> 36:50.894
and points in a certain
direction.

36:50.889 --> 36:53.329
And if you use that as a
direction of a certain field,

36:53.327 --> 36:56.227
you can draw these pictures and
you sort of know what it's going

36:56.226 --> 36:57.006
to look like.

36:57.010 --> 37:01.940
They look like this.

37:01.940 --> 37:04.470
So this means if you put a
compass needle here,

37:04.467 --> 37:06.277
the north will point like that.

37:06.280 --> 37:09.800
That's what I mean.

37:09.800 --> 37:11.700
That's another phenomenon.

37:11.699 --> 37:15.229
Then once you found out about
electric currents,

37:15.230 --> 37:18.640
you also found that if you took
a coil of wire,

37:18.639 --> 37:23.179
then a compass needle somewhere
here began to respond to a

37:23.184 --> 37:26.514
magnetic field,
which if you plot,

37:26.510 --> 37:29.240
seemed to look like this.

37:29.239 --> 37:33.519
It seemed to look just like a
magnet with a north pole here

37:33.518 --> 37:35.288
and a south pole here.

37:35.289 --> 37:36.279
There are no magnets at all.

37:36.280 --> 37:40.510
You just had a coil of wire
carrying the current in a

37:40.505 --> 37:42.125
certain direction.

37:42.130 --> 37:44.350
It goes like this.

37:44.349 --> 37:49.499
Then the magnetic field lines,
which you plot by moving the

37:49.503 --> 37:52.263
compass needle,
they do this.

37:52.260 --> 37:55.670
So all I'm trying to say is
there are several phenomena

37:55.670 --> 37:58.830
going on,
and the clue that you get is,

37:58.829 --> 38:03.629
why didn't I need this before,
and why do I need it now?

38:03.630 --> 38:07.230
What is new in this problem
compared to the problems I've

38:07.231 --> 38:07.811
solved?

38:07.809 --> 38:12.539
Can you see what's making this
problem outside the realm of

38:12.541 --> 38:14.011
what we studied?

38:14.010 --> 38:16.400
What's the one feature you
notice from electrostatics?

38:16.400 --> 38:17.220
Yes?

38:17.219 --> 38:18.439
Student:  The charges
are moving.

38:18.440 --> 38:21.640
Prof: Everything is
moving.

38:21.639 --> 38:26.119
The wire carries charges which
are moving and this little guy

38:26.123 --> 38:29.863
who got attracted or repelled,
he's also moving.

38:29.860 --> 38:35.270
So magnetism is caused by
moving charges and it's felt by

38:35.268 --> 38:36.908
moving charges.

38:36.909 --> 38:39.589
They've got to move to play
this game.

38:39.590 --> 38:48.540
So magnetism is caused by
moving charges and it's felt by

38:48.536 --> 38:55.086
moving charges,
whereas in electrostatics,

38:55.086 --> 39:00.036
we didn't have that motion.

39:00.039 --> 39:01.029
Yes?

39:01.030 --> 39:03.780
Student:  What about
reference frames?

39:03.780 --> 39:06.160
Prof: Yes,
her question was,

39:06.159 --> 39:09.869
how about reference frames,
namely moving according to

39:09.869 --> 39:10.989
whom, right?

39:10.989 --> 39:14.149
So I will come to that near the
end when we do relativity,

39:14.150 --> 39:17.790
when I remind you of some ideas
of relativity and see what that

39:17.793 --> 39:20.443
has to say about electricity and
magnetism.

39:20.440 --> 39:22.700
But you are free,
even in the relativistic

39:22.702 --> 39:25.522
theory, to take the view that
you are not moving.

39:25.518 --> 39:28.708
As long you're an inertial
frame, that means a frame in

39:28.710 --> 39:32.700
which Newton's laws are valid,
you can apply all the laws of

39:32.702 --> 39:35.172
physics as if you were not
moving.

39:35.170 --> 39:38.400
I'm just saying for that
observer, who's inertial,

39:38.400 --> 39:41.480
it's found that when charges
are moving according to him or

39:41.478 --> 39:43.718
her,
they produce currents which

39:43.715 --> 39:47.445
produce a field,
and the charges in that field

39:47.449 --> 39:49.079
will also respond.

39:49.079 --> 39:51.779
Now you can say,
"What's so special about

39:51.780 --> 39:52.140
you?

39:52.139 --> 39:55.379
I will go to a new frame of
reference,right?"

39:55.380 --> 39:58.000
I'll come back to that later,
but I cannot resist telling you

39:57.998 --> 40:00.048
at least the answer to some of
the questions.

40:00.050 --> 40:03.770
For example,
if this charge is moving at a

40:03.771 --> 40:08.131
speed v and I'm
completely stymied by this

40:08.126 --> 40:13.116
velocity, because I don't know
how to deal with it.

40:13.119 --> 40:16.929
There's one way to deal with
it, which is to get on a train

40:16.932 --> 40:20.092
that goes at the same velocity
as this charge.

40:20.090 --> 40:24.990
Then this charge is at rest and
if it still bends towards the

40:24.987 --> 40:27.047
wire,
which it will--you agree that

40:27.050 --> 40:29.980
if you go on a moving train,
a charge attracted to the wire

40:29.976 --> 40:32.076
will continue to be attracted to
the wire.

40:32.079 --> 40:33.549
You've got to say,
"How come?

40:33.550 --> 40:35.270
How do you explain that?"

40:35.268 --> 40:39.508
Neutral wire attracting a
charge.

40:39.510 --> 40:41.830
Either you can say it's all
happening because you're in a

40:41.827 --> 40:44.837
moving train,
but relativity tells you,

40:44.835 --> 40:50.085
people in a moving train are
entitled to the same laws of

40:50.092 --> 40:54.132
physics as people in a
non-moving train.

40:54.130 --> 40:55.660
Now it's true for Amtrak.

40:55.659 --> 40:58.119
As long as you can get the
train to move,

40:58.123 --> 41:00.223
you can make the same
statement.

41:00.219 --> 41:03.869
So how does a person in a
moving train explain it?

41:03.869 --> 41:04.909
What do you think happens?

41:04.909 --> 41:05.619
Yes?

41:05.619 --> 41:07.909
Student:  The wire looks
like it's moving back in that

41:07.911 --> 41:08.371
direction.

41:08.369 --> 41:10.399
Prof: The wire may look
as though it's moving backwards,

41:10.400 --> 41:14.780
but still, a neutral thing
moving forwards or backwards

41:14.784 --> 41:19.964
shouldn't matter,
right?

41:19.960 --> 41:23.980
A neutral rod, let it move.

41:23.980 --> 41:26.600
Why does it matter?

41:26.599 --> 41:28.019
So what do you think happens?

41:28.018 --> 41:31.058
First of all,
do you think it will be

41:31.063 --> 41:35.293
attracted to the wire,
even in the moving train?

41:35.289 --> 41:36.779
Yes, no?

41:36.780 --> 41:39.070
It will be attracted,
because if I see it moving

41:39.072 --> 41:40.982
towards the wire,
you can go on a train,

41:40.976 --> 41:43.366
you can go on a plane,
you will also see it's moving

41:43.369 --> 41:43.939
towards it.

41:43.940 --> 41:46.200
Maybe the rate will be
different and so on,

41:46.202 --> 41:48.952
but the fact that it's getting
closer to the wire is

41:48.951 --> 41:49.761
undeniable.

41:49.760 --> 41:51.620
And there's going to be no new
physics.

41:51.619 --> 41:54.169
You're just going to rely on
good old electromagnetism,

41:54.173 --> 41:57.063
because electromagnetism is
supposed to work for everybody.

41:57.059 --> 42:00.439
In fact, that's how the
relativity--

42:00.440 --> 42:02.380
even though electromagnetism
was discovered before

42:02.376 --> 42:04.416
relativity,
it obeys all the principles

42:04.422 --> 42:06.252
demanded by relativistic theory.

42:06.250 --> 42:09.670
So everybody should have the
same laws of motion.

42:09.670 --> 42:12.810
So have you found a way out now?

42:12.809 --> 42:13.689
Yes?

42:13.690 --> 42:17.010
Student:  Would the wire
look like a bar magnet?

42:17.010 --> 42:19.160
Prof: No,
because bar magnet means you're

42:19.164 --> 42:20.454
going to magnetism,
right?

42:20.449 --> 42:23.849
So let's say I don't know any
magnetism, because the charge is

42:23.851 --> 42:24.411
at rest.

42:24.409 --> 42:27.239
A charge at rest doesn't care
if it's near a bar magnet.

42:27.239 --> 42:27.989
Yes?

42:27.989 --> 42:29.529
Student:  ________ lines.

42:29.530 --> 42:31.010
Prof: So what should be
happening to the wire,

42:31.007 --> 42:31.317
you guys?

42:31.320 --> 42:33.050
Tell me?

42:33.050 --> 42:34.130
Yes?

42:34.130 --> 42:37.100
Student:  If you're
moving and then the charges in

42:37.099 --> 42:39.879
the wire are moving,
say you're moving at the same

42:39.882 --> 42:41.772
speed as the charges on the
wire,

42:41.769 --> 42:42.889
it would seem stationary.

42:42.889 --> 42:46.859
So it's just like the point
charge being attracted to a

42:46.862 --> 42:48.362
bunch of--
Prof: Right.

42:48.360 --> 42:51.140
Why is the point charge
attracted to a neutral wire?

42:51.139 --> 42:55.929
That is my question.

42:55.929 --> 42:56.839
Yes?

42:56.840 --> 42:59.020
Student:  The wire isn't
neutral,

42:59.018 --> 43:02.528
because you're moving along
with the negative charges,

43:02.530 --> 43:05.920
so they stay there,
but then the positive charges

43:05.923 --> 43:07.473
keep--
Prof: That's correct.

43:07.465 --> 43:08.575
So let me explain what he said.

43:08.579 --> 43:12.759
In fact, the answer is,
the wire will not be neutral.

43:12.760 --> 43:14.810
So how did that happen?

43:14.809 --> 43:16.659
You might say,
"Oh, there's a Lorentz

43:16.655 --> 43:18.495
contraction,"
meaning the lengths get

43:18.500 --> 43:20.460
contracted,
but if the wire is neutral,

43:20.456 --> 43:22.696
the positives and negatives all
get compressed,

43:22.699 --> 43:26.679
it still should look neutral,
but that's not how it is.

43:26.679 --> 43:29.549
Remember, in a real conductor
you've got positive charges,

43:29.552 --> 43:32.632
which are at rest and negative
charges which are moving in the

43:32.626 --> 43:34.136
opposite direction,
right?

43:34.139 --> 43:38.209
So I do that by taking a rod,
which is positively charged,

43:38.206 --> 43:41.846
and I take another rod,
which is negatively charged,

43:41.847 --> 43:44.057
and I simply move this rod.

43:44.059 --> 43:46.859
That will produce a current.

43:46.860 --> 43:50.090
Now if the wire is neutral,
I think you guys can understand

43:50.090 --> 43:52.730
that the density on this,
in its own rest frame,

43:52.731 --> 43:55.341
will be different from the
density of charges here,

43:55.340 --> 43:57.550
because lengths get contracted.

43:57.550 --> 44:01.940
It is a contracted wire that
should have the same density as

44:01.938 --> 44:05.268
the positive wire,
therefore in reality,

44:05.266 --> 44:07.946
the contracted wire,
length contracted with

44:07.952 --> 44:11.032
increased density,
should match the density of

44:11.025 --> 44:11.535
this.

44:11.539 --> 44:16.019
But if I go to the moving
frame, what will happen is this

44:16.023 --> 44:19.393
guy will freeze,
and that guy will move the

44:19.387 --> 44:20.747
opposite way.

44:20.750 --> 44:22.980
This guy will stop moving;
the other will move the

44:22.983 --> 44:23.703
opposite way.

44:23.699 --> 44:26.329
But then you can show--I will
do that calculation for you

44:26.327 --> 44:28.297
later--
that the densities that

44:28.300 --> 44:31.270
previously canceled will no
longer cancel,

44:31.268 --> 44:33.278
and the wire will have a net
positive charge.

44:33.280 --> 44:35.230
The point is,
there's a lack of symmetry

44:35.230 --> 44:37.280
between the positive and
negative charges,

44:37.280 --> 44:38.980
because one of them is moving.

44:38.980 --> 44:42.500
In fact, if you took the simple
example where the charges in the

44:42.501 --> 44:45.241
wire are moving at the same
speed as this guy--

44:45.239 --> 44:47.449
let's imagine positive charges
can move--

44:47.449 --> 44:49.589
and if they're moving at the
same speed as this guy,

44:49.590 --> 44:51.650
if you stop this,
you will also stop those,

44:51.650 --> 44:53.780
and they will go the opposite
way.

44:53.780 --> 44:57.120
But if they go the opposite
way, they will contract,

44:57.119 --> 44:59.519
and when they contract,
they will no longer be balanced

44:59.519 --> 45:01.719
by this,
and the positive charge will be

45:01.719 --> 45:05.109
attracted to the negative wire,
and not only attracted.

45:05.110 --> 45:09.830
It will be the same force that
you will get when you learn

45:09.829 --> 45:11.319
about magnetism.

45:11.320 --> 45:14.840
So I'll do this in detail,
but I'm just telling you that

45:14.835 --> 45:18.665
you don't have to worry about
which frame of reference you're

45:18.672 --> 45:19.122
in.

45:19.119 --> 45:21.909
The laws of physics are
guaranteed to work for all

45:21.914 --> 45:24.714
frames of reference in uniform
relative motion.

45:24.710 --> 45:27.460
So when I say velocity is
v, I mean according to

45:27.460 --> 45:28.430
any one observer.

45:28.429 --> 45:31.749
That could be you.

45:31.750 --> 45:34.150
So you've got all this
phenomena.

45:34.150 --> 45:38.330
Now I'm going to give you the
fundamental equations of

45:38.331 --> 45:41.411
magnetostatics,
that will explain to you

45:41.409 --> 45:44.329
everything I've described so
far.

45:44.329 --> 45:45.909
So it will have two parts.

45:45.909 --> 45:50.359
The first part will be,
what is the force felt by a

45:50.356 --> 45:54.446
charge that's in motion in a
magnetic field?

45:54.449 --> 45:57.559
The next thing is,
how do you produce a magnetic

45:57.559 --> 45:58.089
field?

45:58.090 --> 45:59.670
Well, the answer is electric
currents.

45:59.670 --> 46:01.880
Then you can ask,
how does a current here produce

46:01.878 --> 46:03.028
a magnetic field there?

46:03.030 --> 46:05.500
What's the analog of Coulomb's
law?

46:05.500 --> 46:08.920
Coulomb's law is charges here
producing the electric field

46:08.920 --> 46:09.400
there.

46:09.400 --> 46:12.400
Magnetism is a current here
producing a magnetic field

46:12.400 --> 46:14.780
there, so you will have the
second part.

46:14.780 --> 46:18.320
So I'm saying that this
formula, q times E

46:18.322 --> 46:19.872
is the electric force.

46:19.869 --> 46:22.839
Then there's the fact that the
E here due to this guy is

46:22.842 --> 46:24.672
1/r^(2) times the
q here,

46:24.666 --> 46:25.046
etc.

46:25.050 --> 46:27.440
You need both parts.

46:27.440 --> 46:28.530
I'm going to give the two parts.

46:28.530 --> 46:31.190
The second part will come
later, but first is,

46:31.188 --> 46:34.138
if you have a charged particle
in a magnetic field,

46:34.143 --> 46:35.743
what's the force on it?

46:35.739 --> 46:37.959
So here's the answer.

46:37.960 --> 46:42.930
The total force of a charge
particle q is this guy

46:42.925 --> 46:46.915
that we already know,
the cross product of the

46:46.918 --> 46:50.198
velocity with the magnetic
field.

46:50.199 --> 46:53.729
The direction of the field is
determined by the compass

46:53.731 --> 46:54.321
needle.

46:54.320 --> 46:56.910
That's how the B is
directed.

46:56.909 --> 47:00.949
But if you want,
this force is called the

47:00.952 --> 47:02.572
Lorentz force.

47:02.570 --> 47:05.970
It is not invented by Lorentz,
but he did so many things in

47:05.974 --> 47:08.034
this field, it's named after
him.

47:08.030 --> 47:11.290
You can take this to be a
postulate,

47:11.289 --> 47:12.909
the summary of years of
experiment,

47:12.909 --> 47:14.689
and you can say,
"I want to begin

47:14.690 --> 47:16.280
here," you can begin here.

47:16.280 --> 47:19.090
You're never going to derive
this, but this is an

47:19.088 --> 47:20.258
experimental fact.

47:20.260 --> 47:22.350
So if I go to a part of the
room and say,

47:22.349 --> 47:25.539
"Find the electric field
here," I think you all know

47:25.536 --> 47:26.316
what to do.

47:26.320 --> 47:27.990
We've done it many times.

47:27.989 --> 47:29.929
Take a coulomb and put it there.

47:29.929 --> 47:34.049
Find the force on it,
and that's actually the

47:34.047 --> 47:35.637
electric field.

47:35.639 --> 47:38.229
If you put 5 coulombs,
you divide the force by 5 and

47:38.228 --> 47:39.648
that's the electric field.

47:39.650 --> 47:41.960
If I tell you,
what's a magnetic field,

47:41.960 --> 47:45.420
you can take a coulomb there,
but you still have to find

47:45.418 --> 47:47.098
out--
by the way, the electric field

47:47.099 --> 47:49.949
problem's very easy,
because the direction of the

47:49.945 --> 47:53.465
electric field is simply the
motion of the charge.

47:53.469 --> 47:56.289
In the magnetic problem,
there are lots of directions

47:56.291 --> 47:56.891
involved.

47:56.889 --> 47:59.879
There's the velocity of the
charge.

47:59.880 --> 48:02.680
There's a magnetic field,
there's a magnetic force,

48:02.684 --> 48:04.484
and they form a cross product.

48:04.480 --> 48:07.630
v x B is the
force.

48:07.630 --> 48:10.260
So if someone says,
"Which way is B

48:10.257 --> 48:11.227
pointing?"

48:11.230 --> 48:15.080
you'll have to shoot a few
particles and find out how they

48:15.077 --> 48:15.547
bend.

48:15.550 --> 48:19.040
If you shot it exactly parallel
to B it won't bend at

48:19.043 --> 48:19.403
all.

48:19.400 --> 48:21.420
That's the direction of
B.

48:21.420 --> 48:24.200
If you sent it perpendicular to
B, it will bend the most,

48:24.195 --> 48:26.305
because the cross product will
be the biggest.

48:26.309 --> 48:30.139
Then you can slowly determine
by some experiments what the

48:30.143 --> 48:31.693
value of B is.

48:31.690 --> 48:35.870
So let's do a couple of simple
problems where we just use this

48:35.869 --> 48:37.789
part, v x B.

48:37.789 --> 48:41.179
But there's a very important
aspect of v x B.

48:41.179 --> 48:44.709
Wherever a force acts on a
body, you know force dot

48:44.706 --> 48:48.796
velocity is the power or the
rate at which work is done.

48:48.800 --> 48:51.720
If you take the electromagnetic
force dot velocity,

48:51.719 --> 48:58.789
you get q times v ⋅
E q times v

48:58.789 --> 49:04.219
⋅ v x B and
that is 0.

49:04.219 --> 49:07.319
So you know why v ⋅
v x B is 0?

49:07.320 --> 49:09.950
It's again a purely
mathematical result.

49:09.949 --> 49:10.889
Yes?

49:10.889 --> 49:12.759
Student:  Because
v cross B is going

49:12.764 --> 49:14.754
to be perpendicular to--
Prof: v x

49:14.753 --> 49:17.493
B is orthogonal to both
v and to B,

49:17.489 --> 49:19.599
so it's coming out--in this
case, if B and v are in the

49:19.601 --> 49:21.411
blackboard,
v x B is outside

49:21.414 --> 49:22.094
the blackboard.

49:22.090 --> 49:25.850
Its dot product with anything
in the blackboard is 0.

49:25.849 --> 49:30.069
So whenever you have a vector
dotting itself cross something

49:30.065 --> 49:31.775
else, the answer is 0.

49:31.780 --> 49:35.380
That means the magnetic field
is always perpendicular to the

49:35.376 --> 49:36.896
motion of the particle.

49:36.900 --> 49:40.020
That means it doesn't do any
work, never.

49:40.019 --> 49:41.389
Not just weekends.

49:41.389 --> 49:44.329
The magnetic field doesn't do
any work.

49:44.329 --> 49:46.199
So you can say,
"Who cares about such a

49:46.199 --> 49:46.809
thing?"

49:46.809 --> 49:48.149
Electric fields do a lot of
work.

49:48.150 --> 49:49.870
They speed up particles,
they slow down.

49:49.869 --> 49:52.679
The kinetic energy of a
particle will never change due

49:52.684 --> 49:53.964
to the magnetic field.

49:53.960 --> 49:56.600
And yet you will see,
even though it's not able to do

49:56.601 --> 49:58.911
anything by itself,
it's extremely useful in

49:58.905 --> 50:01.745
getting a lot of things done,
like generators and so on.

50:01.750 --> 50:03.490
They rely on the magnetic field.

50:03.489 --> 50:05.919
So we'll see that,
but at the moment,

50:05.920 --> 50:06.730
it's fact.

50:06.730 --> 50:10.660
The force is always
perpendicular to the velocity

50:10.664 --> 50:12.554
for a magnetic field.

50:12.550 --> 50:17.870
So now we are going to do one
or two simple problems.

50:17.869 --> 50:21.049
First problem,
let me take the easiest one.

50:21.050 --> 50:24.670
I want to find a way to select
from a beam of particles,

50:24.668 --> 50:28.418
which are all going from left
to right, those which have a

50:28.418 --> 50:29.798
certain velocity.

50:29.800 --> 50:33.350
I want a velocity filter,
and I cannot seem them

50:33.349 --> 50:35.579
microscopically;
they are little guys.

50:35.579 --> 50:38.289
But they're moving,
they're all moving like this.

50:38.289 --> 50:42.069
Here is how we can make a
velocity filter.

50:42.070 --> 50:47.610
You take two parallel plates,
charge them up so the electric

50:47.606 --> 50:49.856
field looks like this.

50:49.860 --> 50:56.630
This particle q will then bend
like this in the electric field.

50:56.630 --> 51:01.280
Now put a magnetic field into
the paper.

51:01.280 --> 51:05.840
Let me make sure I got this
right.

51:05.840 --> 51:08.340
Yes.

51:08.340 --> 51:11.450
Put a magnetic field into the
board that's shown by this

51:11.449 --> 51:11.959
symbol.

51:11.960 --> 51:16.910
Magnetic field coming out
towards you is shown by dots,

51:16.905 --> 51:21.485
and coming away from
you--coming towards you is the

51:21.485 --> 51:25.695
dot, and going away from you is
the cross.

51:25.699 --> 51:26.919
That comes from the arrow.

51:26.920 --> 51:29.700
If you have an arrow,
if you look at the arrow from

51:29.704 --> 51:32.664
behind, it looks like this,
and if it looks like this,

51:32.655 --> 51:33.875
you run like hell.

51:33.880 --> 51:36.530
These are the two things.

51:36.530 --> 51:39.200
So I want to show you that
B is going into the

51:39.202 --> 51:39.872
blackboard.

51:39.869 --> 51:42.179
Now what's the force due to
B?

51:42.179 --> 51:44.129
Take the cross product--that's
what I was trying to do in my

51:44.126 --> 51:45.446
head,
make sure I got it

51:45.445 --> 51:48.665
right--v x B is
turning a screwdriver from

51:48.672 --> 51:50.082
v to B.

51:50.079 --> 51:55.299
Let's see, B is into the
board, v x B will

51:55.304 --> 51:56.574
be like this.

51:56.570 --> 52:03.310
Therefore the total force on a
charge will be q times

52:03.309 --> 52:06.789
E,
which is downstairs--I mean,

52:06.786 --> 52:11.606
pointing down,
and v x B which

52:11.610 --> 52:13.610
is pointing up.

52:13.610 --> 52:17.980
You can see then for a given
value of E and B,

52:17.981 --> 52:21.981
for any randomly chosen
velocity, these forces will not

52:21.983 --> 52:22.803
cancel.

52:22.800 --> 52:27.680
So some guys may bend like
this, some guys may bend like

52:27.679 --> 52:31.759
this, or some guys will go
straight through.

52:31.760 --> 52:36.990
They are the ones for whom
E = v x B

52:36.990 --> 52:43.250
or the magnitude of the velocity
is E divided by B.

52:43.250 --> 52:47.430
So only those particles with
that velocity will make it.

52:47.429 --> 52:51.089
In other words,
imagine this thing is very,

52:51.085 --> 52:52.125
very long.

52:52.130 --> 52:55.710
Some things will go hit it
there, others may hit it here.

52:55.710 --> 52:59.440
If you are just right,
you will make it to the other

52:59.436 --> 53:01.916
side, so it's a velocity filter.

53:01.920 --> 53:04.970
Now you should be able to
answer the following question

53:04.969 --> 53:06.889
without doing long calculations.

53:06.889 --> 53:09.539
The guys hitting the top,
are they faster than the

53:09.541 --> 53:10.951
desired speed or slower?

53:10.949 --> 53:16.089
Student:  Faster.

53:16.090 --> 53:20.190
Prof: Because what?

53:20.190 --> 53:21.110
You understand that?

53:21.110 --> 53:24.290
If they're going up,
the magnetic force is winning.

53:24.289 --> 53:27.009
The magnetic force is the only
one who cares about your

53:27.007 --> 53:28.967
velocity, so the velocity is too
big.

53:28.969 --> 53:31.339
If they're falling down,
electric force is winning.

53:31.340 --> 53:35.040
Electric force is a constant,
magnetic force is velocity

53:35.041 --> 53:35.851
dependent.

53:35.849 --> 53:40.659
So the fast guys go here,
slow and the right one comes

53:40.655 --> 53:41.285
here.

53:41.289 --> 53:43.599
Here's another exercise you can
do.

53:43.599 --> 53:50.429
There is a magnetic field
coming out of the blackboard and

53:50.425 --> 53:53.535
it's uniform in density.

53:53.539 --> 54:01.719
And I send a particle here with
the velocity like this.

54:01.719 --> 54:05.709
What will it do?

54:05.710 --> 54:09.510
So B is coming towards
me.

54:09.510 --> 54:10.200
B is here.

54:10.199 --> 54:12.479
v x B is a force
in this direction.

54:12.480 --> 54:15.590
It will bend;
it will go there.

54:15.590 --> 54:18.530
So it's constantly being
applied a force perpendicular to

54:18.527 --> 54:20.937
its velocity,
so it's like planetary motion.

54:20.940 --> 54:26.170
It will go in a circle.

54:26.170 --> 54:28.860
Never speeding up.

54:28.860 --> 54:30.960
It's not speeding up,
because the force is always

54:30.963 --> 54:32.193
perpendicular to velocity.

54:32.190 --> 54:35.410
You don't change the kinetic
energy, but you change the

54:35.409 --> 54:37.139
actual direction of motion.

54:37.139 --> 54:38.729
So this is the way to trap a
particle.

54:38.730 --> 54:41.240
If you want to trap particles,
you put them in a magnetic

54:41.244 --> 54:42.954
field, you don't have to touch
them.

54:42.949 --> 54:44.199
They will not go anywhere.

54:44.199 --> 54:48.249
They'll form an orbit.

54:48.250 --> 54:51.640
So let's find out what we can
say about this orbit.

54:51.639 --> 54:56.339
If that distance is R,
the dynamical equation is

54:56.344 --> 54:59.844
mv^(2)/R,
is the force you need to bend

54:59.844 --> 55:02.714
something into a circle of
radius R,

55:02.710 --> 55:05.220
if it's going at speed v.

55:05.219 --> 55:08.039
And we can take V to be
constant, because v is

55:08.038 --> 55:09.088
not going to change.

55:09.090 --> 55:13.770
And that's going to be equal to
the magnetic force,

55:13.768 --> 55:18.448
which is v times
B times q.

55:18.449 --> 55:20.719
I didn't write the cross
product, because everything is

55:20.719 --> 55:21.979
perpendicular to everything.

55:21.980 --> 55:23.900
The magnetic field and v are
perpendicular,

55:23.898 --> 55:26.458
so the cross product and
magnitude is just v times

55:26.456 --> 55:27.046
B.

55:27.050 --> 55:30.520
The direction of the force is
of course towards the center.

55:30.518 --> 55:37.148
If you balance these two
equations, you find

55:37.152 --> 55:43.632
v/R is
qB/m.

55:43.630 --> 55:45.200
So what is v/R?

55:45.199 --> 55:49.749
Let's look at what's v
over R.

55:49.750 --> 55:57.770
If the particle has a velocity
v, the time period due in

55:57.766 --> 56:02.806
orbit will be
2ΠR/v.

56:02.809 --> 56:07.129
The distance divided by speed
is the time period.

56:07.130 --> 56:15.340
So v/R = 2Π/T
or 2Πf.

56:15.340 --> 56:18.940
That's called the omega or the
angular speed.

56:18.940 --> 56:23.610
It goes around and around the
circle with an angular speed,

56:23.614 --> 56:24.424
ω.

56:24.420 --> 56:34.960
So v/R = ω.

56:34.960 --> 56:36.860
So this fact was known for some
time.

56:36.860 --> 56:39.490
It's not very hard to find.

56:39.489 --> 56:44.769
But do you know who made a
living out of this?

56:44.768 --> 56:47.008
Notice something very
interesting.

56:47.010 --> 56:52.800
The angular frequency does not
depend on the speed or on the

56:52.804 --> 56:53.694
radius.

56:53.690 --> 56:57.810
It just depends on the charge
to mass ratio of the particle in

56:57.811 --> 56:59.501
a given magnetic field.

56:59.500 --> 57:01.930
So that circle,
that circle,

57:01.929 --> 57:05.349
they all go around at the same
rate.

57:05.349 --> 57:07.429
As long as the particle has a
given charge,

57:07.429 --> 57:09.569
a given mass,
it's in a given magnetic field,

57:09.570 --> 57:12.270
you can launch them in orbits
of various velocities,

57:12.269 --> 57:16.549
they will all be in synch.

57:16.550 --> 57:19.080
Now do you know who used this?

57:19.079 --> 57:26.049
Does it look like anything,
these guys?

57:26.050 --> 57:32.050
That person was actually
associated with Yale.

57:32.050 --> 57:34.910
That tell you anything?

57:34.909 --> 57:37.239
Okay, his name was Lawrence.

57:37.239 --> 57:40.639
Do you know what Lawrence
invented?

57:40.639 --> 57:43.159
No?

57:43.159 --> 57:44.309
Pardon me?

57:44.309 --> 57:46.949
He invented the cyclotron.

57:46.949 --> 57:50.509
I'll tell you what--in fact,
he had the ideas when he was a

57:50.510 --> 57:54.010
young faculty member here,
but he didn't build it here.

57:54.010 --> 57:57.050
He went off to Berkeley and he
built the cyclotron there.

57:57.050 --> 57:59.310
Then he built bigger and bigger
cyclotrons.

57:59.309 --> 58:04.019
And I want to tell you what the
idea is behind a cyclotron.

58:04.019 --> 58:05.279
Here's what you want to do.

58:05.280 --> 58:08.270
You want to accelerate
particles so you can smash them

58:08.271 --> 58:09.571
against other things.

58:09.570 --> 58:15.470
So one thing you can do is you
can take a little battery here,

58:15.467 --> 58:18.077
connect it to two plates.

58:18.079 --> 58:21.639
This is positive,
this is negative.

58:21.639 --> 58:25.169
If you release a proton here,
for example,

58:25.166 --> 58:28.086
it will fall down the
potential.

58:28.090 --> 58:32.370
So it will lose a potential
energy, q times V.

58:34.768 --> 58:36.328
And just when it hits the
bottom plate,

58:36.329 --> 58:39.049
if you have a hole there,
it'll go through the hole and

58:39.054 --> 58:42.084
you have a particle accelerator
whose output velocity will be

58:42.083 --> 58:47.033
v,
given by this equation.

58:47.030 --> 58:51.120
But now if you want to
accelerate it to a higher and

58:51.123 --> 58:54.663
higher energies,
you've got to get bigger and

58:54.655 --> 58:56.175
bigger voltage.

58:56.179 --> 58:58.839
So how are we going to do that?

58:58.840 --> 59:01.540
Here's what Lawrence did.

59:01.539 --> 59:04.729
So take a magnetic field.

59:04.730 --> 59:08.750
In fact, there are two metallic
halves called Ds,

59:08.751 --> 59:11.021
because of the way they look.

59:11.018 --> 59:15.448
You send a particle here and at
the instant you release it,

59:15.449 --> 59:17.799
imagine that this plate is
positive and this plate is

59:17.797 --> 59:20.017
negative,
and the particle is positively

59:20.023 --> 59:20.533
charged.

59:20.530 --> 59:23.350
That means the whole plate has
one potential,

59:23.353 --> 59:26.373
and the other plate has a
negative potential.

59:26.369 --> 59:32.259
So this particle will speed up
when it comes here.

59:32.260 --> 59:34.710
Now the whole thing is immersed
in a magnetic field,

59:34.713 --> 59:36.063
which I'm not showing you.

59:36.059 --> 59:42.029
So this guy will bend and come
out of this side.

59:42.030 --> 59:45.540
Now what will happen to it?

59:45.539 --> 59:49.249
What will it do now?

59:49.250 --> 59:50.990
If you don't do anything,
what will happen to the

59:50.994 --> 59:51.544
particle now?

59:51.539 --> 59:53.159
Student:  It will slow
down.

59:53.159 --> 59:55.489
Prof: It will slow down,
so that's not the accelerator.

59:55.489 --> 59:58.219
You've got one gain and
immediately lost it.

59:58.219 --> 1:00:02.959
But suppose when it's here,
you very cleverly change the

1:00:02.960 --> 1:00:04.600
polarity of this?

1:00:04.599 --> 1:00:07.839
Just when it's leaving this
hole and going to the other

1:00:07.844 --> 1:00:11.394
D, you change the
polarity, so it's falling again.

1:00:11.389 --> 1:00:14.249
So it picks up a lot of speed
and goes on a bigger spiral.

1:00:14.250 --> 1:00:19.490
Here is doesn't experience any
field, goes around and comes

1:00:19.487 --> 1:00:20.117
here.

1:00:20.119 --> 1:00:24.809
When it comes there,
you quietly change it one more

1:00:24.813 --> 1:00:28.573
time, so it's always falling
downhill.

1:00:28.570 --> 1:00:30.600
So every time it's ready to
cross the gap,

1:00:30.599 --> 1:00:31.739
you flip the voltage.

1:00:31.739 --> 1:00:33.429
How do you do that?

1:00:33.429 --> 1:00:36.089
You're not going to stand there
with the battery and keep

1:00:36.085 --> 1:00:37.455
switching the wires,
right?

1:00:37.460 --> 1:00:38.910
So what do you think you do?

1:00:38.909 --> 1:00:43.009
Student:  Alternating
current?

1:00:43.010 --> 1:00:44.770
Prof: Put an alternating
voltage.

1:00:44.768 --> 1:00:49.588
Take the power supply from your
house, 60 cycle a second,

1:00:49.588 --> 1:00:53.978
denoted by this symbol,
AC voltage, will reverse its

1:00:53.976 --> 1:00:56.296
polarity automatically.

1:00:56.300 --> 1:01:00.310
And the beauty of this process
is the following:

1:01:00.306 --> 1:01:03.456
once it's got a certain
frequency,

1:01:03.460 --> 1:01:07.090
it will produce a cyclotron
orbit whose omega matches that

1:01:07.094 --> 1:01:07.864
frequency.

1:01:07.860 --> 1:01:09.240
Even though the particle is
speeding up--

1:01:09.239 --> 1:01:12.769
here's the whole point--even
though the particle is speeding

1:01:12.773 --> 1:01:16.513
up and going in bigger circles,
the ratio of the speed to the

1:01:16.514 --> 1:01:20.044
radius is such that it takes the
same time to go around these

1:01:20.039 --> 1:01:22.229
semicircles,
no matter how big they are,

1:01:22.228 --> 1:01:24.168
no matter how fast the
particle's moving.

1:01:24.170 --> 1:01:28.050
It's the fact that v
over R does not depend on

1:01:28.048 --> 1:01:30.218
R, R on v.

1:01:30.219 --> 1:01:33.669
So that's the omega that you
will take for this guy.

1:01:33.670 --> 1:01:37.030
Then it will keep on picking up
speed and at the very end,

1:01:37.034 --> 1:01:40.044
it will do a few more such
things, then it will come

1:01:40.043 --> 1:01:41.523
shooting out of here.

1:01:41.518 --> 1:01:43.668
So you don't need--in the end,
you can get 1 million volts,

1:01:43.666 --> 1:01:45.366
but you don't have a 1 million
volt battery.

1:01:45.369 --> 1:01:47.259
You have a 1 volt battery.

1:01:47.260 --> 1:01:49.450
It gives it a kick a million
times.

1:01:49.449 --> 1:01:50.989
A million times you cross the
midpoint.

1:01:50.989 --> 1:01:55.859
It's a very clever way to make
the particle accelerate.

1:01:55.860 --> 1:01:59.260
The Stanford linear accelerator
is somewhat different.

1:01:59.260 --> 1:02:03.040
There you have a lot of
cylinders and charged particles

1:02:03.039 --> 1:02:04.929
go from one to the other.

1:02:04.929 --> 1:02:07.549
But once again,
the polarity is reversed,

1:02:07.548 --> 1:02:10.888
so every time it goes there,
it's constantly falling

1:02:10.887 --> 1:02:11.737
downhill.

1:02:11.739 --> 1:02:15.639
It's just this thing taken out
and made into a straight line,

1:02:15.641 --> 1:02:19.091
that's about two miles long,
I think, in Palo Alto.

1:02:19.090 --> 1:02:22.630
And that was the machine that
discovered a lot of great

1:02:22.634 --> 1:02:25.854
things, including what's called
the J particle.

1:02:25.849 --> 1:02:30.609
So the linear accelerator's
one, but this is the cyclotron.

1:02:30.610 --> 1:02:36.930
So the next question you want
to ask is--these are all

1:02:36.934 --> 1:02:43.974
examples of forces on a single
charge--what's the force on a

1:02:43.974 --> 1:02:45.054
wire?

1:02:45.050 --> 1:02:49.420
Sometimes you do microscopic
experiments with tiny particles.

1:02:49.420 --> 1:02:53.270
Other times you do macroscopic
experiments with wires carrying

1:02:53.268 --> 1:02:53.898
current.

1:02:53.900 --> 1:02:55.270
So I want to look at that
problem.

1:02:55.268 --> 1:03:04.508
So let's look at the wire
carrying current.

1:03:04.510 --> 1:03:09.140
So here's a piece of wire,
and I want to find just a

1:03:09.139 --> 1:03:12.589
little segment,
which I write as vector

1:03:12.588 --> 1:03:13.858
dl.

1:03:13.860 --> 1:03:17.010
There's some current going here
and the whole thing is bathed in

1:03:17.010 --> 1:03:19.860
a magnetic field B,
which is constant over the tiny

1:03:19.860 --> 1:03:20.460
segment.

1:03:20.460 --> 1:03:23.470
We want to know what's the
force on the little segment.

1:03:23.469 --> 1:03:25.379
There's going to be a force
because there are guys moving in

1:03:25.382 --> 1:03:25.742
the wire.

1:03:25.739 --> 1:03:26.649
That's it.

1:03:26.650 --> 1:03:28.520
Each one feels a v x
B force.

1:03:28.519 --> 1:03:31.939
You've got to add it up.

1:03:31.940 --> 1:03:35.390
So if this has got length
dl, how many charges am I

1:03:35.391 --> 1:03:36.361
talking about?

1:03:36.360 --> 1:03:40.140
The number of charges is the
density of carriers times the

1:03:40.144 --> 1:03:41.544
charge of each one.

1:03:41.539 --> 1:03:44.929
That's the number of charges,
per unit volume.

1:03:44.929 --> 1:03:47.719
If the cross section of the
wire is A,

1:03:47.724 --> 1:03:50.974
the length is dl,
that's how many charges are

1:03:50.965 --> 1:03:52.105
involved here.

1:03:52.110 --> 1:03:53.170
Do you understand?

1:03:53.170 --> 1:03:55.280
This cross section is A.

1:03:55.280 --> 1:03:58.260
A times dl is the
volume of the cylinder.

1:03:58.260 --> 1:04:01.410
That times the number per
volume times the charge for each

1:04:01.405 --> 1:04:02.835
guy is the total charge.

1:04:02.840 --> 1:04:10.700
And each one of them
experiences the force v x

1:04:10.697 --> 1:04:12.357
B.

1:04:12.360 --> 1:04:13.990
Now I'm going to do a little
switch here.

1:04:13.989 --> 1:04:20.159
I'm going to write it as
Anev dl x

1:04:20.155 --> 1:04:21.595
B.

1:04:21.599 --> 1:04:23.549
So you see if you follow that.

1:04:23.550 --> 1:04:27.020
The velocity of the carriers is
along the wire.

1:04:27.018 --> 1:04:29.478
dl is also along the
wire.

1:04:29.480 --> 1:04:32.250
Therefore either you can take
the magnitude of this vector

1:04:32.248 --> 1:04:34.778
times the vector v,
or the magnitude of vector

1:04:34.775 --> 1:04:36.375
v times the dl.

1:04:36.380 --> 1:04:37.450
It's the same thing.

1:04:37.449 --> 1:04:40.809
They're both the vector
parallel to either one.

1:04:40.809 --> 1:04:44.879
But if you write it this way,
if you go back to what we did

1:04:44.875 --> 1:04:48.235
earlier, this is just the
current in the wire.

1:04:48.239 --> 1:04:53.969
So we have a nice formula that
says the force on a segment of

1:04:53.971 --> 1:04:59.611
wire carrying current I
is I times dl x

1:04:59.608 --> 1:05:00.848
B.

1:05:00.849 --> 1:05:01.769
So here's an example.

1:05:01.768 --> 1:05:05.488
Suppose you've got a magnetic
field like this.

1:05:05.489 --> 1:05:06.409
And I've got a wire.

1:05:06.409 --> 1:05:08.109
I don't know where it's coming
from, I don't know where it's

1:05:08.108 --> 1:05:08.338
going.

1:05:08.340 --> 1:05:10.940
Let's say it's just carrying a
current.

1:05:10.940 --> 1:05:14.000
And imagine this is in the real
vertical plane,

1:05:14.000 --> 1:05:17.260
so that there are some weights
hanging from it.

1:05:17.260 --> 1:05:19.330
It's trying to fall down.

1:05:19.329 --> 1:05:24.589
You can balance that by
applying a force of magnetism to

1:05:24.585 --> 1:05:29.935
that wire, because the magnetic
force will be dl x

1:05:29.936 --> 1:05:31.176
B.

1:05:31.179 --> 1:05:35.799
Let's see, so dl is this
way, B is out of the

1:05:35.802 --> 1:05:39.392
board, so let me set the current
like this.

1:05:39.389 --> 1:05:45.749
Then dl is this way,
cross B will be a force

1:05:45.753 --> 1:05:46.873
upwards.

1:05:46.869 --> 1:05:50.799
There'll be an upward force of
B times the length of the

1:05:50.802 --> 1:05:52.392
wire times the current.

1:05:52.389 --> 1:05:55.409
And if that's equal to the mass
of the wire times g,

1:05:55.409 --> 1:05:57.389
then the wire won't fall up or
down;

1:05:57.389 --> 1:06:00.269
it will just stay there.

1:06:00.268 --> 1:06:03.308
So you can hold up a piece of
wire by driving a current

1:06:03.313 --> 1:06:05.853
through it and putting in a
magnetic field.

1:06:05.849 --> 1:06:08.889
But here's another simple
homework problem,

1:06:08.891 --> 1:06:12.151
or some problem you can find in
a textbook.

1:06:12.150 --> 1:06:15.100
This is a semicircular wire.

1:06:15.099 --> 1:06:18.039
Now this wire cannot just start
and end here because the current

1:06:18.038 --> 1:06:20.278
has to come from somewhere and
go from somewhere,

1:06:20.277 --> 1:06:21.627
so maybe it's doing this.

1:06:21.630 --> 1:06:25.280
But I'm just focusing on this
section.

1:06:25.280 --> 1:06:29.720
And imagine there is a magnetic
field going like this.

1:06:29.719 --> 1:06:31.789
I want to find the force on the
semicircle.

1:06:31.789 --> 1:06:33.529
This is supposed to be a
semicircle.

1:06:33.530 --> 1:06:38.180
What's the force on it?

1:06:38.179 --> 1:06:40.809
So take a portion of the wire
here.

1:06:40.809 --> 1:06:42.959
This is my dl.

1:06:42.960 --> 1:06:44.340
This is my B.

1:06:44.340 --> 1:06:46.220
Take the cross product of dl
with B.

1:06:46.219 --> 1:06:48.089
Can you see which way it will
go?

1:06:48.090 --> 1:06:50.790
Do the old screwdriver,
your favorite thing.

1:06:50.789 --> 1:06:51.849
Do this.

1:06:51.849 --> 1:06:56.499
It's coming out of the
blackboard.

1:06:56.500 --> 1:07:00.630
And what's the force that's
coming out of the blackboard due

1:07:00.630 --> 1:07:04.970
to the little section = I
times dl times B

1:07:04.969 --> 1:07:07.629
times the angle between these
two.

1:07:07.630 --> 1:07:11.090
And that will turn out to be a
sine theta.

1:07:11.090 --> 1:07:13.780
If you want,
you can measure theta from here

1:07:13.775 --> 1:07:16.705
and you can see that dl
is like that,

1:07:16.710 --> 1:07:19.560
B is like that,
and the angle between the

1:07:19.557 --> 1:07:23.257
horizontal and the radial is the
same as the angle between the

1:07:23.255 --> 1:07:26.585
perpendicular to the radial and
the perpendicular to the

1:07:26.588 --> 1:07:27.618
horizontal.

1:07:27.619 --> 1:07:30.569
So this theta is the same as
that theta.

1:07:30.570 --> 1:07:36.020
Then you integrate the force,
you get I.

1:07:36.018 --> 1:07:39.888
dl will be just
2ΠR,

1:07:39.891 --> 1:07:41.721
times B.

1:07:41.719 --> 1:07:45.709
Sorry, dl will be
ΠR time B,

1:07:45.710 --> 1:07:49.080
times integral of
sinθ

1:07:49.083 --> 1:07:53.703
dθ,
which is -cosθ

1:07:53.701 --> 1:08:02.411
from 0 to Π,
that will give you a 2.

1:08:02.409 --> 1:08:07.559
So that will be the force
coming out of the blackboard on

1:08:07.559 --> 1:08:08.939
this section.

1:08:08.940 --> 1:08:24.210
I should say theta from 0 to
Π.

1:08:24.210 --> 1:08:26.880
I think there's something wrong
with this formula.

1:08:26.880 --> 1:08:28.970
Hold on for a second, guys.

1:08:28.970 --> 1:08:30.940
It should not be done this way.

1:08:30.939 --> 1:08:36.379
dl = Rdθ.

1:08:36.380 --> 1:08:42.140
There's I times B
and Rdθ

1:08:42.136 --> 1:08:48.336
sinθ from 0 to
Π, and that will give you

1:08:48.344 --> 1:08:50.154
2IBR.

1:08:50.149 --> 1:08:52.989
Do you understand that the
length of the wire,

1:08:52.990 --> 1:08:55.520
this segment dl,
is R times

1:08:55.516 --> 1:08:56.776
dθ.

1:08:56.779 --> 1:08:59.669
Theta is measured in radians.

1:08:59.670 --> 1:09:07.050
Anyway that is a force and this
section tells you how to

1:09:07.045 --> 1:09:10.125
calculate that force.

1:09:10.130 --> 1:09:13.290
All right, the last thing which
I will start but which I will

1:09:13.287 --> 1:09:16.187
not finish today,
and you can try to look at some

1:09:16.188 --> 1:09:19.168
good pictures before you come to
class next time.

1:09:19.170 --> 1:09:21.970
There is just no way I can get
this right.

1:09:21.970 --> 1:09:25.410
So here is a current loop.

1:09:25.409 --> 1:09:28.929
And the current goes like this.

1:09:28.930 --> 1:09:32.330
And it's in a magnetic field
like this.

1:09:32.328 --> 1:09:37.468
The question is,
what happens to it?

1:09:37.470 --> 1:09:40.800
So you've got to go to the four
sides of the loop,

1:09:40.796 --> 1:09:44.736
in this side the v x
B, your current is going

1:09:44.737 --> 1:09:45.617
this way.

1:09:45.619 --> 1:09:46.839
B is going straight up.

1:09:46.840 --> 1:09:48.870
v x B will act
like this.

1:09:48.868 --> 1:09:50.648
There the v x B
will act like this.

1:09:50.649 --> 1:09:55.639
Here it will act like this and
here it will act like this.

1:09:55.640 --> 1:10:01.190
And this is the normal to the
plane, it's the area vector.

1:10:01.189 --> 1:10:04.099
So let me draw you a side view,
because that's the only thing I

1:10:04.095 --> 1:10:05.075
can even try to do.

1:10:05.078 --> 1:10:09.408
In the side view,
it looks like this.

1:10:09.408 --> 1:10:12.978
These two edges are getting
pulled.

1:10:12.979 --> 1:10:15.669
This edge that you can see here
is getting pulled out of the

1:10:15.673 --> 1:10:16.043
board.

1:10:16.038 --> 1:10:18.218
The other one's getting pulled
away from the board.

1:10:18.220 --> 1:10:20.920
They just try to distort the
loop, but they cancel each other

1:10:20.916 --> 1:10:22.216
and they don't do anything.

1:10:22.220 --> 1:10:26.660
But these two are equal as
forces, but as a torque,

1:10:26.658 --> 1:10:31.628
they are not equal and
opposite, but they're additive.

1:10:31.630 --> 1:10:35.320
So they together produce a
torque and let's find out what

1:10:35.315 --> 1:10:36.365
the torque is.

1:10:36.368 --> 1:10:40.348
The force on this segment is
B times l times

1:10:40.353 --> 1:10:42.803
I, where l is
this.

1:10:42.800 --> 1:10:45.960
Let the other dimension be
w, which stands for

1:10:45.962 --> 1:10:46.452
width.

1:10:46.449 --> 1:10:48.909
This is w here.

1:10:48.908 --> 1:10:55.438
Then the torque will be that
times wsinθ.

1:10:55.439 --> 1:11:00.099
That is the torque.

1:11:00.100 --> 1:11:04.110
But l times w is
the area of the loop.

1:11:04.109 --> 1:11:07.939
B times A times
sinθ.

1:11:07.939 --> 1:11:10.769
If I represent the area as a
vector,

1:11:10.770 --> 1:11:15.740
pointing on this direction,
then the torque is simply given

1:11:15.743 --> 1:11:20.413
by μ x B,
where μ is a vector

1:11:20.414 --> 1:11:25.244
whose magnitude is equal to the
current in the loop times area.

1:11:25.239 --> 1:11:30.519
I forgot the I.

1:11:30.520 --> 1:11:37.540
And this is called the magnetic
moment of the loop.

1:11:37.538 --> 1:11:40.218
You call it a magnetic moment,
because it's just like a dipole

1:11:40.215 --> 1:11:40.605
moment.

1:11:40.609 --> 1:11:42.649
You might remember,
if I take a dipole,

1:11:42.649 --> 1:11:46.429
put it in a field,
the torque on it is p x

1:11:46.430 --> 1:11:51.080
E and the energy of it is
-p⋅E.

1:11:51.078 --> 1:11:55.468
So magnetic loop looks like a
dipole in a magnetic field.

1:11:55.470 --> 1:11:58.940
In other words,
if there were magnetic charges

1:11:58.938 --> 1:12:01.468
in nature,
which there aren't,

1:12:01.465 --> 1:12:06.195
then this guy behaves exactly
like a magnetic charge and a -

1:12:06.201 --> 1:12:08.851
on the other side of the loop.

1:12:08.850 --> 1:12:12.460
So if magnetic charges existed,
this would be simply a magnetic

1:12:12.457 --> 1:12:14.027
dipole,
aligning itself with the

1:12:14.034 --> 1:12:15.694
magnetic field,
but in reality,

1:12:15.686 --> 1:12:17.516
there are no magnetic poles.

1:12:17.520 --> 1:12:21.050
But there is just the loop that
behaves like a dipole.

1:12:21.050 --> 1:12:24.630
And the dipole moment of the
loop, which is the analog of

1:12:24.631 --> 1:12:28.341
distance times the charge is the
current times the area.

1:12:28.340 --> 1:12:30.980
So we'll take it from here next
time.

1:12:30.979 --> 1:12:35.999