Prof: So,
I've got to start by telling you the syllabus for this
term--not the detailed one, just the big game plan.
The game plan is:
we will do electromagnetic theory.
Electromagnetism is a new force
that I will introduce to you and go through all the details.
And I will do optics,
and optics is part of electromagnetism.
And then near the end we will
do quantum mechanics. Now, quantum mechanics is not
like a new force. It's a whole different ball
game. It's not about what forces are
acting on this or that object that make it move,
or change its path. The question there is:
should we be even thinking about trajectories?
Should we be even thinking
about particles going on any trajectory?
Forget about what the right
trajectory is. And you will find out that most
of the cherished ideas get destroyed.
But the good news is that you
need quantum mechanics only to study very tiny things like
atoms or molecules. Of course the big question is,
you know, where do you draw the line?
How small is small?
Some people even ask me,
"Do you need quantum mechanics to describe the human
brain?" And the answer is,
"Yes, if it is small enough."
So, I've gone to parties where
after a few minutes of talking to a person I'm thinking,
"Okay, this person's brain needs a fully quantum mechanical
treatment." But most of the time everything
macroscopic you can describe the way you do with Newtonian
mechanics, electrodynamics. You don't need quantum theory.
All right, so now we'll start
with the brand new force of electromagnetism.
But before doing the force,
I've got to remind you people of certain things I expect you
all to understand about the dynamics between force,
and mass, and acceleration that you must have learned last term.
I don't want to take any
chances. I'm going to start by reminding
you how we use this famous equation of Newton.
So you've seen this equation,
probably, in high school,
but it's a lot more subtle than you think,
certainly a lot more subtle than I thought when I first
learned it. So I will tell you what I
figured out over these years on different ways to look at F =
ma. In other words,
if you have the equation what's it good for?
The only thing anybody knows
right away is a stands for acceleration,
and we all know how to measure it.
By the way, anytime I write any
symbol on the board you should be able to tell me how you'd
measure it, otherwise you don't know what
you're talking about as a physicist.
Acceleration,
I think I won't spend too much time on how you measure it.
You should know what
instruments you will need. So I will remind you that if
you have a meter stick, or many meter sticks and clocks
you can follow the body as it moves.
You can find its position now,
its position later, take the difference,
divide by the time, you get velocity.
Then find the velocity now,
find the velocity later, take the difference,
divide by time, you've got acceleration.
So acceleration really requires
three measurements, two for each velocity,
but we talk of acceleration right now because you can make
those three measurements arbitrarily near each other,
and in the limit in which the time difference between them
goes to zero you can talk about the velocity right now and
acceleration right now. But in your car,
the needle points at 60 that's your velocity right now.
It's an instantaneous quantity.
And if you step on the gas you
feel this push. That's your acceleration right
now. That's a property of that
instant. So we know acceleration,
but the question is can I use the equation to find the mass of
anything. Now, very often when I pose the
question the answer given is, you know, go to a scale,
a weighing machine, and find the mass.
And as you know,
that's not the correct answer because the weight of an object
is related to being near the earth due to gravity,
but the mass of an object is defined anywhere.
So here's one way you can do it.
Now you might say, "Well,
take a known force and find the acceleration it produces,"
but we haven't talked about how to measure the force either.
All you have is this equation.
The correct thing to do is to
buy yourself a spring and go to the Bureau of Standards and tell
them to loan you a block of some material,
I forgot what it is. That's called a kilogram.
That is a kilogram by
definition. There is no God-given way to
define mass. You pick a random entity and
say that's a kilogram. So that's not right and that's
not wrong. That's what a kilogram is.
So you bring that kilogram,
you hook it up on the spring, and you pull it by some amount,
maybe to that position, and you release it.
You notice the acceleration of
the 1 kilogram, and the mass of the thing is
just one. Then you detach that mass.
Then you ask--Then the person
says, "What's the mass of something else?"
I don't know what the something
else is. Let's say a potato.
And you take the potato or
anything, elephant. Here's a potato.
You pull that guy by the same
distance, and you release that, and you find its acceleration.
Since you pulled it by the same
amount, the force is the same, whatever it is.
We don't know what it is,
but it's the same. Therefore we know the
acceleration of 1 kilogram times 1 kilogram is equal to the
unknown mass times the acceleration of the unknown
mass. That's how by measuring this
you can find what the mass is. In principle you can find the
mass of everything. So imagine masses of all
objects have been determined by this process.
Then you can also use F =
ma to find out what forces are acting on bodies in
different situations, because if you don't know what
force is acting on a body you cannot predict anything.
So you can go back to the
spring and say, "I want to know what force
the spring exerts when it's pulled by various amounts.
Well, you pull it by some
amount x. You attach it to a non-mass and
you find the acceleration, and that's the force.
And if you plot it,
you'll find F as a function of x will be
roughly a straight line and it will take the form F =
-kx, and that k is called a
force constant. So this is an example of your
finding out the left hand side of Newton's law.
You've got to understand the
distinction between F = -kx and F = ma.
What's the difference?
This says if you know the force
I can tell you the acceleration, but it's your job to go find
out every time what forces might be acting on a body.
If it's connected to a spring,
and you pull the spring and it exerts a force,
someone's got to make this measurement to find out what the
force will be. All right, so that's one kind
of force. Another force that you can find
is if you're near the surface of the earth,
if you drop something, it seems to accelerate towards
the ground, and everything accelerates by
the same amount g. Well, according to Newton's
laws if anything is going to accelerate, it's because there's
a force on it. The force on any mass m
must be mg, because if I divide by m
I've got to get g. So the force on masses near the
earth is mg. That's another force.
Something interesting about
that force is that unlike the spring force where the spring is
touching the mass, you can see it's pulling it,
or when I push this chair you can see I'm doing it,
the pull of gravity is a bit strange,
because there is no real contact between the earth and
the object that's falling. It was a great abstraction to
believe that things can reach out and pull things which are
not touching them, and gravity was the first
formally described force where that was true.
And another excursion in the
same theme is if this object gets very far,
say like the moon over there, then the force is not given by
mg, but the force is given by this
law of gravitation. For every r near the
surface of the earth, if you put r equal to
the surface of the earth you will get a constant force that
is just mg, but if you move far from the
center of the earth you've got to take that into account,
and that's what Newton did and realized the force goes like 1
over r^(2). So every time things accelerate
you've got to find the reason, and that reason is the force.
Many times many forces can be
acting on a body, and if you put all the forces
that are acting on a body and that explains the acceleration,
you're done, but sometimes it won't.
That's when you have a new
force. And the final application of
F = ma is this one. If you knew the force,
for example, on a planet,
and here's a planet going around the sun and it is here.
This is the sun,
and you know the force acting on it given by Newton's Law of
Gravity you can find the acceleration that will help you
find out where it will be one second later,
and you repeat the calculation, you will get the trajectory.
So F = ma is good for
three things, that's what I want you to
understand: to define mass, to calculate forces acting on
bodies by seeing how they accelerate,
and finally to find the acceleration of bodies given the
forces. This is the cycle of Newtonian
dynamics. And what I'm going to do now is
to add one more new force, because I'm going to find out
that there is another force not listed here.
I'm going to demonstrate to you
that new force, okay?
Here's my demonstration.
The only demonstration you will
see in my class, because everything else I've
tried generally failed, but this one always works.
So, I have here a piece of
paper, okay? Then I take this trusty comb
and I comb the part of my head that's suited for this
experiment, then I bring it next to this,
and you see I'm able to lift that.
Now, that's not the force of
gravity because gravity doesn't care if you comb your hair or
not, okay? And also when I shake it,
it falls down. So you're thinking, "Okay,
maybe there is a new force but it doesn't look awfully strong
because it's not able to even overcome gravity,
because it eventually yielded to gravity and fell down,"
but it's actually a mistake to think so.
In fact this new force that I'm
talking about is 10 to the power of 40 stronger than
gravitational force. I will tell you by what metric
I came up with that number, but it's an enormously strong
force. You've got to understand why I
say it is such a strong force when, when I shook it the thing
fell down. So the reason is that if you
look at this experiment, here's the comb and here's the
paper, the comb is trying to pull the
paper, but what is trying to pull it
down? What is trying to pull it down?
So here is me,
here is that comb, here's the paper.
The entire planet is pulling it
down: Himalayas pulling it down, Pacific Ocean,
pulling it down, Bin Laden sitting in his cave
pulling it down. Everything is pulling it down,
okay? I am one of these people
generally convinced the world is acting against me,
but this time I'm right. Everything is acting against
me, and I'm able to triumph against all of that with this
tiny comb. And that is how you compare the
electric force with the gravitational force.
It takes the entire planet to
compensate whatever tiny force I create between the comb and the
piece of paper. To really get a number out of
this I'll have to do a little more,
but I just want to point out to you this is a new force much
stronger than gravitation. So I want to tell you a few
other experiments people did without going into what the
explanation is right now, but let me just tell you if you
go through history what all did people do.
So one experiment you can do:
You take a piece of glass and you rub it on some animal that's
passing by, water buffalo. That's why I cannot do all the
experiments in class. You rub it on that guy,
then you do it to a second piece of glass,
and you find out that they repel each other,
meaning if you put them next to each other they tend to fly
apart. Then you take a piece of hard
rubber and you rub that on something else.
I forgot what,
silk, Yeti, some other thing. Then you put that here.
So I'll give a different shape
to that thing. That's the rubber stick.
And you find when you do that
to this, these two attract each other.
Sometimes they repel,
sometimes they attract. Here's another thing you can
do: Buy some nylon thread. You hang a small metallic
sphere, and you bring one of these rods next to it.
It doesn't matter which one.
Initially they're attracted and
suddenly when you touch it and you remove it,
they start repelling each other.
What's going on?
That's another thing you could
do. Last thing I want to mention is
if you took two of these things which are repelling each other,
let's say. Let's say they're attracting
each other like this. Then you connect them with a
piece of nylon and you take it away, nothing happens.
If you connect them with a
piece of wire and take away the wire, they no longer attract
each other. So these are examples of
different things. I'm just going to say,
you do this, you do this,
you do that, then finally you need a theory
that explains everything. So that's the theory that I'm
going to give you now. That's the theory of
electrostatics. And I don't have time to go
into the entire history of how people arrived at this final
formula, so I'm just going to tell you
one formula that really will explain everything that I've
described so far, and that formula is called
Coulomb's Law. Even though Mr. Coulomb's
name is on it, he was not the first one to
formulate parts of the law, but he gave the final and
direct verification of Coulomb's Law that other people who had
contributed. So Coulomb's Law says that
certain entities have a property called charge.
You have charge or you don't
have charge, but if you have charge the charge that you have,
you meaning any of these objects, is measured in
coulombs. Remember, that was not
Coulomb's idea to call it coulomb.
Whenever you make a discovery,
you're breathlessly waiting that somebody will name it after
you, but it's not in good taste to
name to after yourself, but it carries Coulomb's name.
So he didn't say call it
coulomb, okay, but he certainly wrote down
this law. The law says that if you've got
one entity which has some amount of charge called
q_1, and there's another entity that
has some amount of charge q_2 they will
exert a force on each other which is given by
q_1q _2 times this
constant which is somehow written as 1 over
4Πε _0.
That's 1 over r^(2).
But r is the distance
between them, and you can ask in this
picture, what do you mean by distance?
I mean, is it from here to
there, or is it from center to center?
We're assuming here that the
distance between them is much bigger than the individual
sizes. For example,
you say, how far am I from Los Angeles, well,
3,225 miles, but you can say are you taking
about your right hand or your left hand?
Well, I'm a point particle for
this purpose so it doesn't matter.
So here we're assuming that
either they're mathematically point charges or they're real
charges with a finite size but separated by a distance much
bigger than the size, so r could stand,
if you like, for center to center.
It doesn't matter too much.
So this is what Coulomb said.
Now, if you look at this number
here, 1 over 4Πε
_0, its value is 9 times 10 to the
9^(th). What that means is the
following: If you take one body with 1 coulomb of charge,
another body with 1 coulomb of charge and they're separated by
1 meter, then the force between them
will be this number, because everything else is a 1.
It'll be 9 times 10 to the 9
newtons. That's an enormous force,
and normally you don't run into 1 coulomb of charge,
but the reason why a coulomb was picked is sort of historical
and it has to do with currents and so on.
But anyway, this is the
definition. But if you want to be more
precise, I should write a formula more carefully because
force is a vector. Also I should say force on whom
and due to what. So let's say there are two
charges, and say q_1 is
sitting at the origin and q_2 is sitting
at a point whose position is the vector r.
Then the force on 2 due to 1 is
given by q_2q _1 over
4Πε_0 times 1 over
r^(2). That's the magnitude of the
force, but I want to suggest that the force is such that
q_1 pushes q_2 away.
So I want to make this into a
vector, but I've got the magnitude of the vector.
As you know,
to make a real vector you take its magnitude and multiply it by
a vector of unit length in that same direction.
The unit vector we can write in
many ways. One is just to say
e_r, e_r_
is a standard name for a vector of length 1 in the
direction of r. But I'll give you another
choice. You can also write it as
r divided by the length of r.
That also would be a vector of
unit length parallel to r.
So there are many ways to write
the thing that makes it a vector.
And F_21 is
minus of F_12. Now, how do we get attraction
and how do we get repulsion? We get it because
q_1 and q_2,
if they're both positive and you if you use the formula,
you'll find they repel each other, but if they're of
opposite signs, you'll do the same calculation,
but you'll put a minus sign in front of the whole thing.
That'll turn repulsion into an
attraction. So you must allow for the
possibility that q can be of either sign;
q can also be 0. There are certain entities
which don't have any electric charge, so if you put them next
to a million coulombs nothing happens.
So some things have plus charge.
Some things have minus charge.
Some things have no charge,
but they're all contained in this Coulomb's Law.
Now, again, skipping all the
intermediate discoveries, I want to tell you a couple of
things we know about charge. First thing is - q is
conserved. Conserved is a physics terms
for saying--does not change with time.
For example,
when you say energy is conserved,
it means particles can come and collide and do all kinds of
things, but if you add that energy
before, you'll get the same answer afterwards,
and whenever that happens, the quantity is conserved.
The claim is electrical charge
is conserved. So electrical charge may
migrate from A to B or B to A, but if you add up the total
charge, say the chemical reaction of
any process, including in big particle
accelerators where things collide and all kinds of stuff
comes flying out, the charge of the final
products always equal to the charge of the incoming products.
But charge conservation needs
to be amended with one extra term, extra qualification.
It's called local.
Suppose I say the number of
students in the class is conserved?
That means you count them any
time, you've got to get the same number.
Well, here's one possibility.
Suddenly one of you guys
disappears and appears here at the same instant.
That's also consistent with
conservation of student number because the number didn't
change. What disappeared there,
appeared here. But that is not a local
conservation of charge because it disappears in one part of the
world and appears in another one.
And it's not even a meaningful
law to have in the presence of relativity.
Can any of you guys think of
why that might be true, why a charge disappearing
somewhere and appearing somewhere else cannot be a very
profound principle? Yes?
Student:
> Prof: Yep?
Student: Well,
if it's in the same instant disappearing from one place and
appearing another place, it's traveling faster than
light? Prof: Well,
we don't know that it was the same thing that even traveled.
It may not have traveled.
It may even be--Here's another
thing. Suppose an electron,
suppose a proton disappears there and a positron appears
here. That still conserves charge,
but we don't think that the proton traveled and became the
positron, right? So it is not that it has
traveled. You are right.
I hadn't thought about that.
It's a good point that it
implies it traveled infinitely fast, but that's not the reason
you object to it. Yep?
Student: It's not
necessarily simultaneous. Prof: That is the
correct answer. The answer is it is not
simultaneous in every frame of reference.
You must know from the special
theory that if two events are simultaneous in one frame of
reference, if you see those same two
events in a moving train, or plane, or anything they will
not be simultaneous. Therefore, in any other frame
of reference, either the charge would have
been created first and then after a period of time
reappeared somewhere, I mean, destroyed somewhere and
appeared after a delay, or the appearance could take
place before the destruction, so suddenly you've got two
charges. So conservation of charge,
which is conserved non-locally, cannot have a significance
except in one frame of reference,
but if you believe that all observers are equivalent and you
want to write down laws that make sense for everybody it can
only be local. So electrical charge is
conserved and it is local, locally conserved.
In other words,
stuff doesn't just disappear. Stuff just moves around.
You can keep track of it,
and if you add it up you get the same number.
The second part of q,
which is not necessary for any of these older phenomena,
is that q is quantized. That means the electrical
charge that we run into does not take a continuum of possible
values. For example,
the length of any object, you might think at least in
classical mechanics, is any number you like.
It's a continuous variable,
but electric charge is not continuous.
As far as we can tell,
all the charges we have ever seen are all multiples of a
certain basic unit of charge, which turns out to be 1.6 times
10 to the -19 coulombs. Every charge is either that or
some multiple of it. Multiple could be plus or minus
multiple. So charge is granular,
not continuous. Okay, so I'm going to give you
a little more knowledge we have had since the time of Coulomb
that sort or explains these things.
I mean, what's really going on
microscopically? We don't have to pretend we
don't know. We do, so we might as well use
that information from now on. What we do know is that
everything is made up of atoms, and that if you look into the
atom it's got a nucleus, a lot of guys sitting here.
Some are called protons and
some are called neutrons, and then there are some guys
running around called electrons. Of course we will see at the
end of the semester that this picture is wrong,
but it is good enough for this purpose.
It's certainly true that there
are charges in an atom which are near the center and other light
charges which are near the periphery, are outside.
All things carrying electric
charge in our world in daily life are either protons or
electrons. You can produce strange
particles in an accelerator. They would also carry some
charge which would in fact be a multiple of this charge,
but they don't live very long. So the stable things that you
and I are made of and just about everything in this room is made
of, is made up of protons, neutrons and electrons.
The charge of the neutron,
as you can guess, is 0.
The charge of the electron,
by some strange convention, was given this minus sign by
Franklin. And the charge of the proton is
plus 1.6 times into -19 coulombs.
There are a lot of amazing
things I find here. I don't know if you've thought
about it. The first interesting thing is
that every electron anywhere in the universe has exactly the
same charge. It also has exactly the same
mass. Now, you might say, "Look,
that's a tautology," because if it wasn't the same
charge and if it wasn't the same mass you would call it something
else. But what makes it a non-empty
statement is that there are many, many, many,
many electrons which are absolutely identical.
Look, you try to manufacture
two cars. The chance that they're
identical is 0, right?
I got one of those cars so I
know that. It doesn't work.
It's supposed to.
So despite all the best efforts
people make, things are not identical.
But at the microscopic level of
electrons and protons, every proton anywhere in the
universe is identical. And they can be manufactured in
a collision in another part of the universe.
This can be manufactured in a
collision in Geneva, the stuff that comes out
identical. That is a mystery,
at least in classical mechanics it's a mystery.
Quantum Field Theory gives you
an answer to at least why all electrons are identical,
and why all protons are identical.
The fact that they're
absolutely identical particles is very, very important.
It also makes your life easy,
because if every particle was different from every other
particle, you cannot make any predictions.
We know that the hydrogen atom
on a receding galaxy is identical to the hydrogen atom
on the Earth. That's why when the radiation
coming from the atom has a shifted wavelength of frequency,
we attributed to the motion of the galaxy.
From the Doppler Shift we find
out its speed. But another explanation could
be, well, that's a different hydrogen atom.
Maybe that's why the answer's
different. But we all believe it's the
same hydrogen atom, but it's moving away from us.
Therefore, one of the
remarkable things is that all electrons and all protons are
equal, but a really big mystery is why
is the charge of the electron exactly equal and opposite the
charge of the proton. They are not the same particle.
Their masses are different.
Their other interactions are
different. But in terms of electrical
charge these two numbers are absolutely equal as far as
anybody knows. That's another mystery.
Two different particles,
not related by any manifest family relationship,
have the same charge, except in sign.
And there are theories called
Grand Unified Theories which try to explain this,
but certainly not part of any standard established theory,
but it's key to everything we see in daily life because that's
what makes the atom electrically neutral.
Okay, now we can understand the
quantization of charge, because charge is carried by
these guys and these guys are either there or not there,
so you can only have so many electrons.
We cannot have a part of an
electron, or part of a proton. Now, let's try to understand
all these experiments in terms of what we know.
First of all,
when you take this piece of glass, and you rub it,
the atoms in glass are neutral. They've got equal number of
protons and electrons, but when you rub it,
the glass atom loses some electrons to whatever you rubbed
it on. Therefore, it becomes
positively charged, because some negative has been
taken out. In the case of the rubber
stick, it gains the electrons and whatever animal you rubbed
it on, it loses the electrons. So actually real charge
transfer takes place only through electrons.
Protons carry charge,
but you are never going to rip a proton out unless you use an
accelerator. It's really deeply bound to the
nucleus. Electrons are the ones who do
all the business of electricity in daily life.
The current flowing in the
wire, in the circuit, it's all the motion of
electrons. So from this and Coulomb's Law,
can you understand the attraction between these two?
How many people think you can,
from Coulomb's Law, understand the attraction
between these two rods? Nobody thinks you can?
Well, why do you think you
cannot? You know why?
Student: Because
they're not point charges? Prof: Okay,
any other reason why Coulomb's Law is not enough?
Well, how will we apply
Coulomb's Law to understand the attraction between these two
rods? What will you have to do?
Student: You'd have to
apply it to F = ma. Prof: No.
Once you got the F, the
a will follow, but can you compute the force
between two rods? One of them has got a lot of
positive charge. One of them has a lot of
negative charge given Coulomb's Law.
Yes?
Student: You don't know
the exact quantities of the charges..
Prof: Pardon me?
Student: You don't know
the exact quantities of the charges.
Prof: Suppose I tell you.
I tell you how many charges
there are. Yes?
Student: You don't
which direction the attraction is.
Prof: No,
we do know, because the plus and minus will be drawn towards
each other. Okay, I'll tell you what it is.
It's an assumption we all make,
but you're not really supposed to make it.
It's not a consequence of any
logic. Coulomb's Law talks about two
charges, two point charges. What if there are three charges
in the universe? What is the force this one will
experience due to these two? This is q_1.
This is q_2.
This is q_3.
Coulomb's Law doesn't tell you
that. It tells you only two at a
time, but we make an extra assumption called superposition
which says that if you want the force on 3 (should read 1),
when there is q_1 and
q_2, you find the force due to
q_2 and you find the force due to
q_3 and you add them up.
The fact that you can add these
two vectors is not a logical requirement.
In fact, it's not even true at
an extremely accurate level that the force between two charges is
not affected by the presence of a third one.
But it's an excellent
approximation, but you must realize it is
something you've got to find to be true experimentally.
It's not something you can say
is logical consequence. Logically there is no reason
why the interaction between two entities should not be affected
by the presence of a third one. But it seems to be a very good
approximation for what we do, and that's the reason why
eventually we can find the force between an extended object,
another extended object by looking at the force on everyone
of these due to everyone of those and adding all the
vectors. Okay, so superposition plus
Coulomb's Law is what you need. Then you can certainly
understand the attraction. How about the comb and the
piece of paper? That's a very interesting
example and it's connected to this one.
See, the piece of paper is
electrically neutral. So let me do paper and comb
instead of this one. It's got the same model.
Here's the piece of paper.
Here's the comb.
The comb is positively charged.
The paper is neutral.
So anyway, there's nothing here
to be attracted to this one, but if you bring it close
enough, there are equal amount of positive and negative
charges, but what will happen is the
negative charges will migrate near these positive charges from
the other end, leaving positive charges in the
back, so that the system will
separate into a little bit of negative closer to the positive,
and the leftover positive will be further away.
Therefore, even though it's
neutral the attraction of plus for this minus is stronger than
the repulsion of this plus with this plus.
That's called polarization.
So polarization is when charge
separates. Some materials cannot be
polarized, in which case no matter how much you do this with
a comb it won't work. Some materials can be polarized.
The piece of paper is an
example of what can be polarized.
We can understand that too.
And in this example,
if you bring a lot of plus charges here,
and you look at what's going on here,
the minus guys here will sit here and the plus will be left
over in the back, and then this attraction
between plus and minus is bigger than this repulsion,
so it will be attracted to it. But once it touches it,
this rod touches that, then what you have is a lot of
plus charges here. They repel each other.
They want to get out.
Previously they couldn't get
out. They were stuck on the rod,
but now that you've made contact, some of them will jump
to that one. Then when you separate them,
you will have a ball with some plus charges,
and you will have a rod with more plus charges,
and they will repel each other. And finally I said if you take
two of these spheres, suppose one was positively
charged, one was negatively charged, they're attracting each
other. If you connect them with a
nylon wire or a wooden stick nothing happens,
but if you connect them with an electrical wire,
what happens is that the extra negative charges here will go to
that side, and then when you are done they
will both become electrically neutral.
Okay, so that's why.
So the point of this one is:
electric charges can flow through some materials,
but not other materials. If it can flow through some
materials, it's called a conductor.
If it cannot flow through them,
it's called an insulator. So real life you've got both.
So when you're changing the
light bulb, if you don't want to get an
electric shock you're supposed to stand on a piece of wood
before you stick your finger in, unless you've got other
intentions. Then, you will find that you
don't get the shock because the wood doesn't conduct
electricity. But if you stand on a metallic
stool, on a metallic floor and put your hand in the socket,
you'll be part of an electrical circuit.
The human body is a good
conductor of electricity, but what saves you is that it
cannot go from your feet to the floor.
Now, there are also
semiconductors, which are somewhere in between,
but in our course either we'll talk about insulators,
which don't conduct electricity, and perfect
conductors, which conduct electricity.
Okay, so a summary of what I've
said so far is that there's a new force in nature.
To be part of that game you
have to have charge. If you have no charge,
you cannot play that game. Like neutrons cannot play this
game. Nothing's attracted or repelled
by neutrons and neutrons cannot attract or repel anything.
So you've got to have electric
charge. It happens to be measured in
coulombs. So let me ask you another
question. Suppose I tell you,
here is Coulombs Law. Let me just write the number 1
over 4Πε _0.
How are we going to test that
this law is correct? Okay, I'm giving you a bonus.
You don't have to discover the
law. I'm giving you the law.
All you have to do is to verify
it, and don't use any other definitions other than this law
itself. How will you know it depends on
q_1 and q_2 in this
fashion? How will you know it depends on
r in that fashion? That's what I'm asking you.
Can anybody think of some
setup, some experiment you will do?
Let me ask an easier question.
How will you know it goes like
1 over r^(2)? Yep?
Student: Vary the
distance between them, and show that the force falls
off. Prof: Well,
you're right that if you vary the distance between them and
show the force falls like that, but how do you know what the
force is? Yes?
Student: Could you use
a spring here? Prof: What was your plan?
Student: Observe
acceleration. Prof: You are right.
Both of you are right.
You can maybe hold this guy
fixed, and let this go, and see how it accelerates.
And if you knew the mass of
this guy then you know the force.
Then you can vary the distance
to another distance, maybe half the distance.
At half the distance if you get
four times the force you verified 1 over r^(2
)law. The other one is with the
spring. You can take a spring.
Say maybe there are two metals,
uncharged objects, then you dump some charge on
this and some charge on that, and then the spring will
expand, and you can see what force the spring expands,
exerts, and see if it is proportional to 1 over
r^(2). That's how Newton deduced the 1
over r^(2) force law. He found the acceleration of
the apple is 3,600 times the acceleration of the moon towards
the earth, and the moon was 60 times
further than the apple, and 60 squared is 3,600.
That's how he found 1 over
r^(2). Now, he was very lucky.
It could have been 1 over
r to the 2.110 or 1.96, but it happens to be exactly 1
over r^(2). Anyway, that's how we can find
even if it's not 1 over r^(2).
If it's 1 over r^(3),
or 1 over r^(4), whatever it is you can find by
taking two charges. See, we don't have to know what
q_1 and q_2 are.
That's what I'm trying to
emphasize here. If all you're trying to see is
does it vary like 1 over r^(2),
keep everything the same except r.
Double the r and see
what happens. And best way is what you said.
Watch the acceleration,
and if it falls to one fourth of the value for doubling the
distance, it is 1 over r^(2).
All right, suppose I got 1 over
r^(2). I want to know it depends on
the charges as the first power of q_1 and the
first power of q_2.
So how should we do that?
And don't say put 10 electrons
once and then 20 electrons because you cannot see electrons
that well. In the old days people did not
even know about electrons, and yet they managed to test
this. So how will you vary the charge
in a known way? Yep?
Student: You could have
many identical spheres, and maybe keep touching them to
each other. Prof: Ah!
Okay, many identical spheres.
Student: And then put
charge on one and then touch it to the second one and you'll get
half as much. Prof: Very good.
Let me repeat what she said.
First you take many identical
spheres. Well, I not going to even try
to draw identical spheres because I haven't learned how to
draw spheres, but let's imagine you've got a
whole bunch of these guys. You put some charge on this.
You don't know what it is, okay?
We don't know what q is.
We're trying to find out.
You don't have to know what
q is. So let this be one of the
objects. That's my q.
For the other object,
keep a fixed-object containing some other q.
This has got charge q.
Don't vary the r.
Question is,
can you change q to q/2,
and her answer was: if it's got some charge,
maybe a plus, bring it in contact with the
second identical sphere. If it really is identical,
you have to agree that when you separate them they must exactly
have half each. That's a symmetry argument.
Because for any reason you give
me for why one of them should have more, I will tell you why
the other one should have more. You cannot, so they will split
it evenly and therefore charge will split evenly to q/2
here and q/2 here. Then you can take this and put
it there--you've got q/2. Then you can do other
combinations. For example,
you can take this q/2 and connect it to the ground so
it becomes neutral. So this has got 0 again.
You can touch that with the
q/2 and separate them. Then each will have q/4.
So in this way you can vary the
charge in a known way, maybe half of it,
double it. I give you some homework
problem where you want to get 5/16 of a coulomb.
By enough spheres you can do
that. Again, what I want you to
notice is that you did not know what q was,
but all you knew is that q went to q/2 when
you brought two identical spheres and separated them.
That's how we can find that it
depends linearly on q_1.
Of course, it also depends
linearly on q_2 because it's up to you to decide
who you want to call q_1,
and who you want to call q_2.
Okay, so I want you people to
understand all the time that you should be able to tell me how
you measure anything, okay?
That's very, very important.
That's why you should think
about it. If you think in those terms
you'll also find you're doing all the problems very well.
If you're thinking of pushing
symbols and canceling factors of Π you won't get the
feeling for what's happening. So everything you write down
you should be able to measure. If you say, "Oh,
I want to measure the force," you've got to be
sure how you'll measure it, and one way is like you said,
find m times a. If you knew the m you
can measure the force. For everything make sure you
can measure it. If I give you a sphere charged
with something, then of course we've got to
decide. Suppose I give you a sphere.
It's got some charge,
and I want you to find how much charge is on that sphere.
This time I want you to tell me
how many coulombs there are. What will you do?
What process will you use?
Well, then you have a problem
because you are not able to figure out,
but if I tell you here's an object,
it is 3 meters long, you can test it because you'll
go and bring the meter stick from the Bureau of Standards and
measure it three times. I'm asking you,
if I give you a certain charge and say how much charge is
there, by what process can we calibrate the charges?
Yep?
Student: Put it in the
vicinity of a reference charge and then measure the
acceleration. Prof: That's correct.
If you knew one standard
charge, somehow or other we knew its value, then bring the
unknown one next to it, put it at a known distance,
right? You know the r.
You know the 4Π.
You know the ε
_0. You find the force,
you can find this charge. So all we need to know is how
to get a reference charge, right?
So how do I know something has
a coulomb? How do I get 1 coulomb of
charge just to be sure? You know what you could do,
because you haven't defined yet the reference,
so you should think about how will I get a coulomb charge,
or any other charge? So I could take these two
spheres that she talked about, each with the same charge
q. We don't know what it is.
I put them at 1 meter distance
and I measure the force, namely how hard should I hold
one from running away to the other one.
Once I got the force,
the only thing unknown in the equation is q times
q. I know r.
I know 1 over 4Πε
_0. I can get q.
So every time you write
something think about how you'll measure it, because in that
process you're learning how the physics is done.
If you try to avoid that you'll
be just juggling equations, and that doesn't work for you
and that doesn't work for me. Anybody who wants to do good
physics should be constantly paying attention to physical
phenomena, and not to the symbols that
stand for physical objects. All right, so the final thing I
want to do in this connection is to give this number I mentioned,
F_gravity over F_electric.
I said gravity is 10 to the -40
times weaker. Well, you have to precise on
how you got the number. See, it's not like selling
toothpaste where you can say it is 7.2 times whiter.
I don't know how those guys
measure whiteness in a unit with two decimal places,
but that's a different game. It's not subject to any rules,
but here you have to say how you got the number.
In what context did you make
the comparison? It turns out the answer does
depend on what you choose. There'll be some variations,
but those tiny variations are swamped by this enormous ratio I
would get. So what you could do is take
any two bodies, and find the ratio of gravity
to electric force. One option is to take two
elementary particles, whichever two you like.
So I will take an electron and
a proton, but you can take an electron and a positron,
or a proton and a proton. It doesn't matter.
These two guys attract each
other gravitationally and electrically.
So I will write the force of
gravitation, which is G, mass of the
proton, mass of the electron,
over r^(2 )divided by q_electron,
q_proton over 4Πε
_0 times 1 over r^(2).
Notice in this experiment,
in this calculation, r^(2 )does not matter,
so you don't have to decide how far you want to keep them,
because they both go like 1 over r^(2 ),so you can
pick any r. So whatever you pick is going
to cancel and you will be left with this number.
A q_1,
q_2 and the 1 over
4Πε_0 is 9 times 10 to the
9^(th). So now we put in some numbers.
So G is 10 to the -11
with some pre-factors, maybe 6 in this case.
I'm not going to worry about
pre-factors. But the mass of the proton is
10 to the -27 kilograms, the mass of the electron 10 to
-30 kilograms. So don't say how come they all
have these nice round numbers. They are not.
There are factors like 1 and 2.
I'm not putting them because
I'm just counting powers of 10. q_1 is 1.6
times 10 to the -19, so two of those q's is
10 to the -38. Then 9 times 10 to the 9^(th)
is roughly 10 to the 10^(th). If you do all of that you will
find this is 10 to the -40, if it is some typical situation
that you took, and you found this ratio of
forces. If there are two elementary
particles, which are like the building
blocks of matter, and you brought them to any
distance you like you compare the electric attraction to the
gravitational attraction. So one question is:
if gravity is so weak, how did anyone discover the
force of gravity? If all you had was electrons
and protons, you'd have to measure the force between them.
Suppose you knew only about
electricity, didn't know about gravitation.
One way to find there is an
extra force is to measure the force to an accuracy good to 40
decimal places, and in the 40th decimal place
you find something is wrong. You fiddle around and figure
out the correction comes from m_1m_2
over r^(2), but that's not how it was done,
right? You guys know that.
So how did anyone discover the
force of gravity when it's overwhelmed?
Yes?
Student: Most things
are neutral? Prof: Yes.
Most things are electrically
neutral. In other words,
electric force, even though it's very strong,
comes with opposite charges. It can occur with a plus sign
or with a minus sign. Therefore, if you take the
planet Earth, it's got lots and lots of
charges in every atom, but every atom is neutral.
You've got the moon,
ditto, lots and lots of atoms, but they're all neutral.
But the mass of the electron
does not cancel the mass of the proton.
So mass can never be hidden,
whereas charge can be hidden. Mass never cancels.
That's the reason why,
in spite of the incredible amount of electrical forces
they're potentially capable of exerting,
they present to each other neutral entities.
Therefore, this remaining force
which is not shielded is what you see, and has a dramatic role
in the structure of the universe, force of gravity.
But in most cosmological
calculations you can forget mainly the electric force.
It's all gravitational force.
That's because electricity can
be neutralized. So you cannot hide gravity.
Everything has mass.
Even photons which have no mass
have energy. They're also attracted by
gravitation. So gravity cannot be hidden,
and that's the origin of something called dark matter.
So how many of you guys heard
about dark matter? Okay?
Anyone want to volunteer?
Someone whose name begins with
T, anybody's name begins with T and also knows the answer to
this? The trouble is,
you people are plagued with one quality which is not good for
being in physics, namely you're modest.
So you don't want to tell me
the answer. So I have to give an excuse for
whoever gives the answer. If your seat has a number 142,
anybody in seat 142? Maybe they're not even numbered.
Look, anybody with a red piece
of clothing knows the answer to this--go ahead.
Yes?
Student:
> Prof: Pardon me?
Student:
> Prof: Right.
Basically there's no way you
can see it, and there's dark matter right in this room,
okay? And there's dark matter
everywhere, but the reason, the way people found out there
is dark matter, do you know how that was
determined? Yep?
Student: The rotation
of galaxies didn't line up with the matter that was visible,
so... Prof: So yes.
Maybe one example I can talk is
about our own galaxy. So here's our visible galaxy,
okay, the old spiral. Now, if something is orbiting
this galaxy just by using Newtonian gravity,
by knowing the velocity of the object as it goes around,
you can calculate how much mass is enclosed by the orbit.
That's a property of
gravitation--from the orbit, you can find out how much mass
is enclosed. So what you will find is,
if you found something orbiting the center of the galaxy at that
radius, you'll enclose some mass.
If you take objects at bigger
and bigger radius, you'll enclose more and more
mass, until you find orbits as big as the galaxy.
Then the mass enclosed as a
function of radius should come and stop, because after that the
orbit's getting bigger, but not enclosing any more
mass. But what people found,
that even after you cross the nominal size of the galaxy,
you still keep picking up mass, and that is the dark matter
halo of our galaxy. So it's dark to everything,
but you cannot escape gravity. That's what I meant to say.
You cannot avoid gravitational
force. So people are trying to find
dark matter. People at Yale are trying to
find dark matter. The thing is,
you don't know exactly what it is.
It's not any of the usual
suspects, because then they would have interacted very
strongly. So you're trying to find
something not knowing exactly what it is.
And you've got to build
detectors that will detect something.
And you go through it everyday
in your lab, and you're hoping that one of
these dark matter particles will collide with the stuff in your
detector, and trigger a reaction.
Of course there will be lots of
reactions everyday, but most of them are due to
other things. That's called background.
You've got to throw the
background out, and whatever is left has got to
be due to dark matter. And again, how do you know it's
dark matter? How do you know it's not
something else? Well you can see that if you're
drifting through dark matter in a moving Earth,
you will be running into more of them in the direction of
motion and less in the other direction,
because you're running into the wind.
So by looking at the direction
dependence, you can try to see if it's dark matter.
Anyway, dark matter was
discovered by simple Newtonian gravitation.
The particles that form dark
matter are very interesting to particle physicists.
There are many candidates in
particle theory, but the origin of the
discrepancy came from just doing Newtonian gravity.
All right, the final thing
today before we break is that there's one variation of
Coulomb's Law. By the way, I do not know your
mathematical training and how much math you know,
so you have to be on the lookout, say,
if I write something that looks very alien to you,
you've got to go take care of that,
in particular, how to do integrals in maybe
more than one dimension. Anyway, what I wanted to
discuss today is the following: we know how to do Coulomb's Law
due to any number of point charges.
So if you put another charge
q here you want the force on this guy due to all these.
You draw those lines,
you take the 1 over r^(2 )due to that,
1 over r^(2 )due to that, add all the vectors.
That's very simple.
But we will also take problems
where the charges are continuous.
So here's an example.
Here's a ring of charge.
The ring has some radius.
You pick your radius r,
and the charge on it is continuous.
It's not discrete,
or it could be in real life everything is discrete,
but to a coarse observer it will look like it's continuous.
So we can draw some pictures
here, charges all over the ring, and λ is the
number of coulombs per meter. Let me see, if you snipped one
meter of the wire it'll have λ coulombs in it.
And you want to find the
electric force on some other charge q due to this
wire. So you cannot do a sum.
And you have to do an integral.
That's what I'm driving at,
and I'm going to do one integral, then we'll do more
complicated ones later. So I want to find the force on
a charge q here. So what I will do is,
I will divide this into segments each of length,
say dl. Then I will find the force of
the charge here, dF.
I will add the forces due to
all the segments. The force of this segment will
be the charge-- this segment is so small,
you can treat it as a point charge,
and the amount of charge here is λ times
dl. That's the q_1.
The q_2 is the
q I put there. Then there's the
4Πε _0,
r^(2), r^(2 )will be this
distance z times this radius r will be--
maybe I shouldn't call it r.
Let me call it capital
R, and it's R^(2 )plus z^(2).
That's the distance.
But now that force is a vector
that's pointing in that direction,
but I know that the total force is going to point in this
direction because for every guy I find in this side I can find
one in the opposite direction pointing that way.
So they will always cancel
horizontally. The only remaining force will
be in the z direction. So I'm going to keep only the
component of the force in the z direction.
I denote it by dF in the
z direction. For that, you have to take this
force and multiply by cosine of that θ.
I hope you know how to find the
component of a force in a direction.
It's the cosine of the angle
between them. That angle is equal to this
angle, and cosine of this is z divided by R^(2
)plus z^(2 )on the root.
That is the dF due to
this segment, and the total force in the z
direction is integral of this, and what that integrate.
λ,
q, all these are constant, R,
z, everything is a constant.
You have to add all the
dl's, if you add all the dl's you will get the
circumference. In other words,
this is going to be λqz divided
by 4Πε _0R^(2
)plus z^(2 )to the 3/2 integral of dl.
Integral of dl is just
2ΠR. In other words,
every one of them is making an equal contribution,
so the integrand doesn't depend on where you are in the circle,
so you're just measuring the length of the circle.
That's the answer.
The force looks like
λ times 2ΠR,
what is that? λ is the charge
per unit length. That, times the length of the
loop, is the charge on the loop. It's the charge you're putting
there divided by 4Πε
_0 divided by R^(2) plus
z^(2) to the 3/2. That's an example of
calculating the force which will be in this direction.
Now, once you've done this
calculation you may think maybe I missed a factor of Π
or factor of e, something.
Can you think of a way to test
this? What test would you like to
apply to this result? Yep?
Student: Put the
z equal to 0 and have it in the middle.
There should be no forces on it.
Prof: Very good.
What he said is,
if you pick z equal to 0 you're sitting in the middle of
the circle, and you're getting pushed
equally from all sides, and you better not have a
force, and that's certainly correct.
This vanishes when z
goes to 0. Anything else?
Any other test?
Yep?
Student: You could put
it underneath by negative z.
The force should be negative.
Prof: Yes,
it will point down and be negative.
That's correct,
but how about the magnitude of the force itself,
rather than just the direction? Yep?
Student: If you go
infinitely far away it should look like a point charge.
Prof: Yes.
If you go very,
very far, someone's holding a loop, you cannot see that it's
even a loop. It's some tiny spec,
and it should produce the field.
So what field should it produce?
It should produce the coulomb
force q_1q _2,
or 4Πε _0 times
distance squared. And when z is much,
much, much bigger than R, this is one kilometer,
this is two inches. You forget this.
You get z^(2) to the 3/2
is then z cubed. That means the whole thing here
reduces to 1 over z^(2) and it looks like the force
between two point charges. So I would ask you whenever you
do a calculation to test your result.
Okay, before going I've got to
tell you something about those who come late.
I realize that you guys come
from near and far, so when you come late let me
give you my preference for doors, okay?
Door number one is that one.
That's the least problematic.
Door number two is this one,
because in the beginning of the lecture I'm usually on that side
of the board, so you guys can come in.
Door number three is that one
where Jude is taking the picture, but do not stand in
front of the camera and contemplate your future.
If you do I will make sure you
don't have a future, okay?
So don't do that.
If you come fashionably late,
never come through that door, maybe this one.
In fact if you come through
that door because I have reached this side of the board,
you are very, very late, so I think you
should take the day off and start fresh next time,
all right? Okay, thank you.
Professor Shankar has a wonderful very dry sense of humour.
An underrated lecturer, definitely. He seems genuinely interested in making sure everyone in the room understands the material.