1. Electrostatics


Uploaded by YaleCourses on 22.03.2011

Transcript:
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.