And that's the concept of electrostatic potential

electrostatic potential energy. For which we will use the

symbol U and independently electric potential.

Which is very different, for which we will use the

symbol V. Imagine that I have a charge Q

one here and that's plus, plus charge,

and here I have a charge plus Q two and they have a distant,

they're a distance R apart. And that is point P.

It's very clear that in order to bring these charges at this

distance from each other I had to do work to

bring them there because they repel each other.

It's like pushing in a spring. If you release the spring you

get the energy back. If they were -- they were

connected with a little string, the string would be stretched,

take scissors, cut the string fweet they fly

apart again. So I have put work in there and

that's what we call the electrostatic potential energy.

So let's work this out in some detail how much work I have to

do. Well,

we first put Q one here, if space is empty,

this doesn't take any work to place Q one here.

But now I come from very far away, we always think of it as

infinitely far away, of course that's a little bit

of exaggeration, and we bring this charge Q two

from infinity to that point P. And I, Walter Lewin,

have to do work, I have to push and push and

push and the closer I get the harder I have to push and

finally I reach that point P.

Suppose I am here and this separation is little R.

I've reached that point. Then the force on me,

the electric force, is outwards.

And so I have to overcome that force and so my force F Walter

Lewin is in this direction. And so you can see I do

positive work, the force and the direction in

which I'm moving are in the same direction, I do positive work.

Now, the work that I do could be calculated.

The work that Walter Lewin is doing in going all the way from

infinity to that location P is the integral going from in-

infinity to radius R of the force of Walter Lewin dot DR.

But of course that work is exactly the same,

either one is fine, to take the electric force in

going from R to infinity.

Dot DR. Because the force,

the electric force, and Walter Lewin's force are

the same in magnitude but opposite direction,

and so by flipping over, going from infinity to R,

to R to infinity, this is the same.

This is one and the same thing. Let's calculate this integral

because that's a little easy. We know what the electric force

is, Coulomb's law, it's repelling,

so the force and DR are now in the same direction,

so the angle theta between them is zero, so the cosine of theta

is one, so we can forget about all the vectors,

and so we would get then that this equals Q one,

Q two, divided by four pi epsilon zero.

And now I have downstairs here an R squared.

And so I have the integral now DR divided by R squared.

From capital R to infinity. And this integral is minus one

over R.

Which I have to evaluate between R and infinity.

And when I do that that becomes plus one over capital R.

Right, the integral of DR over R squared I'm sure you can all

do that is minus one over R. I evaluate it between R and

infinity and so you get plus one over R.

And so U, which is the energy that -- the work that I have to

do to bring this charge at that position,

that U is now Q one. Times Q two divided by four pi

epsilon zero. Divided by that capital R.

And this of course this is scalar, that is work,

it's a number of joules. If Q one and Q two are both

positive or both ne- negative, I do positive work,

you can see that, minus times minus is plus.

Because then they repel each other.

If one is positive and the other is

negative, then I do negative work, and you see that that

comes out as a sign sensitive, minus times plus is minus,

so I can do negative work. If the two don't have the same

polarity. I want you to convince yourself

that if I didn't come along a straight line from all the way

from infinity, but I came in a very crooked

way, finally ended up at point P, at that point,

that the amount of work that I had to do is exactly the same.

You see the parallel with eight oh one where we dealt with

gravity. Gravity is a conservative force

and when you deal with conservative forces,

the work that has to be done in going from one point to the

other is independent of the path.

That is the definition of conservative force.

Electric forces are also conservative.

And so it doesn't make any difference whether I come along

a straight line to this point or whether I do that in an

extremely crooked way and finally end up here.

That's the same amount of work. Now if we do have a collection

of charges, so we have pluses and minus charges,

some pluses, some minus, some pluses,

minus, pluses, pluses, then you now can

calculate the amount of work that I, Walter Lewin,

have to do in assembling that. You bring one from infinity to

here, another one, another one,

and you add up all that work, some work may be positive,

some work may be negative. Finally you h- arrive at the

total amount of work that you have to

do to assemble these charges. And that is the meaning of

capital U. Now I turn to electric

potential. And for that I start off here

with a charge which I now call plus capital Q.

It's located here. And at a position P at a

distance R away I place a test charge plus Q.

Make it positive for now, you can change it later to

become a negative. And so the electrostatic

potential energy we -- we know already, we just calculated it,

that would be Q times Q divided by four pi epsilon zero R.

That's exactly the same that we have.

So the electric potential, electrostatic potential energy,

is the work that I have to do to bring this charge here.

Now I'm going to introduce electric potential.

Electric potential. And that is the work per unit

charge that I have to do to go from infinity to that

position. So Q doesn't enter into it

anymore. It is the work per unit charge

to go from infinity to that location P.

And so if it is the work per unit charge, that means little Q

fweet disappears. And so now we write down that V

at that location P, the potential,

electric potential at that location P,

is now only Q divided four pi epsilon zero R.

Little Q has disappeared. It is also a scalar.

This has unit joules. The units here is joules per

coulombs. I have divided out one charge.

It's work per unit charge. No one would ever call this

joules per coulombs, we call this volts,

called after the great Volta, who did a lot of research on

this. So we call this volts.

But it's the same as joules per coulombs.

If we have a very simple situation like we have here,

that we only have one charge, then this is the potential

anywhere, at any distance you want, from this charge.

If R goes up, if you're further away,

the potential will become lower.

If this Q is positive, the potential is everywhere in

space positive for a single charge.

If this Q is negative, everywhere in space the

potential is negative. Electro- electric static

potential can be negative. The work that I do per unit

charge coming from infinity would be negative,

if that's a negative charge. And the potential when I'm

infinitely far away, when this R becomes infinitely

large, is zero. So that's the way we

define our zero. So you can have positive

potentials, near positive charge, negative potentials,

near negative charge, and if you're very very far

away, then potential is zero. Let's now turn to our

Vandegraaff. It's a hollow sphere,

has a radius R. About thirty centimeters.

And I'm going to put on here plus ten microcoulombs.

It will distribute itself uniformly.

We will discuss that next time in detail.

Because it's a conductor. We already discussed last

lecture that the electric field inside the sphere is zero.

And that the electric field outside is not zero but that we

can think of all the charge being at this point here,

the plus ten microcoulombs is all here, as long as we want to

know what the electric field outside is.

So you can forget the fact that it is a -- a sphere.

And so now I want to know what the electric potential is at any

point in space. I want to know what it is here

and I want to know what it is here at point P which is now a

distance R from the center. And I want to know what it is

here. At a distance little R from the

center. So let's first do the potential

here. The potential at point P is

an integral going from R to infinity if I take the electric

force divided by my test charge Q dot DR, but this is the

electric field, see, this force times distance

is work, but it is work per unit charge, so I take my test charge

out. And so this is the integral in

R to infinity of E dot DL -- DR, sorry.

And that's a very easy integral.

Because we know what E is. The electric field we have done

several times. Follows immediately from

Coulomb's law and so when you calculate this integral you get

Q divided by four pi epsilon zero R which is no surprise

because we already had that for a point

charge. So this is the situation if R,

little R, is larger than capital R.

Precisely what we had before. We can put in some numbers.

If you put in R equals R, which is uh oh point three

meters, and you put in here the ten microcoulombs,

and here the -- the thirty centimeters, then you'll find

three hundred thousand volts.

So you get three times ten to the fifth volts.

If you um take R equals sixty centimeters, you double it,

if you double the distance, the potential goes down by a

factor of two, it's one over R,

so it would be a hundred and fifty kilovolts.

And if you go to three meters, then it is ten times smaller,

then it is thirty kilovolts. And if you go to infinity which

for all practical purposes would be

Lobby seven, if you go to Lobby seven,

then the potential for all practical purposes is about

zero. Because R is so large that

there is no potential left. So if I, if I,

Walter Lewin, march from infinity to this

surface of the Vandegraaff, and I put a charge Q in my

pocket, and I march to the Vandegraaff,

by the time I reach that point, I have done work,

I multiply the charge now back to the potential,

that gives you the work again, because potential was work per

unit charge, and so the work that I have done then is the

charge that I have in my pocket times the potential,

in this case the potential of the Vandegraaff.

If I go all the way to this surface,

which is three hundred thousand volts.

If I were a strong man then I would put one coulomb in my

pocket. That's a lot of charge.

Then I would have done three hundred thousand joules of work.

By just carrying the one coulomb from Lobby seven to the

Vandegraaff. That's about the same work I

have to do to climb up the Empire State Building.

The famous MGH, my mass times G times the

height that I have to climb. So I know how the

electric potential goes with distance.

It's a one over R relationship. Now I have arrived at the

Vandegraaff, I am at the surface, with my test charge,

and now I go inside. And I slosh around inside,

I feel no force anymore. There is no electric field

inside. So as I move around inside,

I experience no force. That means I do no work.