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• We're going to talk about again some new concepts.

• 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