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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