字幕表 動画を再生する 英語字幕をプリント 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 center. So let's first do the potential here. The potential at point P is