Electric Potential

Let's talk about electric potential. Electric potential is another quantity that's very very very useful to define and it's kind of along the same lines of electric field. So if I'm moving a charge through an electric field it requires work and that's because the electric field is going to put forces on the electric charge. So electric force is a conservative force and that means that any work that's done by it can be expressed in terms of a potential energy function. Work equals negative change in the potential energy.

Now, just like with electric fields, I want to treat all test charges the same. I want to treat them all at the same time and I don't want to have to look at them all separately. So what we're going to do is we're going to define the electric potential as the electric potential energy divided by the test charge. And so this is exactly the same thing that we did for electric field. Electric field is electric force divided by test charge. Alright the electric potential function is going to tell us how much a positive charge would rather be at one place than at another place. So, just like we understand from just basic mechanics, whenever the potential energy is high, it means that the object doesn't really want to be there. So whenever the electric potential is high, it means that a positive charge doesn't want to be there. It would rather be where the electric potential is low. Now, a negative charge would rather be where the electric potential is high. And would rather not be where the electric potential is low. Alright.

The unit of electric potential is the volt and this is exactly the same volt that you know well from batteries. So the volt, one volt is equal to one joule per coulomb. So it tells us how much energy is required for a charge to be at a certain location per coulomb of charge. So if I've got an electric potential of three volts, or whatever I've got, it would cost twice as much energy for me to put double the charge there. Joules per coulomb. Alright? Alright.

Now, we know from Coulomb's law that the electric potential associated with a large charge, capital q, will be kq over r and this is exactly the same as you'll remember from gravity where the electric, sorry where the gravitational potential energy is gmm over r. Whereas the force was gmm over r squared. So this is the same thing. The coulomb's law tells us kqq over r squared but the energy potential energy, is just over r. And then when I divide by the other charge, I get kq over r. And so what that indicates to us is that when you're close to a positive charge, the potential the electric potential is high. It's positive. When you're close to a negative charge, the electric potential is negative. So it's just the simplest thing that it could possibly be and of course it makes sense. If I try to get a positive test charge close to a positive big charge then that's going to cost energy. It doesn't want to be over next to that positive charge. So that means that the electric potential is going to have to increase and that's exactly what coulomb's law tells us.

Alright. Let's do an example. So suppose that I've got a battery and it's got a potential difference of 3 volts. And I want to know how much energy will it cost for me to take 2 microcoulombs of charge and move it from the negative side to the positive side of the battery. Now of course it's going to cost me energy to do that because this 2 microcoulombs is positive. That means it wants to be on the negative side. It wants to be next to all the negative charge because that's who it likes. It doesn't want to be up on the positive side. So I want to know how much energy it's going to take. Well, jeez. Energy is equal to charge times voltage. The charge is 2 microcoulombs, notice we're doing everything in SI units. The voltage is 3 volts and therefore, when we multiply we'll just get 6 times 10 to the -6 and then it's coulombs times volts which is joules. So it will cost me 6 microjoules of energy to move this charge from the negative side to the positive side. Alright.

Now, let's do some qualitative analysis. And we're going to bring some electric fields in when we do this also. I think that this is a great excercise and it just helps us really understand what's going on with electric potential and electric fields. And I want to just kind of draw some lines around here to separate. I've got 3 different examples. Alright. So here I've got 2 positive charges and I've got 3 points, a, b and c. And I want to know at each of these 3 points, I don't want to know exactly what the potential is. I don't want to know exactly what the electric field is. I just want to know is the electric potential positive, negative or zero. And what direction does the electric field point in? Alright.

So let's go ahead with the first example. Positive, positive and I got my point a right in here. Alright. Let's do electric potential first. So a is close to 2 positive charges. Each of these positive charges contributes a positive electric potential. And then [IB] the electric potential is positive. Alright. What about the electric field?

Well, I got to think about it. This positive charge wants to push any positive charge at a to the right. This positive charge wants to push any positive charge at a to the left. Same charge, same distance, they cancel. So the electric field at a is zero. It doesn't point at any direction at all. Alright. Let's do b.

At b again, the electric potential's got to be positive because the only contributions are coming from positives. So we've got positive electric potential. What about the electric field? Well, in this case the this positive charge is pushing everybody off that way. This positive charge is pushing everybody off that way. And when I add these two vectors together, it's going right into my calculation from before but the electric field is going to point straight up. Alright. Let's do point c. Obviously again the electric potential's positive. Now in this case both of the positive charges are pushing to the left and that means electric field to the left. Alright, great. Let's do the second example. This one right here.

Now, I've got a positive charge and a negative charge. So let's look at point a. Well, at point a, same distance from the positive and from the negative. So what do you think about the electric potential? Well, it's going to be zero. So v=0 there. Alright. Let's look at the electric field. The positive charge wants any test charge at a to move to the right, the negative charge wants any test charge at a to move to the right. So what do you think? It's going to be to the right. Alright? Let's do point b.

Now, for point b, again it's the same distance from the positive as it is from the negative. And that again means that the electric potential will be zero. Alright. What about electric field? Well, the positive charge wants the test charge to move like that. The negative charge wants a test charge to move like that. When I add these two vectors together, electric field is directly to the right. Alright. What about point c?

Point c is closer to the positive charge than it is to the negative charge. So the positive charge is more, it's stronger. It's more important to the electric potential at point c. And that means that the is positive. Alright. Now let's do electric field. Well at point c the positive charge is pushing to the left. The negative charge is pulling to the right. So is the electric field going to be zero? Well, let's think about it. The positive charge is closer, that means it wins. There we go. Alright. Electric field to the left. Alright. Now let's do this last one.

Now, here I've got 4 charges. Positive, negative, negative, positive. Alright. So let's go ahead and look at a. Electric potential. Well, point a is the same distance from 2 positive charges and the same distance from two negative charges. So the electric potential is going to cancel. It will be zero, alright? What about the electric field? Well, what's going on here is I've got this negative charge which is pulling like that. I've got this positive charge which is pushing like that, but then I've got this positive charge which is canceling that one out and I've got this negative charge which is [IB] So that means that the electric field is also zero. This situation is what's called a quadropole moment. Where we have the electric potential and the electric field, both zero. Alright. What about point b?

Well, point b is the same distance from this positive charge as it is from that one. So as far as the electric potential is concerned, these guys don't even matter. Alright? What about here and here? Well, again same distance. So the electric potential again is zero. What about the electric field? Well, let's look. The electric field here is pulling toward the negative. The electric field here is pushing away from the [IB] so they're agreeing with each other. Now these guys down here are going to do the opposite thing, but they're further away. So they're not going to be able to cancel this contribution out entirely and the electric field will point in that direction, to the right. What about point c? At point c again you can see for yourselves that the electric potential will be zero and the electric field will again be to the right.

Alright. So that's electric potential.