Electric potential

0:00- [Instructor] Let's explore the idea of electric potential in this video. Suppose we have a system of two charges. Think of this Q as a big charge, it's fixed, and think that there's this q, think of it as a test charge and assume that it can move, okay? But right now, they are separated by some distance, r.

0:18Then we know that this system has electric potential energy given by this particular expression. Now, we define electric potential at a specific spot, for example, at this specific spot, as the electric potential energy of the system per unit charge. So the electric potential at this specific spot, that would be, well, just it's going to be this divided by q, so the q cancels out and we just end up with 1 or 4 pi epsilon naught, Q divide by r.

0:52And the first time I saw this, I was really confused because it looks very similar to electric potential energy and I was wondering, "Why do we have two separate quantities? What's the difference between them?" So, okay, let's understand the difference between them clearly.

1:07First of all, electric potential energy is an energy, right? So it's measured in joules. Whereas electric potential is... Notice, it's energy per coulomb and joules... So its units are gonna be joules per coulomb, and joules per coulomb is what we call volt, okay?

1:27And so you can see the difference between these two Vs over here. This V which is italicized, that symbolizes electric potential, that's a symbol for electric potential. This V, which is not italicized over here, that is the unit for electric potential, which is volt, and volt is basically joules per coulomb.

1:47Okay, but conceptually, what's the difference between these two? Well, electric potential energy is a number that is assigned to a system, right? Every system, every configuration, every arrangement of charge has, you know, an electric potential energy number, right?

2:05But electric potential is a number that is assigned to a point in space. Every point in space, you can assign an electric potential value. But electric potential energy is a number that's assigned to a system, not a point in space, not a single charge, but to a system.

2:23But electric potential, that's a number that can be assigned to every single point in space. So for example, electric potential at this point in space, that is given by this particular expression. And notice, that electric potential, the source of that is this charge.

2:40This is just a test charge. This is the true source of this electric potential that we have at this point, which is given by this expression, and so you can think of this as our source charge. But again, why do we need this quantity in the first place? Why isn't electric potential energy enough?

2:56Well, here's why. Let's take an example. Suppose we plug in the value of Q and r over here, and let's say, for the sake of taking simple numbers, the, you know, electric potential at this specific point happens to be 100 volt. What does that number mean? Well, remember, volt is just joules per coulomb.

3:16So that means if there was a unit charge, a charge of one coulomb placed over here, then that system of two charges would have an electric potential energy of 100 joules. That's what this number tells us. 100 joules of electric potential energy if we had kept a one coulomb of charge.

3:39So you can see this is an indicator of what the system's potential energy would be. Quick note, one coulomb is actually a very large amount of charge. Electrons and protons and even the most charge objects we come across carry far less than that. But we're using one coulomb here because electric potential is per unit charge, so it just makes the idea clearer.

4:02And that per unit charge idea is what lets us transfer the result to any amount of charge. So it doesn't matter how realistic this charge really is, okay? What if I had kept say three coulombs of charge at this point? Then the electric potential energy of this system would be three times as much, 300 joules.

4:24Okay, now, remember how electric potential is a number that can be assigned at every point in space? So let's consider somewhere over here. What would the electric potential at this point would be due to this source charge over here? Well, it's gonna be, you know, since we're going farther away from that charge, you would expect the number to be slightly lower than 100.

4:43Let's assume, okay, somewhere over here, the electric potential happens to be 80 volts. Again, we're just taking simple numbers. That means if I had kept, you know, one coulomb of charge at this point, then that system of two charges would have an electric potential energy of 80 joules, okay?

5:00Now, think about this. What if a charge over here, say one coulomb of charge over here, were to move from here to here? Then we know the system's potential energy changes from 100 joules to 80 joules, right? So we can immediately say that if a coulomb of charge were to move from here to here, then the system would have lost 20 joules of electric potential energy.

5:33Where does that electric potential energy go? Well, that now can be connected to some other form of energy. For example, if there are no other forces acting, then this charge, one coulomb charge would have gained some kinetic energy, right? And that 20 joules lost as electric potential energy would get converted into kinetic energy.

5:52And so look, I can now predict what the kinetic energy of that charge would be when it comes from here to here. That is the whole point of electric potential. Because it tells me what the electric potential energy would be when I keep any charge over there, then that charge moves from one point to another, I can predict how the potential energy of the system changes, and so I can predict what the kinetic energy or, you know, sometimes that energy gets converted to heat energy, I can predict what those energy changes would be.

6:26That is the whole reason we introduce this idea of electric potential. Okay, but you might say, wait a second, if that's what we're really interested in, when a charge moves from one point to another, how much the electric potential energy changes, if that's what we're really interested in, then what we really care about is not the individual, you know, electric potential values at this point and this point, what we really care about is the difference between the electric potential between two points.

6:56Because at the end of the day, that difference is what's gonna tell me how much the electric potential energy will change, right? And you will be absolutely right. And so now, we can introduce another useful quantity called electric potential difference. What is this?

7:13Well, that's just the difference between the electric potential of two points. So electric potential difference delta V, it's basically the difference in the electric potential energy of a system when a test charge moves from one point to another. That makes sense, right?

7:30And so what would be the unit of electric potential difference? Well, since it's just a difference in electric potential, it'll have the same units. So it'll also be joules per coulomb, it'll also be measured in volts. So for example, over here, the electric potential difference between these two points, delta V, equals 20 volts.

7:52The delta over here represents that we're talking about a difference now. And again, what does it mean when I say electric potential difference is 20 volts? It just means that if a coulomb of charge were to move from one point to another, then the system of charges would lose or gain, you know, 20 joules of electric potential energy.

8:16Now, whether it loses or gains depends on, you know, which direction it goes, whether it goes from here to here or here to there. But what's important is that the change in the electric potential energy of the system would be 20 joules per coulomb. That's the whole idea.

8:31Finally, what if instead of one charge, we had multiple charges? How do you calculate electric potential at a specific point due to multiple charges? Well, we just calculate the electric potential due to individual charges and then add them all up, superposition principle.

8:48Now, remember, electric potential is an indicator of energy, electric potential energy. So it's a scalar quantity, and that's great news for us because we just add them up. So there could be some positive numbers, there could be some negative numbers. So when you add them all up, you can end up with some positive value or negative value or you can even end up with zero, for example.

9:08And so in general, the electric potential at a specific point due to multiple charges, well, it's just going to be summation of, you know, electric potential due to individual (indistinct), and so we can represent it this way. Okay, now, we can zoom out. We have four important quantities and it can be very easy to get confused.

9:26So let's see the differences and similarities between them. First is the electrostatic force. The electrostatic force between any two point charges is given by this expression, Coulomb's Law. And the Coulomb's Law gives us the magnitude. What about the direction of the electrostatic force?

9:43Well, that is always going to be either straight towards or away from each other. That's decided by the sign of the charges. Light charges will always repel, unlike charges will attract. Okay, what about electric field? Electric field is electric force per unit charge.

10:03So electric field is an indicator of what the electric force would be if we had kept a positive test charge over there. Again, this gives us the magnitude, and the direction is a direction that a force would act on a positive charge at that particular location.

10:22So due to a positive charge, the electric field is radially outwards everywhere. Due to a negative charge, the electric field will be radially inwards everywhere. And look, the units will be units per coulomb, right? Okay, then comes electric potential energy.

10:36This is a number, as we just saw, assigned to a system of charges. And energy is a scalar quantity, so it's just a number, but the number can be positive, negative, or zero. And the sign depends on the product of the charges. But remember, that sign does not indicate direction because it doesn't have any direction.

10:58It just indicates, you know, how much that number is relative to our zero that we have defined as infinite separations. And finally, electric potential is, basically, electric potential energy per unit charge. And just like electric potential energy, it's a scalar, so it's only a number, but it can positive, negative, or zero, and again, the sign depends on the charge.

11:24But, of course, the sign does not indicate direction because there is no direction. And if you take the difference in electric potential between two points, we find electric potential difference. That electric potential difference is what we, in general, call voltage.

11:43You've probably heard of the word voltage everywhere. That voltage, at the end of the day, is electric potential difference. So they are synonyms, and more often than not, we'll just use the term electric potential difference because it's just much clearer. One more thing, the expression in this summary are for point charges specifically.

12:05One point charge for electric field and electric potential and a system of two point charges for electric force and electric potential energy. But, of course, we can always extend them to systems of more charges using the superposition principle.