About this video
This video defines the electric potential energy of a system of charges as the work an external force does to bring the charges together from infinity, where the energy is chosen to be zero. For two charges it is (1/4π ε0)·Q1Q2/r, which comes out positive for like charges and negative for unlike charges, and the result does not depend on the path used to assemble them. For three or more charges you add the energy of each pair.
You'll be able to
- define electric potential energy using work done from infinity.
- explain why infinity is chosen as the zero reference.
- predict the sign of the potential energy from the signs of the charges.
- find the total potential energy of three charges by summing over pairs.
Key words
- Electric potential energy
- The energy stored in an arrangement of charges, equal to the work needed to assemble them from infinity.
- Zero reference
- The chosen arrangement with zero potential energy; here it is the charges infinitely far apart.
- Path independence
- The potential energy depends only on the final arrangement, not on how the charges were brought together.
- Superposition (for energy)
- Total potential energy of several charges is the sum of the energies of each pair.
Transcript
- [Narrator] Let's explore electric potential energy. Any arrangement of charges can be assigned an electric potential energy value. That'll depend on the charges themselves, the magnitude and the signs of the charges, and it'll also depend upon exactly how they're arranged.
So if you change that arrangement, the electric potential energy value changes. So in this video, let's see how to calculate the electric potential energy of a given system of charge. Let's start with two charges, Q1 and Q2, separated by some distance R. How do we define the electric potential energy of this system?
We define it as the amount of work required by an external force to bring them to this current position from infinity. Okay, let's see what this means because this whole bringing them from infinity always bothered me. What's going on over here? Well, the key thing to understand is that if we want to assign electric potential energy to a particular arrangement, the first thing we need to do is decide what arrangement will have zero electric potential energy.
Here's what I mean. Let's say we decide that when the two charges are exactly separated by this distance, their electric potential energy is zero. This is not something that the physics is telling us. This is something that we are deciding. Just like how when you want to measure positions on a graph paper, for example, you first decide where your origin is, right?
And you're completely free to decide where that origin is going to be. In a similar way, our first step would be to decide what arrangement of charges gets zero electric potential energy, and that's completely up to us. Now, how do we figure out the electric potential energy of this arrangement when they are this distance apart?
Well, all we need to do is figure out how much work an external force needs to do to bring them from this arrangement to this arrangement. Think about it. If the external force has to do 100 joules of work, then that means 100 joules of energy has been added to this system, right?
Now, if we do that work, that external force does that work in such a way that the kinetic energy does not change. Let's assume that they were at rest to begin with and they are at rest finally as well. Then that energy did not change the kinetic energy, so the energy added must have changed the potential energy, the electric potential energy of the system.
And therefore, the electric potential energy has increased by 100 joules, but it was zero before, and so now it is zero plus 100, which is 100 joules. This is how we get the electric potential energy of a system. Okay, so now the question is, what arrangement should we call the electric potential energy to be zero?
And that is completely our choice, just like how on a graph paper, you are completely free to choose your origin. We usually do that depending on what is convenient for us, right? Similarly here, the convenient choice happens to be an R equals infinity. In other words, when the two charges are infinitely far apart, or for practical purposes, think that when they're very far apart, we like to call that electric potential energy of that particular arrangement as zero.
And now this definition makes sense. So electric potential energy of this system would be the amount of work done by an external force in bringing the charges from infinity, because that is our chosen reference for what zero is to this particular configuration.
Hopefully that makes sense now, okay? And so now we can figure out an expression for this. We can do the math, which we're not gonna do over here, but if you do the math and it turns out the amount of work necessary to bring these charges from infinity to, you know, a distance of R, that work happens to be 1/4 pi epsilon nought Q1Q2/R.
And so that is the expression for potential energy for the two charges separate by distance R, provided we choose this as our zero potential energy. Now notice that this expression is very similar to Coulomb's law, but instead of 1/R squared, we get a 1/R. That makes sense because this is not a force, this is energy, so you should have different units.
And if you're thinking why is it 1/R, well, kind of sort of like, you know, a crude way to think about it is that work is basically force times distance. So force is 1/R squared, multiplied by distance R, the R cancels and you'll get 1/R. But, of course, that's not how you derive this.
The mathematics is much more involved over here, but this kind of sort of helps you see why it should be 1/R. Another thing to notice is that if both the charges have the same sign, they're both positive or they're both negative for example, then the product becomes positive and so the electric potential energy becomes positive.
Does that make sense? Well, yeah. If they have the same sign, then they will repel. Like charges repel, right? And so now to bring them closer, the external force needs to push them together and move them in the same direction that the external force is acting, which means the external force would have done a positive work.
And so we would have added energy to the system. And if you add energy, you know, begin with, we had zero energy. When you add energy, it becomes positive. It increases, right? So it becomes positive, so it makes perfect sense. And the situation would be identical if both charges were negative, right?
I mean, again, they would be repelling each other. Again, an external force would have to do a positive work in assembling them from infinity and therefore the electric potential energy would again be positive. And equation agrees, negative times negative is positive.
And by the way, none of what we discussed depends on how we assemble the charges. I mean, over here, I've shown two external forces pushing them together. So both the external forces are doing work, right? That's one way to assemble charge. Another way to assemble the charge would be to just hold this charge, for example, over there at rest and bring the second charge close to it.
In this case, only the second, you know, the external force on this charge would be doing work and the force on this charge wouldn't be doing any work. But that doesn't matter because the total log done eventually by the external force would be the same. It's independent of how we assemble the charges, what path we use to assemble the charges.
None of that matters, which is awesome for us. It doesn't matter how we assemble the charges. What matters is what the initial arrangement was and what the final arrangement was. Anyways, now what if one of the charges were negative? Now notice they will attract each other naturally.
Now, if you want to bring them from this to, you know, this configuration to this configuration, and we wanna do that in such a way that they stop over here, the kinetic energy does not change, we have to now keep them from accelerating. We have to push them in the opposite direction, right?
Because they're naturally tending to, you know, they're naturally getting attracted. And so now notice we have to do negative work. We are removing energy from the system, but it was zero to begin with, and so the final energy ends up becoming potential, electric potential energy becomes negative.
And again, the equation agrees, a positive times negative is negative. And therefore, the potential energy sign comes out to be negative. And now we can see why this particular reference for zero electric potential energy is so convenient. Because of this, if the electric potential energy of two charges happens to be positive, I know immediately the two charges must have the same signs, done.
If it turns out to be negative, I know immediately one of them must be positive, the other one must be negative. Done. This wouldn't be true if we had chosen some other reference, okay? So that's why this is the most convenient one and we stick to this. And by the way, remember that Newton's law of universal gravity also goes down as 1/R squared, which means the work done by the external force will be very analogous to it and it should also have similar expression.
The only thing is when it comes to masses, gravity is always attractive, and therefore, the gravitational potential energy will always be negative. Now hopefully that makes sense. Again, there's nothing inherently, you know, special about gravity that makes it negative.
The reason gravitational potential energy happens to be negative is because when, you know, when they're infinitely far apart, the gravitational potential energy is zero. And now when they come closer, you always end up doing negative work. The external force always ends up doing negative work.
You're always removing energy from the system, and therefore, the gravitational potential energy ends up becoming negative. And notice when it comes to gravitational potential energy, we have to use a negative sign because masses are always positive. So to represent that the energy is going to be negative, I have to put a negative sign.
But when it comes to electric potential energy, we don't have to use a negative sign because charges do have signs. So when we're dealing with the unlike charges, one of the charges will have a negative sign, the product will be negative, and that's how the potential energy will end up with a negative sign.
So the signs of the charges will take care of the sign of the electric potential energy. Anyways, the last question we could have is, this is the expression for when we have two charges. What if we have three or more charges? What do we do then? What is the expression for electric potential energy?
Well, let's say we have three charges like this, okay? Then to calculate the total electric potential energy of this system, we can use a superposition principle. How do we do that? Well, first, we consider electric potential energy, for example, only due to these two charges, ignore the rest.
Then you consider electric potential energy only due to these two charges, ignore the rest. And then due to these two charges, ignore the rest, and then add them all together. Superposition principle, you take one pattern at a time. It'll be a great idea to pause the video and see if you can write out the expression for you, you know, the electric potential energy for this system now.
Okay, let's do this. So first, let's only consider these two charges. The electric potential energy will be 1/4 pi epsilon nought Q1Q2/R12. Next, let's only consider, for example, these two charges. So it'll be 1/4 pi epsilon nought Q2Q3/R23. And next, let's only consider these two.
It's going to be 1/4 pi epsilon Q1Q3/R13. Some of them will be positive, some of them would be negative depending on the charges and the strength and magnitude. And together when we add them all up, you'll get some positive and negative answer. And, of course, I can take 1/4 pi epsilon nought common and this now becomes the expression for the total electric potential energy of the system.
And we can extend this to any number of charges, right? And so there we have it. This is how we calculate electric potential energy for a given system of charges.
Video © Khan Academy, used under CC BY-NC-SA. Watch on Khan Academy. Puzzicle is not affiliated with Khan Academy.