 We've learned how to visualize electric field by drawing field lines. In this video, let's explore how to visualize electric potentials. And the way to do that, or at least one way of doing that, is by drawing something called equi-potential surfaces. So what exactly are these? Well, as the name suggests, these are surfaces, and these are three-dimensional surfaces, over which the potential at every point is equal, equi-potential surfaces. Let me give you an example. So if we come over here, let's say from this charge, I go about two centimeters far away over here. There will be some potential at that point. Let's call that as 10 volt. Let's imagine that to be 10 volt. Now if I went two centimeters over here from the charge, what would the potential there? It should also be 10 volts. What about two centimeters from here? That should also be 10 volt. In fact, I could draw a circle of two centimeters, and two centimeters is just an example, okay? And everywhere on that circle, the potential would be equal, 10 volt. So that circle would be an equi-potential surface. And since it's a three-dimensional, you have to imagine this actually is not a circle, but it's a sphere. So let me just draw that nicely. So I could draw a sphere, let's say here it is, a sphere, and you have to imagine this is a three-dimensional sphere, where on every point of it, the potential is 10 volt equal. And so this would be my 10 volt equi-potential surface. Can I draw more? Of course. If I go a little farther away, maybe two and a half or three centimeters far away, I would draw another sphere that would be another equi-potential surface. Let me draw that. If I go farther away, the potential will decrease, right? So let's say this is another equi-potential surface. Why is this equi-potential? Because on every point of it, the potential is equal, and it's equal to seven volt. Can I draw more? Yes. More spheres. Every sphere you draw will be an equi-potential surface. In fact, if I go a little farther away and I draw another one, I might get a nine-volt equi-potential surface. If I go a little farther away and I draw another one, I might get an eight-volt equi-potential surface, and so on and so forth. Now before we continue, you may immediately notice that the surfaces are closer here, and they're going farther and farther away. Why is that? Well, it's got something to do with the strength of the electric field. Close to the charge, the field is very strong, and that's where the equi-potential surface would be closer to each other. As we go far away from the charge, the field weakens. And so the surfaces go farther and farther away from each other. But why? Why is it that if the field becomes weaker, the equi-potential surfaces go farther away? Can you pause and think a little bit about this? All right, here's how I like to think about it. Consider a tiny test charge kept over here on the 10-volt equi-potential surface. What will happen if I let go of it? Well, the electric field will push it, and it will accelerate, and it will move from this equi-potential to another, the nine-volt equi-potential. Now, because the force over here is very strong, because you are in the strong electric field region, it will accelerate very quickly. It will gain kinetic energy very quickly. And as a result, it will lose potential energy very quickly. And it's for that reason, in a very short distance, it would have reached from 10-volt to nine-volt equi-potential surface. However, what would happen if I were to keep that same test charge over here? Well, now the field is very weak, or weaker compared to here. And so the force acting on it is very weak, and so it will accelerate slowly. And so it's going to take more distance for it to pick up the kinetic energy. And so it's going to lose potential energy more slowly. And as a result, it's going to take a longer distance before it reaches, it loses one volt now. And so what do you think will happen for the six-volt equi-potential? It'll take even larger distance to reach six volts, and so it'll be even farther away. Does that make sense? It's kind of like if you take a ball and drop it on, say, Jupiter, where the gravitational field is very strong, then it'll accelerate very quickly. And so it'll gain kinetic energy very quickly. So it will lose potential energy very quickly. But on the other hand, if you were to drop that same bowling ball on, say, moon, because the gravitational field is very weak, it's going to accelerate very slowly, gain kinetic energy very slowly. And so therefore lose potential energy very slowly. So in weaker fields, you lose potential very slowly. And so the potential surfaces are farther away. All right, let's take another example. And I want you to take a shot at drawing equi-potential surfaces. Let's say we have a long, infinitely long sheet of charge, big sheet of charge, which has, let's say, negative charge. Then we know we've seen before it produces a uniform electric field. Can you think of what the equi-potential surfaces here would look like? Can you try drawing a few potential surfaces over here? Pause the video and think about this. Use the same approach as we did over here. All right, just like over here, let me go at some distance, say about two centimeters from this sheet. It'll have some potential because it's a negative charge. Maybe there is some, I don't know, negative 10 volt potential. Now, if I go two centimeters from here, I should get exactly the same potential as here. And the same would be the case over here as well. That means I can draw, connect all these lines. And if I do that, now my equi-potential surface would look somewhat like this. So this would be my minus 10 volt equi-potential surface. I can draw another, if I go a little farther away, maybe I will draw, get another, let's say minus nine volt equi-potential surface. If I go farther away, maybe I'll get another minus eight volt equi-potential surface and so on and so forth. Over here, I hope you agree that the equi-potential surfaces will be equidistant because the field lines are all, the electric field is uniform. And again, just to reiterate, this is not a line, this is a surface, so you have to imagine this in three dimensions, help you visualize that. If you could see this in three dimensions. So if you look at them in 3D, you can now see that now the equi-potential surfaces are plain surfaces. So over here we got spheres, over here we're getting plain surfaces. All right, but here's a question. These were simple cases. What if we have to draw equi-potential surfaces in general? What if I have some random electric field line due to some complicated network of charges? Something like that, I don't know, just randomly drawing. How would we draw equi-potential surfaces then? We may not be able to use the same approach like here, but what we can try to do is see if there is some geometrical relationship between electric field lines and equi-potential surfaces. So let's come over here. Can we see any relationship between these field lines and the potential surfaces? If you look very closely, you can see that these equi-potential surfaces are perpendicular to the field lines, and that makes sense, right? Because in general, over here the field lines are forming the radius, and the radii are always perpendicular to the spheres or the circles. So over here we are seeing that the two are perpendicular to each other. Hmm, let's look at over here. Hey, here also we are seeing that the field lines are perpendicular to the equi-potential surfaces. Hmm, interesting. So can we say that this is true in general, that equi-potential surfaces and field lines must always be perpendicular to each other? We can't just say that using two examples, we could say that might be a coincidence. So is this true in general? Well, if you and I were in the same room, maybe you would have an interesting dialogue over here, but I don't wanna take too much time, and I'll go ahead and tell you that it turns out that this is true in general. So let me just write that down. Equi-potential surfaces are always, always perpendicular to electric field lines. I can just say perpendicular to field or field lines. Always, regardless of how complex the field lines are. And again, the final question for us in this video is why this is true? And I want you to again pause and ponder upon this. This is a deep question. But I'll give you one clue. Think in terms of contradiction. What would happen if the equi-potential surfaces were not perpendicular to the field lines? What gets broken? Think a little bit about that. Like I said, it's a deep question. Don't expect it to get right away, and it's okay if you don't get it right away, but the idea is just to think a little bit about it before we go forward. All right, let's see. There are multiple ways to think about this. The way I like to think about it is again, bring back my test charge. So here's my test charge. Now imagine we move this charge along the equi-potential surface, say from here to here. Now, because it's an equi-potential, at every single point, the potential is the same. That means the potential energy of this test charge will remain the same as you move it. Let me write that down. No change in potential energy, no change in potential energy as you move along the equi-potential by definition. All right? Okay, what does that mean? Well, if the potential energy is not changing, it automatically means no work done by the electric field. No work by the electric field. Now think about it for a second. Why should this be true? Because whenever electric field does work, whether positive work or negative work, very automatically potential energy would change. For example, let's bring back gravity because gravity helps in understanding this. What happens when you drop a ball? Gravitational field does positive work. What happens to the potential energy? It loses it. What happens when you throw a ball up? Gravity does negative work. What happens to the potential energy? It gains it. So notice, whenever gravity does work, this ball would either lose or gain potential energy. Same would be the case over here. If electric field did work, the charge would have gained or lost potential energy. But we are seeing that it is not changing its potential energy. Means that as you go from here to here, electric field must be doing zero work. But how is that possible? Electric field is definitely pushing on the charge. It's putting a force on the charge and the charge is moving. So how can work done be zero? Ooh, work done can only be zero if the force and the direction of motion are perpendicular to each other. So in short, as you move a test charge along the potential surface, its potential energy should not change. That can only happen if the electric field does no work. And that can only happen if and only if electric fields are perpendicular to the equipotential surfaces. Now, if you find this a little hard to digest this right away, it's completely fine. It took me also a long time to do that. So keep pondering, keep thinking about it. It'll eventually make sense. So long story short, this basically means if you have been given some random field lines and if you wanna draw equipotential surfaces, just start drawing perpendicular, drawing them perpendicular to the field lines. This is how you might do it. And of course, nobody's gonna ask you to do that, but you know, we usually use computers to do that, but that's the idea. But equipotential surfaces must always be perpendicular to the field line. All right, let's summarize. And I want you to summarize. And the way to do that is I'm gonna ask you three questions and see if you can explain it to a friend. What are equipotential surfaces? That's question one. Second question, why over here, these surfaces are going farther and farther apart from each other, but over here, the surfaces are equidistant. And third one, why are equipotential surfaces always perpendicular to the field lines?