1

concept

## Equipotential Surfaces

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Hey, guys. So by now we should be pretty familiar with electric potentials and potential differences. So we're gonna talk about a concept called Quit Potential Surfaces because you need to know them and how they relate back to things like electric feel that we've seen before. So let's go ahead and take a look. So basically what quit potential surfaces mean? Well, equity means equal on potential means potential. So it means these air basically just surfaces of constant potential and the relationship between the electric field and this potential we've actually seen before. In a different video. We know that the E is equal to the negative Delta V over Delta X. So any negative change potential across some distance is actually with the electric field is equal to now. By the way, you might actually see this in a different form. Some of you might actually see this. A Delta V equals negative e times, Delta X or Delta D or some some distance variable. So some of you may actually see it like this, and that's perfectly fine. It's just a different way to express this now, when it comes to echo potential surfaces, there's two very important rules to follow. The first is that an electric field can Onley exist when the potential changes. This is actually a direct consequence of this equation right here. So if Delta V is changing, that means you have an electric field. If this Delta V is equal to zero, then you have no electric field. So if this goes away, then you don't have any electric field. That's the first rule. The next rule has to do with the direction of the electric field, and the rule to remember is that E is always going to be perpendicular to echo potential surfaces, and they're always going to point along decreasing changes in potential. So, in other words, what happens is that if I have this eco potential surface right here, and I have another echo potential surface, and I'm just gonna make up some numbers right here, let's say I have This is 20 volts. Let's say I have anywhere on the surface is 20 volts and anywhere along this surface is 15 volts. Then what happens is that the electric field has to be perpendicular to the surface is right here. Now. Which way's this electric field going to point. Well, the decreasing potential is going to be in this direction, so that means this is where the electric field is going to point, all right? And we have a changing potential. So that means that there definitely is going to be some electric field in between these two sections right here. So this is basically the two rules. Now, as we go along the echo potential surface, what is the amount of work that's done on the charge? In other words, where if I were to put a little electron right here or a positive charge and I want to move it along this surface, how much work would that take? Will remember that the work that a charge experiences across a potential difference is equal to this equation right here. Q times Delta V. So it ends up happening is that if you are on an equal potential surface in which the voltage is not changing, then that means that the work across the equal potential surface is equal to zero because w equals negative Q times. Delta V. So if you have no change of potential, then you you have no work. So in other words, it would take zero amount of work to get this but this positive charge from here all the way over here. But it would take some work if you actually wanted to move in between these two potential surfaces. Okay, Hopefully that makes sense. Let me know if you guys have any questions about that now. The other important part about equal potential surfaces is we need to know what the diagrams look like for these things. We need to know visually what they look like. So we're gonna go ahead and do the drawing for the exponential surfaces for a point charge and an electric dipole. Okay, so for a point charge, we know that the potential for a point charge is cake You over are now. There's two rules we need to follow. We know that the electric field has to be perpendicular, and we also know that it has to change. So for instance, we know that if we were to try to analyze all the points with constant potential, then the only thing that's gonna matter really is gonna be the distance away. Because these K's and queues never changed. So what ends up happening? Is that the echo potential surface for a point charge. Go ahead. Make that in blue is actually gonna look like this. They're basically Wow, that was terrible. They're basically just going to be perfect spheres. Uh, let's go ahead and pretend that's a sphere. So you're gonna have an echo potential surface really close like this, you're gonna have an echo potential surface a little bit farther away and then a little bit farther away like this. So we know that the electric field lines are always going to point outwards and perpendicular to that surface, which makes sense, and it satisfies our rule. We also know that as you go outwards in this direction, So as your distance away from this point charge increases, then we know that the potential has to decrease. So the fact that these field lines actually point outward like this totally make sense. So that means that these are the locations of our equal potential surfaces. And by the way, I don't know. I don't know exactly what numbers you You would point here. Um, you basically have to calculate what the distance between that point charge and your echo potential services in order to figure that out. Okay, so let's go ahead and look at an electric dipole. Now, in a DI poll, we have that the electric field lines anywhere between them are gonna point exactly horizontally or exactly towards the left in this case. So what that means is that anywhere along this line here, the echo potential is gonna be a straight line because thes electric field lines have to be perpendicular. Now, any time you get just a tiny bit closer toe one of those charges now you're gonna have a little bit of an arc and your your echo potential because we know that this has to be perpendicular. So you're gonna get these swooping arcs like this. Those are gonna be your echo potential lines. And then as you get closer in towards one of the charges oven electric dipole, that basically just starts to look like a point charge. So what happens is these eco potential surfaces look like slightly slightly oblong spheres because, or, like, sort of like a blade spheres, or like ovals, because basically, as you get farther from one of the point charges, the other point charge starts to influence that potential. So these potentials aren't perfect spheres. They actually form these little ovals like this. And we also know that the potentials of the electric fields have to be perpendicular along all of these points right here. And by the way, the exact same thing would happen over here. You would get these ovals of equal potential surfaces so they would start to look like this. And we also know that the electric field lines point inwards this way. Because as you move closer towards the negative charge, you get higher and higher negative potentials. So all of these electric field lines makes sense. Alright, so let's go ahead and take a look at an example where we're working with equal potentials. What is the distance from a one micro Coolum charge to an echo potential surface of 150 volts? Well, this is gonna be fairly straightforward because we know we're working within a point charge. We know that the potential for a point charge is cake. You over are so that means if I have this little point charge right here, I know that the electric field lines are gonna point outwards like this and that the equal potential surfaces is basically just going to be a perfect sphere, which I can't draw ever. And we know that basically anywhere along this specific surface, this equal potential or the potential of the surface is gonna be 150 volts. So anywhere you look along the surface, the potential here it could be 150 volts. So, in other words, what we need to find is essentially what is the distance away from a point charge so that the potential is equal to 100 50 volts? And that's pretty straightforward. All we have to do, just move the distance upwards. We have to move the potential downwards, and we get that r is equal to K, which is 8.99 times 10 to the ninth. Now we've got the Q, which is one micro cool. Um, and now you've got the the potential, which is 150 volts. You gotta plug this in. You should get 60 m. That's how far you have to be away from point charge so that the potential is equal to 150 m. And basically, if you were to walk in a circle of 60 m radius. The potential would always be 100 50 volts. You would never change. Alright, guys, let me know if you guys have any questions with this and I'll see you the next one.

2

example

## Field due to Equipotential Surfaces

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What's up, folks? Let's work out to this example problem with equal potential surfaces. So we've got these eco potential surfaces and we're told with the potentials of each of those surfaces, are were also given with the spacing between them is and using that information. We're supposed to figure out what the magnitude and the direction of the electric field due to those eco potential surfaces. Remember that there's two rules that we need to remember. The hours are perpendicular to the equal potentials, which means that we know the electric field is gonna point along this direction. But they also point in decreasing potentials. They point along decreasing potentials. So that means that the electric field has two point in this direction, which actually takes care of the second part of our problem. What is the direction of the electric field and in more precise terms, if we know that the electric field is gonna point in this direction, we could say, is that if this angle makes a 60 degree angle with the X axis, that means our electric field is actually gonna be pointing. We're gonna need to know this angle right here. So this is 30 degrees because these two things they're supposed to be perpendicular. So in other words, as for the direction so that's gonna be direction. Wow, I can't spell. So the direction the E field will point degrees below the negative X axis and that is the answer. As for the direction So how do we calculate the magnitude? How do we calculate what the electric field is due to potential surfaces or changing potentials? We use the equation. E is equal to negative. Sorry, Negative Delta V over Delta X. But in this case, we're Onley interested in the magnitude of the electric field because we already took care of the direction down here. So the magnitude of the direction of the electric field all we have to do is figure out what the change and potential differences is or the potential difference. And we have to figure out the spacing. So we already know that the between each successive sector here are each little specific segment who were going. We're changing five volts and we're doing that across the spacing of one centimeter, so it's gonna be 0.1 So these distances are 0.1 meters. So that means that the electric field is gonna have a constant magnitude off 500 volts per meter. By the way, if you're not familiar with this volts per meter unit, this actually, if you work out with the units of revolt is this actually also turns into Newtons per Coolum, just a different way to sort of express that. Okay, so that's how you figure out with the magnitude. And the direction of the electric field is due to eco potentials. So first, figure out what the potential differences are the spacings, and then just figure out your direction by going in towards towards decreasing potentials and perpendicular to those eco potential services. All right, let me know if you guys have any questions.

3

Problem

Draw the electric field that corresponds to the equipotential surfaces shown in the following figure. Note that the potential is decreasing in the upward direction.

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