Electric potential and electric field

0:00- [Instructor] Suppose we have a uniform electric field, meaning the electric field strength and the direction is the same everywhere. We know it represents, it's an indicator of how much force acts on a unit charge kept over there. So for example if the electric field is 30 newtons per coulomb, it means that if I had kept a coulomb of charge inside this field, it would experience 30 newtons of electric force, right?

0:21Now the question we want to try and answer in this video is, can we think about this in terms of electric potential? What does the electric field tell us about electric potential? How do we connect the two? That's what we want to explore, so let's do that. Okay, so how do we bring electric potential into the picture?

0:40Well, there are multiple ways to do it. My favorite method is to just imagine letting go of this charge. What's going to happen? Well, because of the force, it's going to accelerate. Let's say it moves for one meter. Then by the time it comes somewhere over here, one meter, once it has moved one meter, it would have gained some kinetic energy, right?

1:03Where did that kinetic energy come from? If I consider the electric field and the charge all part of my system, there are no external forces acting on it. So where did the kinetic energy come from? Well, it must have come from the potential electric potential energy of the system, right?

1:19So the electric potential energy got converted to kinetic energy. So we can say that the system lost electric potential energy. Okay. How much electric potential energy did it lose? Well, that is equal to the amount of work done by the electric field, right? How much work did the electric field do?

1:38Well, it put a 30 newtons of force and it moved it by one meter. Work done is force times displacement when the force is constant over that entire displacement. And since the force and the displacement are in the same direction here, this work is positive. So it's just force times displacement.

1:53So 30 joules of work was done. 30 joules per coulomb per meter was done. And so we can say the electric potential energy dropped or reduced by 30 joules per coulomb per meter. In other words, electric potential drops by 30 volt, because remember joules per coulomb is just volt, 30 volt per meter.

2:21Now think about this. What if this electric field strength was 100 newtons per coulomb? Then the force, electric force over here would be 100 newtons and the electric potential would have dropped by 100 volt per meter. In other words, electric field strength is an indicator of how much the electric potential drops per meter.

2:42If the electric field strength is 30 Newtons per coulomb, it suggests that in one meter, if I move along the electric field, the potential, the electric potential drops by 30 volts. So that's another perspective we can have. It indicates how much the electric potential drops per meter as I move along the field.

3:04And so we can now write that electric field strength is a measure of how much electric potential drops or electric potential difference per meter. And to be more precise, we should actually put a negative sign. The negative sign tells us the electric field points in the direction where the potential decreases.

3:24It's basically the potential drop we just talked about and its units can also be though of as volts per meter. Okay, one question we could have is how can electric field have two different units? How does that make any sense? Well, these are actually the same units.

3:42Here's one way to see it. If I multiply and divide by meters over here, newton meter is basically work done, right? Force times distance. So that's joules and joules per coulomb is volts. So there you have it. It's basically volts per meter. So volts per meter is nothing but Newtons per coulomb.

4:01Now, if we only care about the magnitude, we can just take the absolute value on both sides and so the negative sign will go away. That's why we will write the magnitude of the electric field as the absolute value of delta V or delta R. Okay, but what about the direction of the electric field?

4:18Well, notice the electric field always points in the direction where the electric potential reduces. Does that make sense? Yeah. I mean, along the electric field, a positive coulomb will always gain kinetic energy. And so the system will always lose potential energy and therefore along the electric field, potential always reduces, right?

4:41And so electric field always points in the direction where the electric potential reduces. But wait, this makes sense if the electric field is uniform, but what if we have a non-uniform electric field? For example, the electric field is stronger here and it's weaker here and its directions are also, it's not the same everywhere.

5:02Again, if I were to let go of a plus one coulomb charge, for example, it would have gained some kinetic energy, which means the system loses potential energy as it goes from here to here, which means again, electric potential drops along the field. And let's say again, the electric potential drops by a total of 30 volts over this meter.

5:20What does this now represent? Well, it can't represent, it can't say the electric field is 30 words per meter because the electric field strength is not a constant. So what does it mean now? Well, now this represents an average value of electric field over that meter.

5:37So somewhere here, the electric field strength is stronger than 30 volts per meter. Over here, it could be weaker than 30 words per meter, but on an average, the electric field strength is 30 volts per meter. So in general, this quantity represents the average value of the electric field strength over that distance delta R.

5:59Long story short, if we calculate how much the potential drops per meter, it gives us an indicator of how strong the electric field is on average over that distance. Okay, here's another question. What if instead of moving along the electric field over here we moved perpendicular to the electric field?

6:17Now how much would the electric potential drop if we moved say one meter? Great idea to pause the video and think about this. All right. Now when you move perpendicular to the electric field, notice that the work done by the electric field is zero because the force and the direction of motion is perpendicular to each other.

6:37In that case, no work is being done. And so the system does not lose or gain any electric potential energy. And so the electric potential neither drops nor grow. There's no change in the electric potential. And therefore when you go perpendicular to the electric field lines, the electric potential stays the same.

6:59And so now if I ever draw a line perpendicular to the electric field, electric field lines, we call it an equipotential line because the electric potential everywhere along the line is exactly equal. We also call it isolines of electric potential and we can draw multiple isolines like this.

7:21And if we assign this isoline zero volts, then this isoline would be 30 volts lower, right? Because we know this is one meter and/or a meter, the electric potential drops by 30 volts. So this is 30 volts lower, which means this isoline would be negative 30 volts.

7:38And since these are equally distant over here, this would be negative 10 and negative 20 volts. Now of course this is just a two-dimensional representation. If you look at the uniform electric field in three dimensions the isolines now become surfaces. You have plane surfaces.

7:55Everywhere along a particular surface, the electric potential is exactly the same. So these are called equipotential surfaces in general. But for the purpose of our video, we can just stick to lines, two-dimensional representations. So now the important question is, what will the isolines look like for this particular case?

8:15Again, it'll be a great time to pause the video and think about this. All right the key feature of the isoline to remember is that it's always perpendicular to the electric field lines at every point. That has to be the case because if we move a charge along the isoline, the electric field should do absolutely zero work.

8:35And the only way that can happen is if the field line is always perpendicular to the isoline, right? So by definition, isolines are always perpendicular to the field lines. So how could that be over here? Well, if the field lines are curved, the isolines must also end up getting curved in such a way that locally to every point, the isolines must be perpendicular to the field lines.

8:59Does that make sense? And another interesting difference you see between these two is that here the isolines are equidistant from each other, but here they are not. They're closer here and they're farther away over here. Why is that? Well, remember here the electric field strength is exactly the same.

9:15So the potential drop per unit distance is exactly the same everywhere. But here, the electric field strength is not the same everywhere. Here the electric field strength is higher. So the potential drop is higher per unit distance. That's the whole point, right?

9:30Electric field is a representation. The electric field strength is a representation of how quickly or how much the potential drops per unit distance, right? And so that's why over here, the 10 volt drop happens over a very short distance because the electric field strength is very high.

9:46The potential drop is much higher per unit distance. But over here, the 10 volt drop requires a much larger distance because the electric field strength is much weaker. That is the whole point. All right, let's take more examples. What does the equipotential surfaces look like for a point charge?

10:03Well, we know electric field line is radial, and so equip potential surfaces must always be perpendicular to the electric field lines, so they end up becoming circles. And again, if I draw more of them, we have the same feature as we saw earlier. Notice close to the charge, the circles are much closer because the field strength is much stronger.

10:22And as I go farther away, the circles are much farther away because the field strength is much weaker. And again, just a reminder this is a two dimensional representation. In three dimensions, we don't get circles, we get spheres. So equipotential surfaces are spherical around a point charge.

10:41But again, let's just stick to 2D representation. And by the way, doesn't this remind us of the topographic maps that we see? I mean, in these maps, a contour like this represents specific height. So for example, every point over here is a height, is at height zero, and every point over here is at height 200 units and then 300 and 400 units and so on and so forth.

11:06And so just by looking at this map, we can immediately say that, "Hey, hey, hey, on this side, look, the height drops very quickly. So it's much steeper over here and you know, the surface is much steeper over here. But on this side, look, the height drops much more gradually, so it's much flatter.

11:26In fact, you can see the drop from 200 to 100 happens over a much larger distance, so it's quite flatter over here." And so we can understand all of these features, right? And so this is analogous to what we saw here. The heights are analogous to electric potentials and the field, the electric field strength is analogous to steepness.

11:49So wherever the equipment surfaces are closer, the electric field is quote unquote "steeper," the strength is higher, and farther away the electric field is, again, quote unquote, "flatter," meaning the strength is lower. And because this is so analogous, we would like to visualize it this way.

12:07Look at this, we can think of a positive charge at the center, and as we go closer and closer to the positive charge, we can think of it as we are climbing the potential. And the steepness is kind of the representation of electric field. So electric field is very strong over here because it's very steep.

12:24And as you go farther away the electric field is much flatter over here so it is much weaker. Of course in reality we're not climbing any mountain over here. It's just an analogy. Anyways, this is for a positive charge. And again, notice that the electric field points in the direction where the electric potential reduces.

12:42Okay. Now, what if you have a negative charge? Well, it looks very similar, except now as you go closer to the charge, you will be climbing down the potential because as you go closer to the charge, now the potential reduces because it's a negative charge, right?

13:00And again, you can see electric field lines point in the direction where the electric potential reduces. Farther from the charge, the electric field is much flatter, much weaker, and closer to the charge, electric field is much steeper, meaning it's much stronger.

13:15This is how we can visualize this equation. It says that the electric field strength is a measure of how steeply the electric potential changes with position. Finally, remember that if you have a charged conductor, the electric field inside the conductor is always zero.

13:34That means if I were to take a coulomb of charge and move it along inside anywhere, there's no work done on it by the electric field. That means the electric potential everywhere inside the conductor and on the surface must be the same, which means the entire conductor is also an equipotential surface.

13:53Now what happens if I were to bring in another conductor and touch it to this? Well, clearly charges will now move from this conductor to this one, but the charges will distribute in such a way, again, the electric field over here would be zero and the whole thing will be a single equipotential surface.

14:14It has to be, you know, the electric potential has to be exactly equal because if it weren't equal, the charges would still be moving from a higher potential region to a lower potential region, right? But the fact that we are an electrostatic equilibrium, once we reach electrostatic equilibrium, the whole thing becomes a single equipotential surface.