Electric fields (part 1)

0:00- [Instructor] You probably know that if you have two point charges separated by some distance, they will exert a force on each other, right? But here's a question. How does this charge know that there's a charge over here to put a force on it? In other words, how exactly does this non-contact force mediate?

0:19Here's how we can model it. We could think that this charge Q produces an influence around itself, and that influence exists throughout the universe. We call this the electric field. And when another charge comes into that field, it experiences a force. So the idea is the charge doesn't directly exert a force on this other charge, but it does so via this electric field.

0:47And similarly, this charge will produce its own electric field. It too extends everywhere. And when this charge comes into that field, it experiences a force. So how exactly do we model this electric field? Let's find out. First of all, since the electric field is an indicator of how much force it produces, we want the field to be a vector quantity.

1:12So now the main question is, how do we figure out the strength or the magnitude and the direction of the electric field at any point? Well, let's start with the strength, the magnitude. We think of electric field as the electric force acting per unit charge. Okay, what does that even mean?

1:32Let's consider a point P, and I wanna know what the electric field at that point is. How do I figure that out? Well, keep some charge at that point. For example, we'll keep plus two coulomb. That's just an example. You can keep any charge you want. Keep plus two coulomb and find out what is the force that is exerted on that charge.

1:53So let's say the force exerted on that charge is 20 newtons. So 20 newtons of force gets exerted on two coulombs of charge. So what would be the force exerted on one coulomb? It would be half of that. It will be 10 newtons, right? And so now we say the electric field at this point P is 10 Newtons per coulomb.

2:14Makes sense, right? Electric force per unit charge. That is how we think about its magnitude. And what about the direction of the electric field? Well, that's in the same direction as a positive charge would experience the force. So notice this number does not mean that there is a plus one coulomb charge, and it is experiencing 10 newtons of force.

2:36No, no, no. It says that if there was a coulomb of charge, it would have experienced 10 newtons of force. It's an indicator of how much force a coulomb would experience. With that indicator, we can now tell what would happen if I kept a plus three coulomb of charge.

2:51It would experience 30 newtons of force. And what if I kept a minus five coulomb charge over here? Ooh, it would experience 15 newtons of force, but since it's a negative charge, it would experience the force in the opposite direction. So there it is, the electric field now gives me complete information of how much force, you know, the magnitude and the direction of the force that gets exerted, you know, when I place any charge over there.

3:13So in general, how do I define the electric field strength mathematically at any given point? Well, it is basically you just keep some charge over there. Technically we call it a test charge because we are using it to test, to probe, to find the electric field at that point.

3:31Ideally, you would want the magnitude of this test charge to be as little as possible so that it doesn't affect the electric field that much. But what's important is it's a test charge because it's used to test how much electric field is present at that point.

3:43This is the source charge because it is the one that is creating the electric field, right? Anyway, so what you do is you keep a test charge, you calculate the force on that test charge, and you divide that force by the test charge. That is the electric field.

3:55That tells you the force per unit charge, right? So electric field at any point is the electric force acting on a test charge divided by the test charge. This works both in magnitude and direction. And as we saw, the unit of the electric field is newtons per coulomb, and that's always the case.

4:14Newtons per coulomb is the unit of electric field. Okay, so can we find an expression for the field, electric field due to a single charge Q at some distance R? Great idea to pause the video and see if you can figure this out yourself. All right. So the magnitude of the electric field, let's only focus on the magnitude because we know the direction.

4:37So the magnitude of the electric field is going to be the magnitude of the force acting on that test charge, which we know from Coulomb's Law is one over four pi epsilon naught times the magnitude of the product of the charges divided by R squared where R is the distance between those two charges, right?

4:54So you keep a test charge over here, you calculate the coulomb's force acting on it, that is this part, and then divide by that test charge, divide by the magnitude of that test charge. So notice the Q cancels out and what you end up with is one over four pi epsilon naught Q over R squared.

5:10That is the magnitude of the electric field due to a single point charge. Okay, so that's the magnitude, but what about the direction in general? Well, the direction around a point charge is always going to be radial. So if it's a positive source charge, it will be radial outwards.

5:28And if it's a negative source charge, it will be radial inward. So now what we can do is think about electric field everywhere. So let's consider the electric field at the same radius everywhere. What will it look like? Well, everywhere it's going to be radially outwards.

5:44What would happen if you had to look at the electric field somewhere closer? Well, notice if you go closer, R is smaller, and therefore the electric field strength would be much higher. So you would expect electric field to be stronger. That makes intuitive sense.

5:59You're coming closer to the charge, so the electric field strength is higher. And so you would now have much stronger electric field strength. The magnitudes would be much higher. And in contrast, what if you go much farther away? Well, the electric field will be much weaker.

6:14And so the magnitude will be much, much smaller. And there we have it. We have created a map of the electric field around this charge. We no longer need that cloud. We now have a rigorous way to visualize electric field. Now, of course, ideally the vector lengths, which are the magnitude should follow the inverse square law, but they aren't properly scaled over here, okay?

6:38But anyways, this is just a two dimensional representation. Remember we live in 3D, so here is a simulation showing the electric field in three dimensions due to a single point charge. It's beautiful, right? Again, you can notice that it's radially outwards. If this was a negative charge, it would be radially inwards.

6:56And notice as we go farther away, the electric field strength dies out as one over R square. And you can see that in the magnitude of the vector becoming smaller and smaller. By the way, this is called the vector field map, and this is one way to represent, you know, how the field varies around the charge.

7:14Here, the magnitude or the strength of the field is represented by the length of the vector. We can also represent the magnitude using brightness instead. Now look, the length of the arrows is the same everywhere, but it's the brightness that changes, and it's the brightness that encodes the strength.

7:32Again, you can see it's very bright close by, and as you go farther away, it becomes much weaker. But now comes my favorite method of depicting the vector field. Are you ready? It's by drawing continuous lines. Here we go. We call this the field lines. Now, again, if it's a positive charge, we'll draw outward arrow marks.

7:52If it's a negative charge, we'll draw inward arrow marks. But I love field lines because they're much easier to draw, much quicker to draw as well. But what about the strength of the electric field? How is it encoded now? What represents that? Well, now that is represented by how crowded the field lines are.

8:10Notice close to the charge, the field lines are very crowded. It's densely packed. Farther away, the field lines are moving, you know, the distance between them is much higher, so it's much more, it's much loosely packed. And so now the density of the lines represents the strength of the electric field.

8:27That's pretty cool, right? So this is the field lines representation. Okay, but so far we considered the electric field due to just one point charge. What if you have more than one point charges? How do you figure out the electric field at any specific point now?

8:41Again, it would be a great idea to pause the video and see if you can think about this because it's analogous to what we did when we're calculating forces due to Coulomb's Law. Okay, well, just like what we did with the forces, we're gonna do the same thing over here.

8:57We're gonna first calculate the electric field at 0.1 only due to this charge, this particular charge, ignoring all other charges. We'll call that as EP1, for example. Then we'll calculate only due to Q2, maybe it's this much along this particular direction, radially away from that.

9:12Let's also imagine this is a positive charge. It would be radially away from that. Let's call that as EP2. And if there are more charges, we will calculate EP3, EP4, and so on and so forth. And then we vectorially add them up. So if you have only two, then you can use the Parallelogram Law to just add them up or we can decompose them, whatever we want.

9:32The idea is you do a vector sum, and now that represents the net electric field at that point. This is called the Principle of Superposition. And it is analogous to what we did with the forces because everything is similar, right? Electric field, at the end of the day, what is it?

9:50It's just a representation of how much electric force is exerted on a unit charge. So of course it'll be analogous to that. Everything is similar to what we did with the force over there. So now let's go back to our simulation and see the electric field due to two point charges.

10:05Charges of equal magnitude and the same sign. This is what it looks like. Again, if they were both positive, the field lines would be moving away. Sorry, it would be outwards. If they were negative, they would all be inwards. And notice that the field lines can also bend.

10:22What does that mean? Well, we draw field lines such that they are paddled to field vectors at every point. So if a field line is bent, that means the field vectors point in different directions along that line. So remember, each point on the field line has its own field vector, and the direction of that vector will always be tangential to that field line.

10:47Okay, one last thing we'll consider, we'll consider two equal charges, but opposite signs. This is called an electric dipole. If the red one is a positive charge, then the electric field lines would be moving away from it, and they would terminate into the negative charge.

11:03It's pretty cool. Finally, instead of just two charges, imagine we had like a line of charge. We had a charge distribution, right? It could be a line distribution. It could be like, I don't know, maybe charges are distributed on a plate or something. It could be a three-dimensional, very complicated distribution, whatever that is.

11:19If you wanna calculate the electric field at any point, all you have to do, all you have to do is calculate electric field due to individual, each of those charges separately and then add them all up vectorially. I know mathematically it sounds very complicated and very tedious, but what's important is it's the math that gets tedious.

11:39What's interesting and what's amazing is that you don't need any new physics. So the principles of superposition is so cool that I can just use this particular physics. I just need field due to a single charge. I just need that expression, and I can use that to calculate the field due to any messy configuration, any messy distribution of charges.

12:00That is pretty awesome.