 When you keep a dipole inside a uniform external electric field, the charges experience equal and opposite force, and so the dipole as a whole experiences a net zero force. But the two forces makes the dipole turn. In this case, it'll tend to turn it this way, and as a result, there will be a torque acting on it. And what does that torque do? Well, it'll always try to align the dipole in the direction of the field. Let me show you. If you have the field to the left, and let's say you keep a dipole this way, the torque will act in a way to try and align it. And once it's aligned, the torque disappears. It doesn't matter how the field is. The torque will always try to align the dipole in the direction of the field. And this is something we've seen before in the previous videos. And so if you need a refresher, feel free to go back and watch our video on intro to torque acting on dipoles. But what we're going to do in this video is figure out how much that torque is in terms of the dipole moment and the electric field. So let's do that. So how do we calculate torque in general? Well, if there are two equal and opposite forces acting on any object, we call that as a couple, then the torque due to that couple always equals the strength of the force. The two, the forces are equal. So strength of the force multiplied by the perpendicular distance between them, which I'm calling as L. And over here, L would be this distance. This is the perpendicular distance, right? This is our L. So now we need to find what this force is and what this length is in terms of what? In terms of the electric field strength, in terms of the charge and the length of the dipole. So let's put all of that. So let's say the charge is Q. And the length of the dipole is, I don't know, let's call that as D. Sometimes in textbooks, we call it as 2A. It doesn't matter. You can take any variable you want. So let's see. Can I calculate what the force is? Well, yeah. I know the charge. I know the electric field. And from that, I can figure out what the force is. And similarly, I know this length is D. That's the length of the dipole. And if I know this angle and you can assume that angle to be, say, theta, then from trigonometry, you can figure out what the length is. Oh, so we can simplify and I want you to pause the video and see if you can simplify this yourself. You have everything that you need. So pause and give it a shot. Don't worry about going wrong. It's fine. But always try to first do things yourself. All right. So let's start by figuring out the force. How much is the force? Well, for that, I look at the electric field. Remember what is the meaning of the electric field? It represents the force acting on one coulomb. So if I keep one coulomb, the force is E. If I keep two coulombs, the force is 2E. So if I keep Q coulombs, the force would be Q times E. So from the definition of the electric field strength, I know the force is going to be Q times E. So that's our force. Okay. Now, how do I figure out this length? Well, for that, let's take this angle to be theta. This is the angle between the dipole moment vector and the electric field vector. And now from this right angle triangle, I know the hypotenuse. I want to figure out what the opposite side is. And so I can use sine theta. So sine theta is going to be L divided by D. So L equals D sine theta. Hey, we found that as well. So let's write that. This is going to be D times sine theta. Okay. Can we simplify this even further? Oh, yes, we can. I see something. I see something. Do you see it as well? Q times D. We've seen that before. What do you call the product of the charge and the length of the dipole or the distance between the two charges? Hey, that's our dipole moment. Remember, that represents the strength of our dipole. And we call it P. So two times D is P and we have E. And so we get P times E times sine theta. And I think we're done. We're actually almost done, not completely, because this represents the magnitude of the torque. But remember, torque is a vector quantity. It has a direction as well, which means we should be able to write this as a vector equation. But how do we do that? Well, I look at the right-hand side and I say, look, I have two vectors. P is the dipole moment, which is given by this vector from negative charge to positive charge. That's a vector. And E is the electric field vector. So I have two vectors being multiplied. So that means we're talking about vector multiplication. And then we have cross products and dot products. And sine theta comes under cross product. That's the vector product. So what I'm trying to say is that we can now write that the torque vector, we can now write it as P cross E. P cross E. Because the magnitude of P cross E is P sine theta. But wait a second. If you remember vectors, you might also say, wait a second, but can't it also be E cross P? Because magnitude of E cross P is also E P sine theta, which is the same thing. The difference between them is in the direction. P cross E and E cross P are in the opposite direction. So the question is, which one is our torque? Should I write it as P cross E or E cross P? How do I figure that out? Well, P cross E represents P turning towards E. And E cross P represents E turning towards P. But in our case, it's the P that's turning towards E, right? So we will not write it as E cross P. We will write it as P cross E. That represents the true direction. And there we have it. This represents our vector equation for torque. So in our example, what is the direction of the torque? Well, we could say the torque is in the clockwise direction, but we need a particular direction for the vector, right? So for that, we use our right hand thumb rule. We use our four fingers. We make sure our four fingers are in the clockwise direction, whichever direction the P is turning. And then the thumb represents the direction of the vector. And so in this case, we say the torque is into the screen. Let me give you another example. Let's say this time we have a dipole, whose dipole moment is to the left, and we have electric field coming out of the screen. Now tell me, what direction will be the torque acting on it? Can you think about it? Well, again, torque is P cross E, which is basically saying that P will get aligned towards E. Now over here, E, it's a little hard to visualize, but E is coming out of the screen. So it's a little hard to visualize that. And therefore, P is going to turn like this. And so what would be the direction of the vector, of the torque vector? Well, again, we use right hand thumb rules such that our four fingers represent this turning. And so for that, it will be this way. And so now thumb points upwards, and so we'll say torque is pointing upwards. That's the direction of P cross E. So that's how you get the direction of the torque. My final question to you before we wrap up is, for what angles do you think our torque is going to be maximum? And for what angles do you think it's going to be minimum? Can you pause and think about that? Okay. For that, I'm just going to look at the magnitude. And since I'm only considering the angle, at what angle is sine theta maximum? Well, sine theta is maximum at theta equals 90 degrees. So that means when the angle is 90 degrees, that's when I get maximum torque. So your P should be perpendicular to the electric field vector. And that kind of makes sense because when P is perpendicular to E, the two forces are farthest apart. And that's why so the L will be maximum. And that's basically the reason why our torque would be maximum. So maximum torque over here. Fine. When do you think our torque would be minimum? The minimum value would be zero. And that is when sine theta becomes zero. That can happen when theta is zero, meaning P and E are aligned somewhat like this. And that kind of makes sense because in this position, the two forces acting on the two charges are kind of like pulling the dipole apart, not producing any torque. The torque is zero. And this is something we've seen before as well. The whole idea is the torque is trying to align the P in the direction of E. And once that has happened, the electric field is happy, everybody's happy. Torque goes to zero. But there is one more position in which torque can be zero. That is if theta is 180 degrees because sine 180 degrees is also zero. And that would look like this. So this is where P, this is P actually, inverted. P and E are exactly opposite to each other. Why is the torque zero over here? Because if you look from the forces point of view, notice even here we have, on this charge there's a force like this, on this charge there's a force like this, and that's not producing any torque. And as a result, you get zero torque. So in both these cases, we have no torques. And therefore we can say in both these cases, our dipole is in equilibrium. Equilibrium is a state where net force and net torque is zero. Now net force is zero in all cases, but net torque is zero in these two cases. So these two are equilibrium cases. However, there is a difference between this equilibrium state and this equilibrium state. Can you think about what that difference is? Well, let's come over here. Let's disturb this equilibrium. Let's say I twisted this, I tilted this dipole a little bit. What's gonna happen? Well, remember this is being pulled this way, this is being pulled that way. A torque will come back, so there'll be a torque acting back on it, and it'll go back to its equilibrium position. If I tilt it this way, something very similar will happen. Torque will come back on it. I mean torque will make sure that this thing comes back to its equilibrium position. So if you disturb this equilibrium, it comes back. Nature likes this. We call such equilibrium stable equilibrium because the name itself tells you, it's very stable. You disturb it, it comes back. But what happens over here? Let's see. If I disturb this one, let's say a little bit, what direction will be the torque? Well, the positive charge gets pushed to the right, negative charge gets pushed to the left. Look, it's gonna turn all the way, all the way like this and come to this position, which means our equilibrium position would be lost. So you disturb it just a little bit. So you disturb it just a little bit. The equilibrium position will be lost. If you disturb it again this way, you'll find the same result. Equilibrium position will be lost. And therefore, this equilibrium is what we call unstable equilibrium. Again, because as the name suggests, you disturb it a little bit, you will not get this equilibrium back. So theta equal to zero represents stable state. Everybody likes that. Everybody's happy. Theta is equal to 180 degrees. Also torque is zero, but nobody's happy. The electric field is not happy. It wants it to align the dipole in the direction. So disturb it a little bit. It gets the opportunity, puts all the torque, aligns it in the stable state. And therefore, this is an unstable equilibrium state.