 When you bring magnets near a compass, the needle turns. Big deal. But remember, current carrying loops also behave like tiny magnetic needles, meaning they can also turn in the presence of magnetic fields. And we can use this concept to build stuff like galvanometers or motors which run on electricity. But to do any of that, we need to know how much turning force or how much torque acts on this current carrying loop. And that's basically what we want to explore in this video. All right. So the torque acting on a current loop is given by this expression. Tau is the torque, m is the magnetic dipole moment which I'll get to in a second, and b is the strength of the external field. So what exactly is this dipole moment? Well, we've talked about that in a previous video but just to quickly recap, any current carrying loop behaves like a tiny magnet. And the strength of that magnet depends on various things like the area, the current, and the number of loops. That strength is what we call the magnetic dipole moment. More the dipole moment, stronger the magnet, this coil is behaving like. And we've already seen that this is a vector quantity and the direction is given by our right hand thumb rule. So you take your right hand and you clasp such that the four fingers represent the direction of the current, then the thumb gives you the direction of the magnetic moment. Now if this is not super clear to you or you need a refresher, no worries. You can go back and check out our video on magnetic dipoles and moments. But anyways, let's get back to our equation. What is our equation trying to tell us? Let's first look at the magnitude of that torque. Then worry about the direction of that torque. So the magnitude says, if you look at the magnitude, the magnitude of tau would be equal to magnitude of M times B times, for a cross product, there's a sign. So there'll be times sine of the angle between M and B. So if we call that angle theta, it'll be sine of the angle between them. So this is saying that if you have more magnetic moment, you'll get more torque. Does that make sense? Well, yeah. If you have stronger current, if it behaves like a stronger magnet, you will expect more torque acting on it. On the other hand, if it behaves like a weaker magnet, less torque. If there is no magnet at all, obviously it will not be affected by the magnetic field, so torque would be zero. So that makes sense. The equation is also saying that the torque would increase with the strength of the external field that we apply. That also makes sense, right? If you apply a very weak field, there'll be a very weak turning force. A strong field would provide a strong turning force. And if you don't provide any field at all, it'll not turn at all. So that also makes sense that the torque depends upon the magnetic field. Finally, it says that the torque also depends upon the sine of the angle between the two. What does that mean? Well, let's look at it. In this particular case, theta, the angle between M vector and B vector, is 90 degrees. So the equation is saying that the torque is maximum. And now it says that in this position, because of the torque, our coil is going to turn. And because it says M cross B, it is saying that M is going to turn towards B. So the torque will act this way. And notice, as the coil turns, the angle between M and B decreases. Can you see that? And as the angle decreases, sine theta decreases, which means the torque is reducing. Reducing torque becomes smaller and smaller and smaller and smaller and smaller and smaller. And when it gets aligned over here, theta becomes zero. The torque would be zero. So once in this position, there'll be no longer a torque acting on it. And if you think about it, that kind of makes sense. That's exactly what we saw in the intro. We saw that when the needle was not aligned with the magnetic field, there was a torque. But once the needle gets aligned, it stays there, meaning the torque becomes zero. So you see what the field is trying to do? The field's job is to align the magnetic moment or to align the magnet in the direction of the field. That's why when it's completely aligned, the torque becomes zero. And when it is at perpendicular, when it is perpendicular, that's when the torque becomes maximum because it is the exact opposite of being aligned, perpendicular. So the field hits it and puts a maximum torque. So here's what we saw. When we have 90 degrees between M and B, the torque is maximum. And when the M is aligned with B, zero degrees, the torque also becomes zero. Now be careful over here. You might think, hey, wait a second. In this case, why are we saying it's zero degrees? The plane of the coil is perpendicular to the magnetic field, right? Well, yeah, but remember the angle is not between the plane and the field. The angle is between the M vector and the field vector. Got it? So that's where I used to always get confused, especially in numericals. So remember, this is the angle between the M vector and the B vector, not the plane and the B vector. And to give you some intuition behind this, let me give you a familiar example. Consider a pendulum. Basically, imagine you have a ruler, you made a hole in that, and you hung it somewhere. Now in this position, you can kind of see that gravity is going to put a torque on it in this direction and tries to turn it over here. But as this ruler turns, let's turn this ruler, as this ruler turns, you can know, you know in your bone that now the torque starts decreasing, decreasing, decreasing. And eventually, when it's aligned in this position, that's when the torque is zero. Exactly same thing happening over here as well. Gravity wants to align the ruler in its direction. And that's why in this case, we have the maximum torque, 90 degrees. And when it's aligned, we get zero torque. The field is trying to align the ruler. Here, the field is trying to align the M vector. That's why it's putting the torque. Okay, what if the angle is more than 90 degrees? In this position, what do you think happens to the torque? Will it be more than when it was perpendicular or less? And can you imagine what will happen if we are to turn it this way and keep it in this direction? Go ahead. Why don't you pause the video and think a little bit about what happens in those cases? Alright, if you're given this a shot, let's see. If the angle becomes more than 90 degrees, again, sign value will start reducing because sign has the maximum value at 90. Sign 90 is one. Right? So, above 90 degrees, again, the torque starts reducing, reducing, reducing, reducing. And in this position, the angle between B and M is 180 degrees, and sign 180 is also zero. So, even in this position, the torque is zero. So, even at 180 degrees, the torque becomes zero. But can you feel the difference between these two cases? In both these cases, torque is zero, but there's some conceptual fundamental difference between these two positions. Again, I want you to pause the video and think a little bit about this. Maybe you can use the ruler example and see if you can figure out what difference you find. Alright, if you think in terms of the ruler, this is exactly like trying to put a ruler and balancing it this way. Even in this situation, the torque on the ruler would be zero. If you can exactly balance it, it would be zero. But you can feel in your bones the difference between these two, right? This is a very stable position. What I mean is, if you were to nudge it a little bit, the torque acting on this will bring it back to this position. This is a very stable equilibrium position, we would like to say. On the other hand, if you disturb this equilibrium even a little bit, the torque will not bring it back. The torque now, that equilibrium position is lost and the torque will turn it all the way to the stable position. And that's why even though this is an equilibrium, we call this as unstable equilibrium. You disturb it a little bit and it's gone. And that's why this gives you that unsettling feeling, right? Because this is a very delicate balance. You disturb it and it's gone. Something very similar is going to happen over here as well. This is an unstable equilibrium position. Meaning, if we were to disturb this even a little bit, then the torque will put it in this position, the stable position. So the big difference is even though the torque is zero in both these cases and they are in equilibrium, this is an unstable equilibrium and this is a stable equilibrium. And finally, I want you to think about energies associated with these two equilibrium positions. Can you pause the video again one last time and think about which of these two positions would have a higher potential energies? And again, the ruler might help you. So pause the video and think about which of them has a higher potential energy. Alright, you might know already that when it comes to gravity, more height means more potential energy. Since the ruler over here is higher than over here, there must be more potential energy over here and there must be less potential energy over here. So you see in the unstable equilibrium state, the objects tend to have higher potential energy. And what is nature trying to do? Nature is trying to put a torque and reduce the potential energy and get it to stability. It's that potential energy which gets converted to kinetic energy or rotation at least in this case and maybe eventually it'll get dissipated as heat. And the same thing applies here as well. When the magnetic moment is in the opposite direction of the field vector, we have at the maximum energy. This has a very high potential energy and nature hates that. And so when you disturb this equilibrium position, then the nature puts a torque on it and goes from unstable equilibrium state to the lowest energy stable equilibrium state. And with this knowledge of torque, we can go ahead and build stuff like kilometers or motors and not just that, they're also applicable in other areas as well. For example, did you know that the same concept is used in MRI scan machines? That's right, state-of-the-art MRI machines also use the same principle without getting into the details. But the whole idea is our body contains a lot of protons from the hydrogen atoms that spin on its own axis, producing a magnetic moment. And the MRI machine produces an external magnetic field because of which there's a torque acting on the proton which turns it into the equilibrium state. And then based on how much energy is required to turn it back into an unequilibrium state and how much energy it releases when it flips back to the equilibrium state, doing all that calculation, one can figure out or one can actually image your entire body just by using the same principle. I mean, yes, things are complicated, but it's the same principle. You should definitely read about it. Amazing, right? So long story short, current carrying loops behave like magnetic dipoles. And when they are kept in an external field, there's a torque that acts on them. And the torque basically tries to align the magnetic moment along the magnetic field, which is why when they are perpendicular, you get the maximum torque. And when it's aligned at zero degree Celsius, you get the minimum torque, stable state, lowest energy. And when we are anti-aligned or anti-parallel, again, you get zero torque, but that's the highly unstable equilibrium state.