Torque acting on current loops

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 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 is a sine so there will be times sine of the angle between m and b and 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 talk 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 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 sign 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 is 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 you know 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 you know 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 let's 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 would it be more than when it was perpendicular or less and can you imagine what will happen if we had 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 all right if you're given this a shot let's see if the angle becomes more than 90 degrees again sine value will start reducing because sine has the maximum value at 90 sine 90 is 1 right so about 90 degrees again the torque starts reducing reducing reducing reducing and in this position the angle between b and m is 180 degrees and sine 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 is 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 all right 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 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 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 all right 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 of 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 are at the maximum energy this has a very high potential energy and nature hates that and so when when you disturb this equilibrium position then the nature puts a torque on it and goes from high energy unequal sorry unstable equilibrium state to the lowest energy stable equilibrium state and with this knowledge of torque we can go ahead and build stuff like galvanometers 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 to 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 you 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 up 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