 Now the nature of this torque if you notice that you know, this is the magnetic field and if you have a magnetic moment like this Okay, and if you slightly rotate This magnetic dipole moment little bit Then what will happen? It will rotate it back and try to achieve its original position Okay, so this is a restoring torque. It tries to restore its position What will happen when it again come back when it again come back when? Theta becomes zero The angular velocity will still be there. It will not be zero So it will further rotate Fine, so it will not stop it will just Rotate and then go till here and then stop Okay, so it will again rotate by angle same theta Then it will again come back So it will oscillate like a pendulum You remember pendulum So it will oscillate like a pendulum. So even though if if you have a pendulum You know like this if you don't touch it, it will remain stable Nothing will happen. But if you Rotate it a little bit like this and leave it It will just start oscillating left and right Same thing here This will also start oscillating. Okay, so if If moment of inertia of dipole moment see dipole moment could be because of the bar magnet So bar magnet will have a moment of inertia. Let us say, okay So it's like a rod rods moment of inertia is ml square by 12. Okay, so if moment of inertia For dipole moment is given if for dipole moment moment of inertia is I Okay, I want you to find out how much will be the time period of oscillation What is the time period of oscillation for this? Anyone got it? Just message. Yes or no if you have got it or not Okay, for this you need to have a basic knowledge of simple harmonic motion Which we have done long back And I hope you still remember those basics So what we do is whenever you have a situation for SHM you slightly Displace it from its equilibrium position right from from this equilibrium position Which is along the magnetic field what I do is that I rotate it slightly by a small angle theta If I rotate by small angle theta torque will be there how much m cross b Which is nothing, but if I take the magnitude m b sin theta Okay, so this is torque and torque about center of mass is m b sin theta This should be equal to i alpha torque is equal to i alpha right now You'll see that the angular acceleration is in this direction, but theta is increasing in that direction So theta and angular acceleration. They are in opposite directions Okay, so if theta is very small, I can write i alpha as mb into theta and in a vector form there will be minus sign coming over there Got time here So alpha will come out to be equal to minus of mb by i Into theta Okay, now you can compare this thing with equation of SHM, which is alpha Should be equal to minus of omega square theta Okay, so when you compare these two this and this you will get omega as Under root of mb by i okay, so time period will come out to be 2 pi under root i by mb Okay, so remember this time period of oscillation for a dipole Inside a magnetic field Okay Right, so suppose you do not know the magnetic field if you are unaware of What is magnetic field then you can conduct an experiment suppose you already know the time period You can you have the instrument to measure the time period of a dipole moment m Then you can use this equation to find the value of magnetic field B Okay, so don't think that this equation will be always useful to find the time period You can use this equation to find out any unknown of this equation given all other unknowns Okay, so this is with respect to the Torque experience by the magnetic dipole moment now. Let us talk about the potential energy potential energy of A dipole inside the external magnetic field Okay, it is very similar to the electrostatic potential energy which was equal to minus of P dot e Right, so magnetic potential energy is written as minus of M dot B. So the formulas are analogous Okay, the way you derive Magnetic potential energy is like this torque is what? mb sin theta, so this is a torque and and potential energy is the work done The change in potential energy is the work done by the field Right, so this will be equal to integral of mb sin theta d theta Okay, so you can integrate this and you will get here as minus of mb cos theta Okay, so if we do it in a you know Proper manner as in if you really want to get into the derivation of it change in potential energy Which is negative of the work done by the field is equal to this now here when field does the work Theta will reduce so d theta will be a negative quantity So when you put it with magnitude negative negative becomes positive Okay, and the limits let us say go from theta 1 to theta 2 Okay, so you will get here as mb minus of sin theta Theta 1 to theta 2 Okay, so this is change in potential energy. Let us say this is potential g at point 2 minus potential g at point 1 so u 2 minus u 1 will be equal to minus of Sorry, this is cos theta integral of sin theta d theta So mb cos theta 2 Okay, minus of minus mb cos theta 1 Okay, now whenever we define potential energy Whenever we define potential g we assume something as zero potential energy Okay, so let us see what is the appropriate position where we can say that potential is zero So if I put theta 2 as let us say zero sorry 90 degree Let us say theta 2 is 90 degree. I'll get potential g at 90 degree minus u 1 is equal to now theta 2 is zero Sorry 90 degree. So mb cos theta 2 becomes zero. So this is zero plus mb cos theta 1 Now if I assume u 90 also to be zero So that is an assumption potential energy when the 90 degree is inclination is assumed to be zero Then potential g at any angle theta will be equal to minus of mb cos of theta Which is nothing but negative of m dot product with b vector Okay, so like this you can find the potential energy and whenever you write the equation How you got the plus? So Rick is asking how we have plus over here see What happens work done by the field is mb sin theta d theta now torque is acting in this way okay, and the theta is Itself a vector. So if you take d theta theta is decreasing Isn't it when torque is been applied theta is decreasing. So it is coming down. Okay, so d theta is a negative quantity Okay, it's like this the force is opposite of displacement. So that is why that negative comes in but here We are dealing with angles. So we check the You know change in theta is negative or positive. So that is why so potential energy is minus m dot b Whenever you use potential energy in any form of problem solving you will always Be dealing with difference in potential energy. You'll never deal with absolute potential energies so that is why you can randomly assume anything as zero potential energy because Difference will be same no matter what you assume to be zero Okay, it will always be like you two minus you one or you one minus you two whenever you solve question Okay, so let us take few numericals and see How we can apply whatever we have learned till now Write down. There is a magnetic needle there is a magnetic needle which has magnetic moment equals to 6.7 Into 10 ratio power minus 2 ampere meter square Okay, this is magnetic moment moment of inertia is 7.5 Into 10 ratio power minus 6 kg meter square Okay, it performs 10 oscillations 10 oscillations in 6.7 seconds Okay, you need to find what is the magnitude of magnetic field B? How much it is B? Kojal, are you using calculator any chance? Please do not use any calculators. It will really You know Effect you when you will write actual exam. Do not Okay, good others also Remember this approximation that pi square is Roughly equal to 10 you can use it. So we know that time period is 2 pi under root i by mb Okay, so under root i By mb is equal to t divided by 2 pi So i divided by mb will be equal to t square divided by 4 pi square So B will be equal to 4 pi square i by t square m Anyone of you got the answer other than kojal? Okay Now what is time period of this oscillation? 10 oscillation are made in 6.7 seconds So time period for one oscillation will be 6.7 divided by 10 So it will be point six seven so point 67 second is the time period of oscillation Okay So when you substitute all the values you will get roughly 0.01 tesla, okay All of you understood some of you are still doing it