 Today, we are going to discuss what happens if we keep a magnet in a magnetic field. What sort of motion it experiences? To do that we need to know a little bit of an vector algebra. So, let us recapitulate our understanding of vectors. All the physical quantities that we come across are of two types some are scalars some are vectors. Scalars are those which has only magnitude and no direction. For example, temperature, mass, density on the other hand vector quantities are those which are magnitude as well as direction. For example, velocity, displacement, momentum. So, to denote a vector quantity we need to describe its direction and how do we describe its direction? We need to have a definite coordinate system. So, the most popular coordinate system is of course, the Cartesian coordinate system and here it is. This is x, y and z. Notice the relative positions of these three coordinates x here and then you go anticlockwise direction to y and then this is the vertical direction. So, this direction is anticlockwise. So, we call this coordinate system by no means this is the only coordinate system possible there are many others, but this is very common. What do you call a right hand coordinate system? If you hold your hand this way and bend this four fingers. So, that it goes from x to y then the thumb points to the z direction. That is why it is called right hand coordinate system. So, here we define a unit vector in this three directions i, j and k. Magnitude is one and directions are along x, y and z direction. So, in terms of this I can write any vector let us say vector a written in this fashion can be written as where a x a y a z are the three components. So, I could have another vector b for example. So, which could be similar written as i b x j b y a z. So, addition of vector gives another vector which is defined to be vector where the components are added. So, this sum of the two vectors will be a sum of these two this is vector addition like multiplication vector. Multiplication is defined two ways one is called scalar multiplication where the resultant is a scalar quantity. We write in this fashion a dot b is defined to be a x b x plus a y b y a z b z. Here we call it also a dot product because we use a dot symbol to write this. So, we can call it scalar product or a dot product. Here this is also given by this vector let us say a and vector b. If this angle is theta then this is given as magnitude of a magnitude of b times cosine theta. The magnitude of a vector is of course a measure of the length of this. This is the magnitude of this and this is similarly the magnitude of b and magnitude is defined this defined by a x square a y square a z square. That is the magnitude similarly we can get the magnitude of b. So, here the scalar product has this magnitude. Another product is called a vector product where the resultant is a vector quantity that is written as a cross b written as i let me check this one. So, this is the vector product it is also called a cross product because we use a symbol cross product to denote this one. Now, it is somewhat difficult to remember that you see I also cannot remember. So, I had to use this note, but this is this can also be written as determinant in this fashion. So, i j k a x a y a z b x a y a z. If you expand the determinant it will give the same expression as this one. Now, here in analogous manner the way you have written this one this can also be written as magnitude of a magnitude of b times sine theta and let us say unit vector k x a y a z b x a y a z. In a direction this direction is defined to be the direction which is perpendicular to both vector a and vector b and follows the right hand rule. So, this is a this is b and this is theta. So, if I use my right hand this way. So, this this is the unit vector pointing this direction. So, when you go from a to b in the right hand fashion then the direction which is perpendicular to both a and b is the direction given by this. Naturally if I change the direction of the multiplication make it b cross a then I should go from b to a. So, it comes in this fashion. So, this will therefore, only equal to minus of. So, you see in a vector product the order is important with this background. Now, let us try to see what happens if we place a magnet in a magnetic field. So, here this panel a shows a magnet. It is a magnetic moment shown by the red line and with the letter m. This is the south pole and the north pole. This is let us say it is held at this pivot here. Now, if I place this magnet in a magnetic field given by the black lines and the letter b and it is tilted at an angle then what happens? This will try to orient itself along the direction of the magnetic field. So, that the north pole points towards this direction south pole points towards this direction. That will be the minimum energy. To do that it has to experience a torque. So, that this can rotate this fashion. So, a magnet of this kind a bar magnet and placed a magnetic field will experience a torque. This is the magnetic field direction and this is the magnet and this is the magnetic moment. So, the torque will therefore, obviously depend on the strength of this magnetic field and the strength of the magnetic moment and which way it is going to rotate is given by a vector product. So, I write torque vector quantity is written as m cross b. So, this shows that the torque depends on the magnitude of this as well as magnitude of this one and also the how the angle is. So, which direction the energy of this magnet becomes minimum when this points towards this. So, that this magnetic moment points in the same direction as this magnetic field direction. And the maximum energy will be when in the opposite direction. This is the b magnetic field direction and the magnet points in the opposite direction. So, the energy in an intermediate condition like this where let us say this angle is theta. This energy theta will be given as b dot n. Why is this so is easy to understand that this is nothing but magnet of b magnet of b times cosine theta. So, you see when theta is 0 which is this and here theta is 180 degree. So, cosine theta gives you the minus 1. So, this energy will be maximum and this goes minimum which is understandable why it should be so. So, importantly that if I start with an arrangement of this kind then let it go then it will try to come to minimum energy configuration which will be trying to align in this fashion. But then because it comes here it also acquires certain kinetic energy. So, it will not come to a rest it will go beyond this. Then when it comes away from the equilibrium position the torque again will act in the direction such that it brings it back in the sense it will go back and forth. So, this is shown in this panel c here. So, this bar magnet is going to oscillate back and forth if it is starting from an arrangement of this kind given in b. So, I have an animation here to show that. So, this magnet is kept in a magnetic field b and pointing at an angle to start with. Now, let us see how it looks. So, we see that this undergoes this sort of movement. So, in this is simple motion we call it oscillation. So, a bar magnet kept in a magnetic field will undergo to and from motion which is oscillation of this kind. Now, what happens if this magnet is not really a bar magnet that we have, but the magnet of elementary particles like electrons or protons. The magnetic movement comes from its motion of orbital in nature or spin angular movement. So, we should see what sort of motion experiences if this magnet has an angular movement on to start with. So, to do that let us first find out how the magnetic movement arises from the orbital motion of some charged particle. Let us say take a circular loop and certain charge q is moving in this direction at a velocity v then you know that movement of charge produces magnetic field. So, it is a circle. So, this will produce a magnetic field this is the magnetic movement of that. So, let us say this radius is and this movement of charge at a given velocity can be represented also by a current I. So, here the magnetic movement that this will produce is given as a current times the area times the direction of the magnetic movement is given by this. This is the direction of this. So, it is again right hand rule that if it is moving in this way that magnetic movement will point in this direction. So, a is the area of the ok. So, this is the magnetic movement that is going to be produced when the current I moves in a circle of radius r. So, area a is equal to of course the r square. Now, this charge is moving at a velocity v. So, what is current I equal to how much? Let us say this charge takes time t to complete this circle. So, if the velocity is v then t times v is equal to 2 pi r the current I is produced by the movement of the magnetic charge q. So, this is q by t. So, t is the time that this charge moves here in to makes one complete circle. So, amount of charge that is flown through this will be I times t is equal to equal to charge because this is a current in nothing but charge following passing at unit time. So, this times this gives that this gives t is equal to 2 pi r y v. So, this gives q v by 2 pi r. Then m becomes magnetic movement becomes I is equal to q v by 2 pi r times a is pi r square times the direction of the unit vector here. So, this is equal to q v r by 2 times a. So, this is the expression which gives the magnetic movement in terms of the charge velocity and the radius here and gives the direction. Now, if the charge happens to be the charge of an electron that is electron is moving in this fashion then the q becomes negative charge of electron. So, magnetic movement m is given by minus e v r by 2 times this is the magnetic movement. Now, let us try to relate this magnetic movement to the angular momentum of this charge. So, how do I do that same way. So, this is moving at a velocity v is the direction. So, angular momentum is l vector is equal to r cross p and p is nothing but mass into the velocity. So, this is actually equal to mass of electron r v into the direction of this one. So, r cross p is actually the same vector product this gives the direction in the same direction as this m. So, I need some more space from this we find that m is equal to minus e by. So, this now connects the angular momentum of electron to the magnetic movement it produces. This is the relationship we are trying to arrive at. Now, in quantum mechanics the angular momentum l is actually measured in units of h by h cross which is actually equal to h cross is defined by h by 2 pi. So, this is the unit of angular momentum. So, this l I can write it as some dimensionless operator l times the unit of angular momentum is this. So, in that unit now this m can be written as e h cross by 2 m by l. So, here these are all fundamental constants. So, this constant is called the Bohr magneton which we have used earlier Bohr. So, this is usually written as mu of v symbolically or beta of e. So, this will be written as let us say beta e l minus of that one. What is the value of this Bohr magneton? Since these are coming from the fundamental constants we can easily calculate them let us do it. Electric charge e is 1.60218 into into per minus 19 coulomb plus 1.60218 into 1.60218 constant h cross by 2 1.05457 into 10 to the power 34 joule second and mass of electron 10 to the power minus 33 kg. If you put all these things here then Bohr magneton mu v appears will come out to be 9.2 second by joule coulomb second per kg. This is all very well, but we will let us try to change this unit to a little more should I say. So, quote unquote usable unit which is will be proving very convenient for us for that we do some dimensional analysis. Dimension of charge how do I write ampere in terms of some other quantities for that if electric charge moves in a magnetic field that electric charge experiences certain force that force is called Lorentz force. Force experienced by in a magnetic field is the Lorentz force now expression is this force experienced by the moving charge is given as the charge moving q velocity and the magnetic field cross product of this one. So, here the dimension of force is Newton which is kg meter and second inverse square charge q is coulomb which is ampere second because current is nothing but charge passing per second. So, charge becomes ampere into second velocity v is meter per second. So, what we have here the force dimension is equal to dimension of this, this and this magnetic field dimension we keep it as it is. So, this gives me the kg meter and second square this unit which is the force is equal to charge which is ampere second. In the velocity meter per second and this dimension of magnetic field. So, this gives me the dimension of the magnetic field in terms of more familiar units kg second square ampere inverse. So, this is the dimension of the magnetic field. Let us call this tesla and write it as t. So, this is simply a short hand notation of this one tesla is this using tesla now dimension of ampere becomes kg second square and tesla minus 1. Skipping in mind that coulomb charge is equal to ampere and second we can now get the dimension of mu b here that mu b. We saw the dimension of Bohr magneton mu b is given as joule coulomb second by kg. Now, if we replace the dimension of charge we can get by ampere second this gives joule ampere second square by kg. Now, we replace the dimension of ampere that is the dimension of by this joule kg second square by kg. This is the dimension of ampere that we have seen. So, this gives this dimension of Bohr magneton to be joule per tesla. So, as I was saying this is sort of easier unit to use or more convenient unit to use because if I know the magnetic moment of a particle in units of Bohr magneton then I can multiply the magnetic strength to this to find out the energy in joule because joule per tesla is here. So, if I multiply by tesla I get the energy in joule very easily. Now, this tesla is a modern unit that is it is Si unit. The older unit of magnetic field strength was Gauss written as G and the relation between the two is 10 to the power 4 Gauss is 1 tesla. As the tesla is a modern unit it is expected that everybody should use it as much as possible but scientific literature has usage of this as well as this one. So, we will continue to use both of them interchangeably in our lecture. So, this also is equivalent to saying that 10 Gauss is equal to 1 milli tesla. I have been offering a coupling constant for example denoted as either milli tesla or Gauss. So, at this stage let us take a break and we will continue a discussion of the interaction of magnetic moment in a magnetic field in a subsequent lecture.