 So, let me sort of quickly review what we are doing and so primarily I wanted to tell you that electric and magnetic fields we know we understand them, but potentials are less understood. But on the other hand if you think about it that the if you talk about for example level of water, now the water level is not the same as water itself. So, in potential in some sense we are talking about a level, so scalar potential we have all been familiar with and that is why you never ask a question, you always assume that whatever is being said Mgh is the potential energy is perfectly correct. You do not always ask why are you saying it is Mgh, why not Mgh plus 10 Mg times 10 Mg, I could have done that, but that is because of certain conventions that people have worked out with respect to the definition of the zero of the potential energy. If you are doing a problem connected with gravitational field, you would say potential energy of an object when it is at infinite distance is zero. If you are in the same gravitational field, but talking about small distance, what happens to a particle if it is dropped from here, now you do not take the zero of the potential energy at infinity, you will say zero of the potential energy is the floor. And even in the books they will not write it down that is because this is something which has been taught to us understood by us from the time we were children that well last time I talked about the a situation where we talked about a potential corresponding to the linear field 1 over r, so we said that potential has to be longer than that, okay. So what my statement about the scalar potential is applicable to all types of potentials you can think of, the potentials themselves are not important. The potential why we work with potential particularly scalar potential is a very useful thing because it is a scalar and I can add them easily, vector potential is not that greatly useful for the simple reason that it has the same level of difficulty as directly dealing with magnetic field itself. But the point is that derived quantities they are important, this is as you know the financial capital of India, so I am sure many of you know what is a derivative, so this derivative is not differentiation in Bombay, in Bombay language a derivative is a financial instrument where the it gets its value, derives its value from something underlaying, okay. So in some sense the quantities themselves are not important, but take a vector potential, vector potential itself is not particularly important but its derivative its current is the magnetic field which is physically important because it can exert a force on a charged particle which I can actually measure. So this is the issue and what we said is though the vector potential seems like something very esoteric mathematical construct, there are ways in realizing that yes I can sort of give one sort of physical meaning and there were lots of discussion because I guess primarily because this is not found in any electrodynamics book. So the thing that I was trying to tell you is that why interference pattern, I was not planning to do it but because of this question had been raised I will actually prove in my optics course that two perpendicularly polarized lights do not interfere, you could easily do it by simple mathematics. Now this is not really part of the normal Young's double slit experiment but if from the same source you split the two lights and on one part of one of the beam you put a polarizer on path of the other beam you put a cross polarizer the effect will be to totally get rid of the interference pattern and the reason is that polarization is an important thing there and this is of course what we are trying to say is that this experiment proves that what the vector potential does is to change the electron wave function's phase. As you know that if the phase itself is changed it will not change the intensity okay but because we are adding phases in an interference experiment the result will be totally different and I will not go through it again but we have had reasonable amount of discussion of this. There is one particular case we calculated in case of solenoid but another particular case I would like to point out what is the vector potential corresponding to a constant magnetic field. Supposing I take a magnetic field in the z direction. Now I have already told you that the question itself is meaningless what is the vector potential? Because vector potential is not a unique one. So for example in this case is written for example as a B cross r by 2 you can sort of take the del cross of that okay and show that this is actually the magnetic field and if you do a divergence of this quantity it turns out to be 0. Now divergence equal to 0 is a Coulomb gauge here I have given you two equivalent expressions for A both of which have the same curve I have told you two functions can have the same derivative that is not a problem. So this is exactly what has happened here it is a three dimension you differentiate this the x component is minus B y by 2 y component is plus B x by 2 do a trivial calculation you will find that the del cross of A happens to be equal to B itself along z direction and alternatively you take just an x component magnetic field only magnetic vector potential only in the x direction and its magnitude is minus B y. You can calculate the curl again and you will find it happens to be still B and both of them satisfy del dot of A equal to 0. So and this reason is what is called a gauge choice that we have and gauge choice comes because of the fact I am going because of this gentleman who came from somewhere I am going to skip the boundary conditions or magnetism because otherwise I will not be able to do it fine. So that is all about vector potential we will come back. Let me then talk about magnetic material remember in the our discussion of the electrostatics we pointed out that we have conductors about which we have not talked much because that is something which is very familiar to us and we have dialectics. Dialectics is just another name for insulators and what we said is that what distinguishes a dielectric from a conductor is at microscopic level the availability of free moving charges. Conductors we have free charges in insulators or dielectrics the charges are bound. The negative charge centers are bound to the positive charge centers but on the other hand there can be certain amount of separation between these two charge centers so that a small sample can have a dipole moment even though it does not have a charge. We talked at some stage about a quadrupole a multiple expansion it is possible to expand things such that it is a collection of dipole moment quadrupole moment octopole moment etcetera etcetera etcetera but we are not going into that. Now in magnetism exactly the similar situation accepting that we know that it is steady currents which give rise to magnets magnetic phenomena the source of magnetism is steady current. But however you have to realize that even inside molecules and atoms there are internal currents right what is current current is basically a moving charge. So even if you take the Bohr model of an atom the electron is moving around the nucleus and the moving current that gives rise to an internal current or atomic current etcetera. Now the question is this. So this atomic currents if you go to little more detail it can arise because of the orbital motion of the electrons around the or about the spin motion. Now remember our picture of atoms and molecules are very classical here. We are not worried about the fact that some of them may not because I am discussing macroscopic system for which classical mechanics is good. So the spin which is a purely quantum mechanical object spin cannot be explained classically let me be very clear it is a purely quantum mechanical object we still you know many textbooks will point out that look what is a spin you know it is something like earth rotating about itself that is a spinning motion whereas earth rotating around the sun is an orbital. In classical physics this is correct but there is no concept of a spin in classical mechanics. Okay that is let us not go there because that is not part of my job. So in the presence of the magnetic so I may have like I told you that in the absence of an electric field the molecular dipoles are randomly oriented. As a result the net dipole moment works out to 0. I make exactly the same statement for the magnetized specimen. I start with randomly oriented magnetic moments it is the same question which people ask that you pick up a piece of iron you all know iron is a magnetic substance but an ordinary piece of iron is not magnetic right it is net magnetic moment is 0 because what happens in a piece of magnet is there are domains there are small regions in it with the net magnetic moment. But at an arbitrary temperature these magnetic moments are aligned half a thirdly so that the net magnetic moment that you get is equal to 0 such a situation is known as a paramagnet. If you apply a magnetic field this will try to persuade these domains to align giving a net magnetic then of course you are aware of the classification between ferromagnet, paramagnet etc. I am not going into all substances are diamagnetic whether they are paramagnetic ferromagnetic etc. not important all substances are diamagnetic because diamagnetism is a consequence of Faraday's law and it has nothing to do with what material you are talking about. Now when we say some substance is ferromagnetic it simply means that the magnetic effect there is lot stronger than the Faraday's law effect. So what we say is that what is meant by a current? The current is the source of my magnetic field and this current as we have been talking about can be the macroscopic charge transport that is bulk transport of charge what happens when you put in a wire connect the wire to a battery. So that is the macroscopic charge transport so that is one part of the current. A second part of the current is lies within it there are these atomic currents and what I look for is an average atomic currents because there are large number of such currents which have to be added up. So this is my net current density like we define polarization vector as the net dipole moment per unit volume I defined the magnetization vector as net magnetic moment per unit volume of the substance. You notice that everything is parallel ok. So let us then ask what about magnetic material? We have of course now learnt about the magnetizer vector potential and I did not quite work it out supposing you took a current loop I have already said that a current loop is equivalent to a magnetic loop you all know this. Now if you calculated the vector potential corresponding to a current loop you get an expression like this. In fact what you do is del cross A if you take you will find the expression for the magnetic field due to a circular current loop. So this is what we start with if I have an elementary magnetic moment this is what it will be. So if I have a large number of magnetic moments I would say well simply integrate this or sum this same expression and so in this case that the r prime is being integrated which means this is over the substance and r is the point of observation. So this is where the magnetic vector potential will be given by an expression. I am not derived it but let me quickly tell you that it is not a very difficult job what you do is you take a circular current loop which you know is equivalent to a magnetic moment which is equal to the current flowing in times area and direction is perpendicular to the plane. So use that to derive the expression for the vector potential. Now once you have done that that a magnetic moment kept at the origin at the position r gives you this vector potential. If you have a distributed magnetic moment at the position r prime then the vector potential due to this trivially must be given by this expression ok. Now just the way we had seen that the polarization gives rise to a volume charge density and a surface density. So we had said that del dot of p was something and p dot n was something here it turns out an identical expression Now this is the del cross b which was equal to mu 0 j. We have said j now has two parts a conducting part j c and atomic current density part which is j m. This j m then is given by del cross m r and the surface one is given by n cross m r remember again parallel thing we had minus del dot of p and p dot n ok. Here there is a reason why the signs are different because of the convention that the direction of the dipole moment is from the negative chance to the constant change, but other than that there is nothing else. The derivations are identical and the conclusions are similar. Now with this let us now take stock of where we start. This stage contains essentially everything that we need to know about the max of the group. First b dot d l integration was known to be mu 0 i this is what we have done my ampere soil. So what happens to that? We say look i is current. So one part is conduction current the another part is the current due to atomic current which I call as the magnetization current. So my integral b dot d l is simply modified like this and according to this definition which I had the magnetization current is simply given by del cross m dot d s which is by Stokes law is m dot d l. Now if you bring this to that side put in i m by this way instead of integral b dot d l being equal to mu 0 i c you get some other quantity dot d l equal to mu 0 i c. So you say that look that what the situation is here is that what happens to be deciding things is not the magnetic field b. Remember again we had said in the dielectric situation we needed to change the electric field. We said there was a change because of the dielectric we defined a new vector called d vector. Here what we do is again we define a new vector which is incidentally in many books you will find b is called the magnetic flux density and this divided by mu 0 minus the magnetization vector. You put it in the equation for the h becomes much simpler which is h dot d l equal to i c. You should remember what you did for the electric case. We defined there a quantity known as the d field and we said just as e field is caused by all types of charges real or induced or everything. If I could switch off or separate somehow the real charges from the fictitious charges the induced charges then it is the d field which is determined by the real charges though in principle there is no way of doing such separation. In actual problem there is no way you can separate. Now here we say exactly the same thing. We said that supposing I have a problem with the magnetization and what I do is I say alright some other mentally I can switch off the magnetization. Then the only current that I have is the conduction current the free charges and the field that is determined by the free charges is this new field h. There is a lot of confusion on the terminology b is sometimes called magnetic field of induction h is called magnetic field. In fact the electrical engineers probably prefer to call h as the magnetic field whereas in physics since we always talk about the field b that has become our standard terminology that b field is the magnetic field. Though in the old days you will find if you had a old book of Halle de Riznik yesterday people talked about Halle de Riznik you will find the magnetic field b was measured in Weber per meter square before the unit Tesla came into the picture and the nomenclature is the flux per unit area ok. So the this is not something which I will be doing because professor Suresh will be talking extensively about at least in one lecture about magnetic materials but if you have linear magnetic material where the magnetization is proportional to this h and you have this expression and paramagnets are characterized by susceptibility becoming greater than 0 and diamagnets of course have susceptibility less than 0 etc. Ok this is what I was supposed to do in the morning.