 So, let us return back, we define flux, I am going to now talk about time varying phenomena, yeah. Sir, in the rectostatic case, we use almost two similar type of between, one is B is equal to some constant epsilon naught into E. That is provided, it is a linear material. Yes, yeah. And plus B, and here it is H and some constant 1 upon mu naught, H B minus M. So, this in one case, there is a plus and in another case, there is a minus. Yes. Is it only a matter of convention that two things are different or there is a physics thing going on? Yes. It is a direction of the dipole moment that was defined as some negative charge from positive charge. You will find that, that is the way it will start now. Okay, sir. If someone changes the convention or if the convention. No, if someone changes the convention, lots of things will change. Had the convention be otherwise, then would we have expected similar type of equations in both the cases? Certainly. But there you know, conventions are very much of interest to physics. You know, I mean, you change from a right-handed coordinate system to a left-handed coordinate system, your right hand rule will change. Right? So, let us not worry about that, but this is the way it is. Okay. But the effect is essentially the same. All right. So, let me then go over. Now, we already know what is a flux. I am trying to recollect for you. Remember that in the first lecture, I had a picture of a Fisserman's net. So, we said that look, so the flux was defined as the b dot n ds. So, let me look at and we had sort of seen that del dot of b equal to 0. Now, look at a situation like this. I have a Fisserman's net here and this is the rim and I have a net outside. So, suppose I am to talk about this surface and then outward normal will be perpendicular to this out of the net. The other one if I take this, then the normal direction is outward like this. Then del dot of b will tell you, this is very important point to which I am coming back, that whether you calculate the flux from this surface or from that surface, the flux is identical. It is a consequence of the fact that the del dot of b equal to 0. The as long as the boundary is the same, on a given boundary, you can make many types of surfaces. For example, take the rim of a cup. You have a cup over it, so that is a surface. And if that rim which is circular, you put a disc on it, that becomes another surface. The flux turns out to be the same. Let us look at what is this electromotive force, ok. Remember motive force, it is a motion is connected. It is a most unfortunate nomenclature in physics because electromotive force is not a force. It is in the past when people had not understood what was happening. People had thought that look, there is something which is forcing these charges, if you connect them to a battery to move in a circuit. So, it was thought to be a force and that is how the name came out, ok. So, this electromotive force, what is electromotive force? So, you know that you define it as integral of e dot del. But essence, the circuits that you have are always closed circuit. An electrostatic force is a conservative force. The integral of e dot dl, contour integral of e dot dl over any closed circuit must be 0. That is the definition of conservative force. So, therefore, I cannot understand if I the only forces I had are the electrostatic force. The integral of e dot dl should be equal to 0, in which case it cannot explain the dissipation that takes place via joule heat and things like that. So, there should be no losses there, it is a conservative force. Now, so what is actually happening there is this, that I have a battery. That battery which is providing if you want me to still use the word motive force. Now, what the battery is doing is to convert chemical energy to electrical energy, ok. And what it does is to set up within that region of battery in non-conservative situation. In non-conservative field, I am calling it let us say e prime. So, my electromotive force if you like is given by the integral of e prime dot dl. Now, you would be saying that if it is only within the battery why did you put a contour integral there? But you see it does not matter because the non-conservative force outside the battery is 0. So, its contribution to the contour integral is equal to 0. So, you could write it that integral e prime dot dl within the battery region or you could simply say in the outside field also, it does not matter. That is not the only interesting thing. If you define the total electric field to be a sum of the conservative part and a non-conservative part e and e prime let us call it e total. So, I can define my emf to be integral of the total et dot dl. The reason is that one part is conservative. So, the like contour integral of that is 0. The second part is not 0, but then second part has a part within the battery and a part outside the battery the outside part is 0. So, therefore, integral of total e dot dl is the same as this integral which is my definition of the electromagnetic force ok. So, let us summarize our understanding of the electromagnetic force. We said that the total electric field is sum of two parts. There is a conservative part which is my electrostatic force and there is a non-conservative part where the battery is playing role. The current density is a sum of both parts. The line integral of the conservative part outside the battery is exactly cancelled by the line integral of the conservative part within the battery and that is the only way total integral can be given. The other part is 0 because the non-conservative part outside the battery. So, that is our understanding and that is in fact I am not do not have time to go for it. That is why there is always a talk of an open circuit voltage. That is between the terminals of a battery there exists a voltage. You do not have to connect it draw currents in order to talk about a voltage. You buy a battery from market and you can say its open circuit voltage is 1.5 volt 9 volt or whatever you pay for it ok. This is simply a method of talking about that what is its ability to draw current ok. Let me quickly come to the first of the time dependent phenomena namely Faraday as well. I will accelerate this a little bit because most of these are very well done by you people. So, firstly the statement is very simple. Faraday had said a changing magnetic flux through a circuit gives rise to an error. Now this what makes a magnetic flux change? Remember that flux is b dot ds. So, either b itself could be changing or the area could be changing, both could be changing, the angle between them could be changing all sorts of things could be happening. But Faraday's observation was it is immaterial what changes? As long as there is a change in flux whatever be its cause it will result in an error. Later on many names were given to it. For example, people say that if it was caused because of a relative motion people called it the EMF to be motion of EMF. Again convention if the magnetic field itself is changing it will also give rise to a EMF and there is no way to distinguish between them. So, Faraday's law was always stated like e is proportional to d5 by dt. Later on you will find it is written as e is equal to minus d5 by dt. This minus sign is very tricky don't take it as a algebraic sign. It is not an algebraic sign because minus sign was introduced to tell you about another law which goes with it and that law is known as Lenge's law. Now the Lenge's law tells you that if what is the direction of current not EMF current if I had actual physical circuit connected by conducting material. Now remember Faraday's law is true even if you had a circuit consisting of wood. I can have a magnetic field varying and I can have let us say circle made of wood and I can talk about how much is the EMF generated in that circuit. Because the definition of EMF has nothing to do with whether I have iron there or wood there. But current there is a need because if you want to talk about current that circuit cannot be wooden circuit but it better be a conducting circuit. So, Lenge's law said that what would be the direction of the current if we are to have a circuit where there were conducting element. And it says that that results in an induced current. The direction of that will be such as to oppose the cause which is making this change. It does not like ok. You try to increase the flux the induced current will be generated in such a direction that is the effect of that induced current will be to produce a magnetic field which is in the reverse way. So, that it will try to decrease the flux. This minus sign was introduced by the people to tell you that there is a law called Lenge's law which tells you that the effect is to oppose changes. The nature does not like changes to occur it wants status quo ok. So, please do not take that minus sign as an algebraic quantity. So, let me give you one example. Supposing I have a circuit which is in a magnetic field. Let us say that magnetic field is coming out of the plane of the board. And let us suppose I stretch this a little bit. So, I have a length here dL and I am stretching it with a velocity V. So, that in time dT this length element gets stretched by V dT. Now, then what I do is this that this has resulted in a change in the area which is given by V dT times this dL or cross because we have seen that is a an area can be regarded as a vector. So, let us look at the Lorentz force again Q V cross B. So, my E n f then is by definition E dot dL is force per unit charge which is V cross B dot dL. So, that is equal to this is A cross B dot dL which you can rearrange using A dot B cross C equal to B dot C cross A etcetera etcetera to a form like B dot del cross V. And del cross V right because V is essentially dX by dT. So, I write it as minus V S by dT and you get this is equal to d5 by dT. So, what has happened here is that the we have said that if this is stretching with a velocity V the area is changing with a velocity V then the cause of this force is there is a motion on the other. Its cause is because of the motion. The next problem I will not be doing because I am sure several times you have asked your students to do it. Basically, it is a loop which is moving in one direction and you can sort of see what actually there is something interesting about it. There is a magnetic field coming out. There is a loop which is moving let us say in some direction. Now, you see if I am going to be taking talking about an integral around this loop there are charges here there are charges are moving assume there are positive charges which are moving. We know that of course the charge that moves in negative charges, but now these charges because they are moving and there is a magnetic field there they are subject to Lorentz force V cross B dot dL. Now, the contribution to the integral from this side and that side will be 0 because corresponding to every point here there is a point here where you are traversing the in the opposite direction. This is an inhomogeneous field not a constant field. So, take any point let us say here and the corresponding point there. The strength of the magnetic field is the same, but because your length element is in the opposite direction the contribution to that integral goes away. On the other hand there will be opposite points even on this two, but the strength of the magnetic fields are not the same. As a result there is contribution from this side and that side and which results in this integral not being equal to 0. So, that is the reason for the EMF here ok. Well, this is the same more or less, but since this is giving me the first of the Maxwell's equation there is let us talk about it. What we are saying is this that the two examples that I gave you had emotional EMF because they arose because of the charges in motion. Charges in motion are subjected to Lorentz forces and because of that we calculated different values of EMF. There is no way of saying or distinguishing whether the it is because of the motion or because of changing magnetic field to the flux changes. So, let us look at this. This is my definition of the Faraday's law. Integral E dot dl is minus d phi by dt, but phi is b dot dl. So, that is what we do. By Stokes law E dot dl is del cross E dot dl which is equal to minus d by dt of b dot dl. Bring everything to one side you get del cross E plus db by dt equal to 0. So, del cross E which was equal to 0 because we said electrostatic force is a conservative. Now, we are saying del cross of E is minus db by dt ok, minus db by dt this is what del cross E gives me. So, this is a consequence of the fact that I have a time varying situation. With the magnetic field whatever the reason is varied then my del cross E equation changes. curl of E is no longer 0. So, another way of looking at it is a changing magnetic field. The statement should be changing magnetic flux. But a changing magnetic field also gives rise to an equivalent electric field and that changes the curly equation. So, curl of E is no longer 0. So, let us look at some of the consequence thereof. Supposing I have two loops and there are current through these loops. Now, one of the things you should realize is that if there is a current in this loop it creates a magnetic field around it. This magnetic field is threaded by this loop as a result there is a flux through this loop and if the current in this loop is changing with time then the magnetic field that it generates also changes with time. As a result the flux that is passing through loop number 2 also changes with time. But if the flux is changing with time there is an emf that will be generated in loop number 2. And I can immediately write down by realizing that the field due to this loop is written as B 0 by 4 pi D L 1 cross pi this is just bias of what is all. And the flux then is B 1 dot B S 2. Remember I have put in one to say that the field is generated by loop 1 and the flux is because there is a an area here. So, notice because this expression has B 1 the flux through the second loop is proportional to the current in the first loop. That tells me that I can write the flux through the second loop as proportional to the current in the first loop and the proportionality constant can be calculated by all this integral. So, Faraday's law will then tell me that the emf generated in this loop due to the changing current in that loop is given by minus Lengis law M 2 1 times Di 1 by dt. This constant is called the mutual conductor. It is always a good idea to talk about mutual conductance before you have introduced self conductance. And because it is much easier for you all realize that mutual conductance being lot more difficult to calculate one does not do it in school. But it may be so, but conceptually mutual conductance is much easier to understand than self conductance. Mutual conductance says the interaction between two bodies. I am telling you you are listening it. So, this is a much easier thing to understand. Here is a person lecturing there is a person receiving. When you say I am lecturing and I am receiving there seems to be some confusion. Now, so this is mutual conductance because it is much easier to understand this is a seat of magnetic field this is threading it. So, the flux through is this changing since the current is changing the flux is changing flux is changing there is an emf in this circuit. Turn the table. There is a current in this circuit which is also changing with time. It will allow you for its flux to be intercepted by this ok. And you can recalculate and show that the mutual inductance is a function of geometry how they are placed. So, having said that now I have a loop I have a magnetic field produced. Now this magnetic field is not only threading the loops that there may be nearby it is also threading its own loop. This is the whole idea that it is also because of the fact it is threading its own loop there is an emf generated in the loop itself I am giving a lecture you are listening to me for almost 3 hours you are getting tired that is mutual inductance. I am talking continuously I am getting tired that is self-inductive. Now, so therefore now you will realize try to do this problem in the class from first principle try to calculate the self-inductance of a long straight line you will realize how much a problem is there ok. You will and do it by normal method not by sign half Li square is the energy so therefore I can calculate energy by some other way and equate it do it by first principle it is not easy. The so therefore this emf its job is actually to reduce the current that is flowing in this and hence it is called a back emf ok. The it opposes the always the inductance or the effect of the Faraday's law is to act as something which opposes the change it does not like changing whether in another person or in yourself I told you to do self-inductance calculation of self-inductance of a long straight wire which I will not do by the time you finish it you go to the December you have not succeeded just send me an email I will try to do it but do it for some simple case let us look at the self-inductance of a solenoid tightly wound solenoid we know the magnetic field is mu zero n i with every term then since there is a constant magnetic field inside I link a flux which is equal to the area of each term times the magnetic field which is mu zero n i so if the total length is L then total number of turns that I have is small n into L so total amount of flux linked is this much so L is d5 by dt which is simply given by this that gives you immediately an expression for what the self-inductance is let us do the same thing suppose I have two solenoids which are tightly wound with each other much easier to calculate so you know the field so you have that you are putting in a current I through one of them and through the other one also look at that if you are changing the current I through one of them how much is the flux linked in the other one and you can get an expression for the mutual inductance we already had found out the expression for the self-inductance of a single solenoid you can check that m12 is simply the geometrical mean of the two self-inductance that is provided everything is tightly wound at things like that mutual inductance is a very sensitive function of geometry let us come to the following thing now when I establish a current in the circuit remember what we do we switch on right the process of switching on is a very interesting thing the current was zero before you switch it on the fact that you have switched on it means in a short time the current went from zero to whatever its value would be after the switch was put on so in other words there is a changing current that is a simple thing which you do every day go to a room switch on the process of switching on changes the current from zero to whatever value that happens then the because the current is established in the circuit you must have an induced EMF developing now so when this induced EMF which I know is a back EMF develops work must be done to opposite because otherwise I can never establish a current to my desired value and of course the same thing that if there are more than one conductors then you have to also change the current in the other circuits because of that let us look at energy of a current this way now what is the flux supposing I have got large number of circuits what is the flux through let us say i th circuit so we said i th circuit flux changes because of current in that circuit self inductance is proportional to the current so I write it as Li ii small i means i th circuit it also there is a component change in the current in the other circuits j not equal to i here a self inductance comes here a visual inductance comes so my EMF which is d phi i by dt e i the EMF on the i th circuit that is simply obtained by writing something over d by dt of this Li di i by dt plus Mi j di j by so the rate at which the work is done is simply e i i and if you write that you can write it this is Li multiplying with i di i by dt which is same as di square by dt and likewise in this expression you have multiplied an i so I have written it as di i ij by dt I need a factor of 1 by 2 because general differentiation ok so this work done ok total work done I can express in terms of say this is my definition of the work done electric field e i and this EMF I have seen is this is expression for that so i i was there i i square and things like that and this is simply the expression for i i phi i I am just going through one of them phi is b dot ds so I convert this to by Stokes law to a dot j now let us come to a different situations so we have talked about changing current now suppose in a circuit I have a current but I also have let us say a capacitor but before that let me ask the following question we said a changing magnetic field gives rise to an electric field naturally people have realized by that time there is a lot of similarity between or symmetry between electric field and the magnetic field so the question naturally arises if a time varying magnetic field gives rise to an equivalent electric field because EMF is nothing but integral e dot ds what would happen if I were to change the electric field with time will it give me a magnetic field or not now this actually comes back to the fact that magnetic effects are not more difficult to observe so changing magnetic field gave rise to an electric field which could be easily demonstrated experiment but it was not known for a very long time whether a changing electric field will give rise to a magnetic field just because it is a small effect and in fact that is basically the contribution of Maxwell into the whole game you notice that we had Gauss's law del dot of b equal to 0 del dot of p equal to rho by epsilon 0 we had a Faraday's law del cross of e equal to minus dv by dt we had Ampere's law but ultimately the whole set of equation came to be known as Maxwell's equation so what did Maxwell actually do if all the work were done by other people you know why did Maxwell get the credit Maxwell was able to get this last mile and last mile is very important I mean you may have a fiber active cable running in the nearby but if you do not have a catfied cable which is coming to your home there is a useless population so that is what he did and he answered it in a very simple way let us look at the following situation supposing I have a current flowing in a circuit in which I have a capacitor now what is happening when I switch on the current in the circuit the capacitor is getting charged now so I have a circuit the capacitor is getting charged the question that he wanted to know is this during the process of charging current is not 0 if it is a DC you have a capacitor because the capacitor there is an air gap the current will not be there but we are not talking about that we are talking about a situation which happens just before the charging takes place the process of charging is delivering the charges from the battery onto the plates of the capacitor during that time charges are moving from the battery through the external circuit to the plates of the capacitor so there is a current at that time now this current we will call as a conduction current because that is what exists in the outside circuit now suppose I considered a loop here threading that area then I can use the ampere slot to say integral v dot dl equal to mu 0i but then what was done is the following we have just now said that the flux that is calculated ok so this is what I am doing I have said that this is supposing this is the surface S1 which is just the plane of that loop so you could convert that equation into del cross b equal to mu 0j by simply using your stokes law now suppose instead you decided that I will still consider the same loop but will not take the disc as the surface but we will imagine that with the same loop as the rim there is a pot and part of that pot goes between the capacitor we have said that the flux is independent of which surface you take but what happens now the problem is that there is that part of the pot is not cutting any current because there is no physical conduction current inside a capacitor but the calculation must be the same so there is a conflict so earlier this would seem to suggest del cross of b equal to 0 because there is no current but earlier we had said del cross of b equal to mu 0j now same physical situation 2 different calculations cannot give me the 2 different results and this is what Maxwell did he said his thinking was this some of these thinking have led to wrong names ok his thinking was imagine instead of vacuum that space between the capacitor had some medium, material medium then since you have set up charges in other words that material medium has now put in an electric field this will lead to charge separation within that and we go back now to how much is the flux the flux through s2 then he calculated as we have done it earlier as surface is s2 but it is d dot ds now in that case by d phi e by dt is d by dt of d dot ds and this is the time differentiation there is a space integration you can bring that in which will become d dot ds is del dot d dq bar so you can show that d phi e by dt is equal to dq by dt because rho del dot d is actual charge rho that is the reason I brought in d real charge but rho dq bar is nothing but q so which tells me that the change in the electric flux is the same as dq by dt which is the current in the outside circuit so now we are left with the last bit of our equations that we need to now change del cross h we are del cross e equation so we are del cross b equal to mu 0 j which is same as del cross h equal to mu j so to that we now needed to add a term which is because of the changing well it is d what was called as displacement field so this tells me well you can immediately see that if you do del dot del cross h it will give rise to continuity equation so with this we have now established all the equations which go in the name of Maxwell's equation good to see them in one place always I have del dot of e equal to rho by epsilon 0 repeat again this e is the real electric field because not just the real charge density rho is the actual charge density real or induced put together del dot of b equal to 0 there is nothing that I can do to it this is these two equations remember this equation is the only one which has remain unscatched in our discussion del dot of e has remained rho by epsilon 0 but our idea of what rho is has changed rho is now not only the free charges but also the induced charges so that equation has changed though it does not look like having changed but it has changed second equation does not change because nature has not provided us with any magnetic monoblock the third equation that is there is what is known as Faraday's law the changing magnetic field is equivalent to an electric field induced electric field so del cross of e and minus sign as I told you is the linear law it opposes the next one is changing electric field though it is always written in terms of d d divided so the thing is that d as we know is our mental idea about the effect that would be produced if somehow or other we could switch off the inducing activity it is not something which I can actually calculate accepting in case of linear material where we say d and e are proportional it is unfortunately their names have stuck but today the fashion is to say vector d without a name and a vector h okay only when you consider linear material then h is proportional to b mu times b that is called permeability but most material are not linear I want to know the physical meaning of current as well as displacement current that is what I am trying to tell you the name is only displacement current is nothing to do with displacement the name came because original idea of Maxwell was to explain it in terms of the separation of charges by putting it in the media so forget about the name the question is what is d is what you are asking okay nowadays no physicist uses the name displacement vector displacement current is used see if you have a current passing in the outside circuit continuity demands that inside the capacitor right there has to be a continuous thing because outside it comes out not that quantity whatever it is has to have the dimension of current right so but however in order to understand it what it is you have to give it a different meaning which is what we did we showed that if you realize what is the changing inside the capacitor right its time rate of change is equivalent to a quantity which has the dimension of current okay and this quantity that you calculate is exactly equal to the conduction current in the outside circuit so that is what is known as the displacement current but unlike the physical current displacement current is not something which is flowing okay it is a name given because it has a dimension of current is it clear okay now so therefore my equations are these delta to be equal to rho by epsilon 0 I told you E is the actual field delta to be equal to 0 del cross E faraday's law this is ampere Maxwell's law del cross H should have been equal to J only because del cross B is mu 0 J if H and B are proportional then only a factor of mu should come but then I have a factor which is due to changing electric field d divided these set of 4 equations are collectively called as Maxwell's equation always along with this I need 2 footnotes okay that is the question was being asked what is d d is being defined d is being defined in terms of measurable quantities d is being defined as epsilon 0 E I can measure E plus the polarization factor and likewise H is being measured in terms of B and the magnetism so they are called constitutive relations because they are introduced to auxiliary field namely d and H the fields that we are we always talk about in physics are always E and B okay but if you are doing electrodynamics these will also be there okay suppose I want to now rewrite Maxwell's equation not in terms of E and B but in terms of the corresponding potentials I have a scalar potential and I have a vector potential definition of scalar potential is minus the gradient of phi is the electric field definition of vector potential is the curl of the vector potential is the magnetic field so in those equations that I had del cross of E equal to minus this is faraday del cross E is minus so if I substitute B to be equal to del cross A then you notice that I can write this equation by taking this term to the other side as del cross of E plus dA by dt equal to 0 right now then since this is the curl of a quantity equal to 0 this entire quantity is expressible as gradient of a potential now please understand what is happening because I am we said in the electrostatic field since the electric field was conservative del cross of E was equal to 0 E could be written as minus grad phi now I am saying something else I am saying E is not conservative if you have time difference but though E is not conservative some quantity called E plus dA by dt seems to be conservative what is that quantity I do not know A is not E is not conservative because del cross of E is no longer there but I have shown that if you add to E some quantity like dA by dt that has a curl equal to 0 but any quantity whose curl is equal to 0 you should be able to write it as let us say minus grad phi so this is what we did so we said minus grad phi plus dA by dt it is immaterial we always use minus because our potential definition is minus gradient of phi but any mathematics book they do not care if del cross of a quantity is equal to 0 that quantity is expressible as gradient of a scalar it is minus del V minus dou A by dt or plus dou A by dt this quantity this quantity E plus dA by dt yes if I write it as I think there is a sign problem minus sign that is minus sign that is minus sign you know whatever such things are pointed out I tell my students that these are the jobs of accountants not physicists so do not worry about it there is an error so there is an error has to be corrected now in the vacuum look at this because that is all that I am going to do del dot of E is rho by epsilon 0 so put it back there so what do you get is an equation for del square V now if you get an equation for del square V but remember del square V is written in terms of A I can look at another equation this I have to flash so if you look at the second equation you find do some manipulation do not make the mistake that I did you will get equations of this type yeah I mean I agree that there is a minus sign difference it does not matter now these equations can be actually decoupled this can be decoupled because we have a lot of liberty with respect to choosing a gauge so what one does remember I talked about Coulomb gauge which said del dot of A equal to 0 but now it turns out that if instead of Coulomb gauge you choose another gauge in which del dot of A is not equal to 0 but you take that del dot of A is equal to minus 1 over C square dv by dt if you do that that is called a Lorenz gauge then you find these two equations become decoupled and these equations are nothing but wave equations so with that I think it is good to conclude thank you very much