 So, good morning and welcome to the fifth lecture of this course. Yesterday, after the end of the lecture, you know, some students asked me to doubts. I will try to explain them with some introductory remarks in these first two slides. First of all, let us clearly understand, you know, open surface and closed surface and the unit normal concept. So, when you have open surface, as I mentioned in the previous lecture, you have two possibilities. You can either have unit normal in this direction or unit normal in this direction. So, you have a choice. But if you are actually considering corresponding line integral, for example, you know, you are applying Stokes theorem, integral a dot dl is equal to, you know, surface integral del cos a dot ds, then this counter integral, corresponding counter integral decides the direction of this unit normal for the open surface. So, that is, so again, here you do not really have choice if you are considering the corresponding line integral. So, that is the first point. For a closed surface, it is always the outward normal. So, here there is no issue. There is always all these, what I have shown here, there are three surfaces I have shown and corresponding normal is the outward normal always. For closed surface, there is no issue. Again, just to, you know, sort of further consolidate our learning about divergence and curl, I was mentioning you in the last lecture, divergence is a flow source. So, something, why it is called as flow source? Because at this positive charge, something is coming out and at the negative charge, this E line is terminating and nothing is going out. So, at positive charge, there is a positive flow source and at negative charge, there is a sort of negative flow source. That means, flow stops at that. So, flow terminates. So, again, it is a negative divergence. So, divergence is little bit easy to understand. But curl, for example, here, you know, the well-known example of curl is, you know, current carrying conductor shown by a red dot and then current is coming out. The convention is always dot means current is coming out, cross is, current is going into the paper. So, now the flux is, I have shown there, flux. Now, these are vortex source. Now, what is a vortex? Basically, it is like, you know, in many books, you will, you know, see this example being given. If you drop a stone in water, then the waves will produce and those waves will basically go outward from that point where the stone was dropped and those intensity of waves will go on reducing as you go away from the point where the stone was dropped. So, exactly the same thing is there. As you go away from this current source, the magnitude of field intensity reduces gradually because, you know, later on we will see h is equal to i upon 2 pi r or 2 pi rho in cylindrical system where rho is the distance from the source. So, another point to be noted here is, now again, you know, here del cross h is equal to j we know generally. These are point form of one of the Maxwell's equations. Now, if I ask you a question that there is, see this, you take this arbitrary point here somewhere and what is del cross h is equal to? So, del cross h, the way I have asked question, it has only two possibilities. Either del cross h is equal to j or del cross h is 0. So, we should remember that since it is, since del cross h is equal to j is point form of equation, when we ask a question, what is the del cross h at a point? If the j exists at that point, del cross h is not 0. So, in this diagram, del cross h is not 0 only inside the conductor and del cross h is 0 outside, although you may feel that there is, you know, encircling of these field lines around the current source and why should curl be, you know, 0 at that point. But remember the basic definition that we saw yesterday of curl is, curl exists if there is a curling in the vicinity of that point because again curl is at a point in space. So, we have to ask a question, is there a curling around that point in question? So, any point outside this current source, there is no local curling effect. What we see here, you know, that, you know, these counter encircling the current source, this is like an integral effect. So, in the definition of curl, in the denominator, we have delta S and we said delta S tends to 0. So, if you take a very close, you know, suppose the third line, third circle here, if I take and draw it here, we have to ask a question, is there a curling very around that point in question? There is no, that is why curl is 0, whereas curl will be non-zero inside the conductor because current and j exists. So, this is a point to be noted. Mathematically also, you can easily prove if you actually calculate curl at any outside point, curl will come identically equal to 0. So, that was the second point. Now, the another thing is that you have this, we discussed about 2D approximation. So, now let us see, when we do 2D approximation, generally, first let us, you know, discuss this figure here. So, the 2D approximation when we do, we always take, you know, cross section which is sort of x y plane and perpendicular to current which is in z direction. Most of the, you know, equipment and machine analysis when we do by 2D analysis, we generally would do this way. So, we take always cross section normal to the current direction, is it not? So, now, we need to visualize this correctly that current here is in z direction, I have shown you z, x and y direction is into the paper. As I said, current is in z direction and we have shown there on the paper x z plane. So, y is into the paper. So, basically x is like this, y is into the paper and z points upward. So, y in this case is into the paper, the way I have shown. So, now, current when it is in z direction, actually, you know, the flux, flux is actually crosses this area. Flux always is identified with the corresponding crossing surface. Flux always flows through surface or crosses the surface. Now, in this diagram here, that area is not visible because one dimension of area is into the paper. Is this point clear? Very, very important. So, that is shown here. So, current is in z direction and then this is, this is that one dimension that we see here. The other dimension is into the paper, into the z direction. So, this is that area. So, the flux always will cross this area. Is this clear? That is what is shown by this hashed area. So, now, coming to 2D approximation, suppose, you know, you have a current source in the vicinity of this metallic bar and suppose you are interested to find how much flux is linked to this bar and how much loss is occurring in that metallic plate and in case of time varying fields. So, you know, remember in this initial introductory, you know, concept that I am explaining, I am not strictly follow, although I am explaining here basic concepts, I am using time varying fields and those things also because, you know, I am not going in a textbook kind of thing, only consider statics then, you know, like that. So, I am taking help of some time varying concepts also while explaining this initial concepts. So, here you have this, when you do 2D approximation for this, what essentially you are doing is in the FEM analysis and all that, you are actually, you know, the z dimension is considered as infinite. That means, whatever you see in this xy plane, this is the xy plane, same thing repeats in the z direction up to infinity. That means, effectively, there are no end effects. So, whatever you see in this xy plane field distribution, same thing will continue through and through in or at any z into the paper. So, that is the meaning of 2D approximation. So, when we do that, there are no end effects and field energy losses, inductance and any other performance parameter is calculated per meter depth. So, you are calculating per meter depth, going further into this 2D approximation. Now, this some arbitrary surface I have shown by this S and corresponding contour shown marked by C. So, again just to reemphasize the point, I have just drawn an arbitrary surface and this is 1 meter depth in z direction. So, this is the xy plane and this is z direction. Now, you know, we know the divergence theorem as applied to 3D in general 3D. We have divergence A dV over volume integral is equal to closed surface integral A dot dS, is it not? That is the original divergence theorem. But here when we are doing 2D approximation, we are actually that dV we are writing as dS into 1. So, that is this dS into 1. So, this is the complete volume. Similarly, on the right hand side, it was actually originally it is A dot dS, but that dS is written as dL into 1. So, that is why the same 2D when you have 2D approximation, actually the surface integral reduces to line integral. Note the difference between this divergence theorem applied with 2D approximation versus the Stokes theorem which is, you know, surface integral, open surface integral is converted to closed line integral. So, here also you are going from surface to line by using Stokes theorem. Same thing in some other, you know, context we are doing wherein we are using divergence theorem and when we are doing 2D approximation, you can again go from open surface integral to closed line integral. Is it clear? Now, you know, again yesterday one of the participants asked question about, you know, this conservative and non-conservative fields and, you know, I thought I will just explain little bit more elaborately about these two fields and then I made a comment that conservative fields generally would not do any work. So, now here these are the three cases where, you know, first capacitor is charged by a battery by closing this switch, current flows and capacitor gets charged. When the current was flowing, the source is transferring the energy to this capacitor, the potential energy through the moment of electrons. So, always remember, as I mentioned to you in the last previous class, the energy gets transferred or work is done when actually electrons move and the current flows. So, during the time when the capacitor is charged, the source is transferring the energy to this, you know, electrons that is the kinetic energy and finally kinetic energy is stored as potential energy when the capacitor is fully charged and when the capacitor is fully charged, current becomes 0. Is it not? Current becomes 0 and there is no longer, you know, transfer of energy or there is no longer work done. So, that is why this second figure in the middle is representative of conservative field where it is an electrostatic field and there is only potential energy which is stored in this capacitor and the corresponding, you know, del cross E is equal to 0. So, it is a conservative field. Now, suppose, you know, this charge capacitor, so it and this as I mentioned, this second figure and the corresponding charge capacitor will not be able to do any work, useful work unless it is, you know, made to, it is connected to some circuit and current flows, right. So, now in the third circuit, that is exactly what we are doing. So, this capacitor now is connected across resistor or resistor and now the capacitor discharges. Now again, you know, this potential energy is converted into the kinetic energy of the, you know, electrons. The electrons move, the current flows and then, you know, work is done and work is done in terms of here heat. Remember that work, manifestation of work in practical life are, you know, at least three which are, you know, first is heat, second is motion in, for example, in rotating machines and those examples and third is light, right. So, these are the manifestations of work. Now, here in this case, this is not probably useful work because it is just, you are wasting the energy. But if it is some heating application, it will be a useful energy, right. So, again, when this capacitor is being discharged for that time period, when the current is flowing, you again have del cross E not equal to 0 because some work is being done. It is non-conservative field. Is this clear? And, you know, this is in sync with, you know, Maxwell's equation that whenever current flow is there, in fact, here, this current that is flowing will be function of time. Is it not? Initially, the current will be high and at the end current will become 0. So, i changing with time, so also b will change with time. So, daba b by daba t will not be equal to 0 during that transient period and that confirms that del cross E indeed is not equal to 0. So, I think we have discussed quite in detail about this and I hope this will be useful to you to remember. Now, we will go further. Again, you know, last time we last lecture, we saw one of the current densities was displacement current density. Now, again displacement current density is really applicable for in case of time varying field case because it is daba d by daba t. But we are preempting that discussion because we are in general discussing the current densities whereas this displacement current density is really useful for time varying, in time varying cases. So, now if you actually see, we just made a comment last time that, you know, Maxwell introduced this daba d by daba t term to make the whole system of equation consistent with continuity equation. So, this is just, you know, in just two steps, it is proved that yes, indeed the consistency is obtained starting from divergence j equal to minus daba rho v by daba t which is continuity equation. Then you know, you in place of rho v, you substitute divergence d and then you just interchange the operators del and dt d by dt and why you can do that because they are sort of independent and whether you do one first or the second first, the result is not going to change. That is the reason that you are able to interchange the operators and that we will do probably quite often during this course. And then, you know, after for the rearranging, you get because divergence is equal to this. So, divergence of this whole thing is equal to 0. Now the Maxwell, the original Ampere equation which was del cos h is equal to j. Now, if you add daba d by daba t term there, so then, you know, and you take divergence and we know divergence of curl is always equal to 0. So, that is why this, you know, divergence of this will come equal to 0 which matches with this. So, that basically makes the last equation, Maxwell's equation consistent now with the continuity equation which is law of conservation of charge and it cannot be violated. Now, you know, let us understand what is this law of conservation of charge and, you know, try to understand because we are all familiar with Kirchhoff's current law and all that. Let us understand this whole thing, displacement current density from the point of view of sort of KCL kind of theorem. So, suppose you take a simple circuit which is, you know, source is connected to load and this is some, you know, maybe transmission line. Now at low frequencies, you can just approximate it as simple R and L in series. And then at low frequency, LF stands for low frequency, say at 50 hertz, divergence j will be equal to 0 because whatever current is coming in here will actually is going out, is it not? So, by definition of divergence, divergence at this point of current or current density will be 0 and that actually sort of gets confirmed because divergence j actually is minus daba gov by daba t. So, this has to be 0 at low frequencies because, you know, because of this fact that current is same. So, yes indeed because in time, in frequency domain d by dt j omega. So, this term basically becomes j omega rho v because d by dt j omega and this indeed is equal to 0 is negligible because the frequency omega 2 pi f is very small because see remember rho v volume charge density charges generally in practice, they are in nano picocoulombs, they are, you know, very, very small. You will never find, you know, charges in practice 1 coulomb, pi coulomb, those kind of charges are not generally possible. So, this is very small number rho v and if this is 2 pi into 50, it is going to become a negligible number. So, it justifies that, you know, divergence j is indeed equal to 0 at low frequency. But now, if I, this, you know, frequencies of this source, I make it very high, say 10 raise to 13, 14 hertz like that, then now my j omega v rho v is not going to be a negligible number. Is it not? So, that means, divergence j is not equal to 0 at high frequencies. Now, the question to be asked is where is this, you know, that means, divergence j at this point is not 0. That means, whatever current is coming in is not equal to the current going out of the point P. Always remember divergence is at a point, you have to analyze because it is a point form because it involves partial derivatives. So, that means, the net current at point P has to go somewhere. So, the only place it can go is through this stray capacitance. So, every line, we know that it has stray capacitance. Why stray? Because this line is at some potential. This line is at some lower potential. So, there is a invisible stray capacitance and, you know, the net current will go through that. So, at an instant, if suppose we assume I1 is greater than I2, that means, you know, this point P, I have just read one here, I1 and I2. So, I1 is greater than I2. So, that means, divergence at this point is negative. Is it not? Higher current was there and lower current is coming out. So, divergence has become negative. Something is depleted. Is it not? So, that actually, that means, divergence J is negative. So, divergence J is, if divergence J is negative, here if you put it, divergence J negative, then daba rho V by daba T will become positive. Is it not? So, daba rho V is positive. That means, this, you know, at this point, the charges are flowing into the capacitor and basically, you know, the rho V is increasing with time. Obviously, because, you know, this current, here less current is going. So, the net, the difference of charges have to flow into the capacitor. And that is why it is positive and positive charge builds up across this stray capacitance. But remember, when we are analyzing all this, we are actually analyzing only at a point in space. Effectively, what you can say is this charge is moving out from this point. And then, in terms of circuits, then we can understand that, yes, indeed it is charging this capacitor. And another issue here is, you know, this, at higher frequencies, this model is definitely not good. Is it not? Because there is a concept called as, you know, transit time effect and electrical line length and so many other things. And the concept of lambda. But since this, that will, I would have discussed in detail if we were discussing high-frequency electromagnetics. But since this course is on low-frequency and electrical machines and equipments, I will make some only passing comments that you have to use here distributed line model. This slide here shows distributed model of a transmission line, wherein this R, L and C, they are the parameters of the line per unit length. And now here, this whole transmission line is divided into number of sections. So, one section length has to be appropriately decided. Now, if this is delta L, this delta L should be less than lambda by 20. Now, what is lambda? Lambda is the wavelength is equal to velocity upon frequency, because velocity is f lambda. And since we are talking, this is in free space because energy and power travels in free space here along the line. Later on, when we actually discuss pointing vector at that point, I will explain you how the power actually flows along the transmission line. So, till that time, you know, let us defer that discussion. But lambda is velocity upon frequency. So, since this is free space, here we have 3 into 10 raised to 8 as velocity of light divided by, if it is say 50 hertz, then these actually in kilometers, this of course in meter, it comes in meters. But if we convert this into kilometers, this will come 6000 kilometers. So, what is the meaning of lambda? Then in 6000 kilometers, there is a phase change of 2 pi radians. But you know, when the transmission line is analyzed from the point of view of transients, then we have to take the highest frequency, transient frequency of interest. And then correspondingly, then this wavelength will reduce. And then this delta L for each entity has to be correspondingly less than lambda by 20 and lambda being the wavelength corresponding to the highest frequency of interest in transient conditions. Now, when this transmission line model is used for high frequency communication circuit, there of course, the frequency of operation is high. And then accordingly, the same rule applies that, you know, the frequency of operation may be in mega in gigahertz or whatever, then you know, the corresponding lambda will be much smaller. And then this delta L should be less than lambda by 20 per accurate results. So, that this circuit analysis is very close to actual field distribution and corresponding field analysis. So, this slide shows core of a transformer which is surrounded by a winding. Now, here you can see, you know, various lump circuit component that will eventually act away in the distributed parameter model of the trans of this transformer. So, you have that is why capacitances between winding and ground between winding and winding and between this two tests, right. All those capacitances and this resistance and inductance of the winding will get represented in distributed parameter model as will be shown next. So, with reference to the series and shunt capacitances and the inductances that we saw in that slide for a transformer, this distributed line parameter model which was for transition line has to be modified for transformer in the following way. So, I am showing just one entity. So, again you know this, there will be resistance of winding, then inductance, then there will be series capacitance here, CS and then ground capacitance CG and this will be that one entity now. So, this is CG, ground capacitance CG, this is L, this is R. Another thing you know, there is one more difference here because you know, you have this other suppose second section here and the corresponding inductance series capacitance and ground capacitance CG, right. So, this is the second section of the distributed model. Now apart from this, you know self-inductance, there will be mutual coupling also between these inductances, right. So, there will be mutual coupling because windings are wound, coils are wound very tightly and there will be mutual coupling between turns and between discs of a winding. So, there will be some mutual inductance also between the coils. So, all these parameters, LM and this CS, series capacitance and ground capacitance, they can be calculated by using finite element analysis and this is one of the very popular applications for, you know, doing FE analysis for transformers. So, the last point on displacement current which, you know, is very interesting is like this that if you have, if you take, you know, again a parallel plate capacitor, these are these two plates, these are the two plates, again supplied by a source alternating in this case and you keep a floating metallic plate in between these two capacitor plates. Floating, the metal, there is a metallic plate and you keep it floating, that means electrically floating, that means it is not connected electrically to any, any potential, right. So, and now actually in the positive half cycle, this upper plate will be positively charged, this will be negatively charged. The corresponding since this is positive, what will happen now in this metallic plate, the positive charges will go down, negative charges will get pulled up, is it not, by, by a coulombic force of attraction, right. So, there will be charge separation and what are these charges here, these are free charges in the metallic plate, they are free charges, is it not. So, actually in each half cycle, now in the second half cycle, what is going to happen, this will become reverse, this will reverse and now it will be positive here, negative here. So, now actually you zoom on to that metallic plate and observe only the moment of those free charges, you will find that those free charges are moving and that will constitute what current, which type of current and current density. So, conduction, is it conduction current density, convection current density or displacement current density, yeah because it is free charges, no. So, it is conduction current density, is it not. So, it is the conduction current density, there is a conduction current here, there is a displacement current here, there is a displacement current here, there is a conduction current here, right. So, continuity is there. So, this gives a very clear visualization of, you know, what displacement current density and the corresponding current density. So, next, you know, topic is fields in dialectics and polarization. So, what we will, in this we will do is, we will analyze first the atomic model and you know, there is a central nucleus with positive charge and then there are negative charges and now if you apply E field, again you know the positive charges. Now E field, that means this is, this is positive and this is negative, right. So, the positive charges will get pulled on this side, negative charges will get pulled on this side and this circular, you know, configuration becomes little bit elliptic and you can see negative charges are more concentrated on this side as compared to this side, right. And this is equivalent, you know, a stress spring and this electric field has done some work and stored potential energy in this, what is known as dipole, electric dipole, right. And then now if you actually, you know, if there are millions of such, you know, atoms and the corresponding dipoles will get formed if the electric field is along this direction and this is some insulating material, then, you know, you have all these dipoles getting formed and for, you know, simplified representation, we can take that this plus minus gets cancelled, this plus minus gets cancelled and that effectively, you know, we can show the dielectric as polarized with only the surface polarization charges, right. So, that is why here minus and plus only we are showing it at the surface. But remember the polarization happens throughout the volume of that dielectric, but it is just a simplified representation. Now same dielectric, now we are putting it in in the inside the capacitor plate, right. And now you have, you know, this polarization which is from minus to plus, remember, polarization vector is from minus to plus, right. As shown here, it is from minus to plus, right. And, you know, you have because of this now dielectric being polarized in such a way, you have the corresponding positive and negative induced charges on this top and bottom plate. Is it not? Moment this polarization charge occurs here, this charge is going to replace further some electrons from this plate into the source. Is it not? So, this plate will get further positively charged. That is what is shown by additional this positive charges in circle. And that we are calling as polarization induced free charge, free charge it is. It is on the conductor. So, that is why it is on the conductor because they are free to move. So, this polarization effect which are, these are all bound charges, those are basically inducing free charges, additional free charges on the conducting plate, right. And that is how you get, you know, additional positive and additional negative polarization induced free charges on these plates. And then you have additional these lines of field, right. And that is why then you can write originally when this dielectric was not there, the governing equation was only d equal to epsilon 0 e, is it not? Now, because of this polarization phenomena, you have got additional lines which are represented by this p vector. So, effectively this additionally induced free charges on top and bottom plates due to polarization and the corresponding lines of field are representing this polarization phenomena, right. And then you know polarization is proportional to e and the constant of proportionality is this electric susceptibility chi e it is called as. And then you know we all know 1 plus this chi e is equal to epsilon 0, epsilon r which is the relative permittivity. And then finally, you get d equal to epsilon e, right. Now, another important point to notice starting with this equation, you have rho, rho divergence d is equal to rho v, right. And that is equal to divergence of epsilon 0 e plus p from this equation, right. So, if you now expand this and then divergence p, you have to write it as minus rho p, why minus because it is from minus to plus. That is why it is divergence p is minus rho p and what is rho p, polarization volume charge density. So, then you know you by rearranging then you get divergence e is equal to rho v which is a free volume charge density plus rho p which is the polarization volume charge density divided by epsilon 0. So, that is the difference. Divergence of d is always related to only free volume charge density, but divergence of e is related to free as well as polarization volume charge density if you know both exist, right. So, this you know epsilon r you know these values are of importance for electrical machines and equipment and for this course. So, typical values of dielectric constant are mentioned here. So, you know vacuum is 1, air is very close to 1, oil mineral oil which is used in oil cooled transomers is 2.2, cellulose paper that is solid insulation again in used in transomers and bushings and some other high voltage equipment it is 3.5 to 3.8, cellulose press board thick press board is 4.4 mica is 6 and all that. So, I think we will stop here and continue the rest of the topics in the next lecture. Thank you.