 Let us begin this Herculean task of condensing 40 lectures in 4 lectures. So, naturally when I try to do such a thing, one has to sort of be very selective on what you want to do. What I have decided to select is partly based on what we found that the students find it very difficult to. So, what my plan is the following that in the next 4 lectures that I am giving you, we will be doing the following. One of the things which I think you should make the students very clear, what is actually meant by a field? We keep on talking about electric field, magnetic field, gravitational field. So, what is a field? What we have found is this that unless you make this clear, the students just think that field is just another vector. So, I would spend a bit of a time in pointing out what is a field? In particular because we will be talking about electric and the magnetic field, I will be talking about vector field and its mathematical representation. Then I will try to do even a much bigger task. See the electricity magnetism course which we give or you will you are giving that requires a good understanding of vector calculus. Now what you will find is that you expect the mathematics teachers to do this job for you and most of the time you will find that they are more interested in if limit epsilon goes to 0, what happens to delta or things like that. And physics people they know could not care less about what happens to epsilon and delta. They assume that if something is to be differentiated, it will be differentiated or it is differentiable. We do not really prove any existence and things like that. So the point is this that vector calculus I have written, I will give you a crash course on vector calculus in half an hour. As you can see that normally a vector calculus course itself is a one semester course. Having done that, I will be in this lecture, I will be talking about electric field and potential, electric flux and Gauss's law, we will be spending some time in discussing properties of conductors and dielectrics. The you must have realized while teaching this course if you have taught it that most of the electricity magnetism course essentially 50 percent of the time goes in teaching electrostatics. And the reason is that the magnetic phenomena, the magnetostatics problem there is no greatly great deal of new mathematics that comes in. Some things does come for example, the concept of a vector potential which is never clear to the students. So, I keep on telling you not what is not clear to you, but I am going to talk about what is not clear to the students. So even if I am repeating something that you know, I am simply giving you my experience of what students find difficult to assimilate. So vector potential is one such concept and that I will be spending a bit of time on. The after that after the electrostatics and the magnetostatics, I will be discussing the time dependent phenomena and then depending upon how much of time I am left with in these four lectures we will be trying to talk about electromagnetic waves. So let us let us proceed with that. Well this is roughly what I told you that my second part will have magnetic field. Charged particle in a magnetic field has fantastic or tremendous applications for engineers in fact you must have all learned because you are all physics people that we talk about LHC, the particle accelerators and things like that. And so the knowing the principle of accelerators how they are applied it requires the behavior of charged particle in a magnetic field. Then I will be talking about force on current current conductors, potential energy of magnetic dipoles, amperes and bio-savartes law and etcetera, etcetera, it is nothing so special about it. So please we as a professor Fatak pointed out we are all teachers so I am not here to teach you but I am essentially giving you a review of electrodynamics and if you have a question pertinent to whatever I am talking about simply raise your hand and then I will take the question and when you ask a question it will be nice if you also identify yourself so that your colleagues can know what it is ok. So let us come to this question of what is the field. I am assuming we all know what are vectors and scalars as you know that vectors are essentially quantities which have the as we learnt in our school which have both magnitude and direction. Now field could be a scalar or a vector and so the difference between a field and a let us say scalar field and a scalar quantity is that the field is defined at every point in space in a in a particular region of space for instance. If I take this room as a region of space of my interest then at every point in this room I define several fields for instance I can define a temperature field. I can define a gravitational field now it is a different matter that the gravitational acceleration due to gravity G it does not vary very much from one point to another but if you recall the definition of the acceleration due to gravity then you will realize there is no reason why the acceleration the gravity is the same here or it is there. Temperature is a scalar quantity there is no particular reason why the temperature should be this have one particular value at this point and another you know the same value at this point which it does not have. So for example now the thing becomes apparent if you are looking at for example your kitchen or to your home now you can find the temperature gradient if you are close to your cooking stove then the temperature there is more you go towards the window the temperature decreases. In other words the room for instance in this example kitchen at every point there there is a scalar quantity defined which I decided to call it as temperature field. Likewise the quantity that I am discussing could be a vector. So for example the gravitational field which in principle has different values at different points though since the differences are so small I might decide that they are roughly the same and that is why you are to you tell your students that the acceleration due to gravity is 9.8 meter per second square. It is not that it has exactly the same value of energy. So in talking about a vector field we associate in addition to a magnitude a direction at every point in space. So gravitational field, electric field, magnetic field or even if you are looking at the motion of liquid in a place then even your flow field of a liquid or a fluid it is an example of a vector field. So what I am going to do is this. So there is something written here on red it is not very important if they cannot switch off the lights so because the numbers are not particularly important. So I am trying to tell you how to sketch a field and this is something which you can ask your students to do. What I have done is I have just said that consider a mathematical field vector field in two dimension which is given by yi minus 2xj. It is a good exercise you ask your students that take a computer and plot it it is a two dimensional plot. So you look at what I have done here which is not visible properly is I have given you that they are good at least something is visible now. You can see at this point which says at the point 0 1 this is if this is 0 and y is 1 the vector is i. Similarly, here minus 2 0 so if x is minus 2 that becomes minus minus plus 4 4j and minus 2i so this is what is written here. So one can take a graph paper and what you do is you plot the vectors by taking some sort of units that look x axis one unit of length will be this much y axis one unit of length will be this much and based on that we will plot this function. So let us do that. So look at this this is a very rough plot. So what I have done here is I have taken a say for example if it is just i so I have taken one i this would be 2i this will be whatever and so you plot an ordinary vector and you start getting the way a vector field looks like. But your students are likely to be more tech savvy than you and me. So what they you would like to do is to tell them that look you do not have to go by a pencil and a paper, but you for example this I have drawn in Mathematica software and you have a vector field plotting facilities in that and maybe you can ask somebody students they do either Mathematica or Maple or whatever they use and see the idea is that they can get a visual impression of what a vector field looks like actually ok. And this is of great importance when you try to teach them for example girl of a vector field the gradient of a vector field and things like that. So what I am going to do now is this I told you that I am going to give you a quick course, crash course on vector calculus there are a few things I need and that is few things that you will be needing if you are teaching electromagnetics. We need definitions of line integral, surface integral, volume integrals are nothing interesting. We also need the concept of gradient divergence and curl and couple of theorems namely the divergence theorem and the Stokes theorem. I think if you spend enough time in teaching your students this much of mathematics so far as your electrodynamics course is concerned I think you will find that they should not have because this is something which we repeat even though they have done a calculus. So we said that a vector field is defined at every point in space go back to your idea of how did you define differentiation and integration. Remember that when we define integration of a single variable or differentiation of a single variable for instance if you are talking about differentiation we told our students that look or we were told by our teachers that take the value of the function at point x subtract from it the value of the function at the point x minus h divided by h let h become very small and the quantity that you get is called the differential of the function F. So this is the way we define it. Now the difference is and how did you define integration? We said that all right split the region into a fine mesh and find out what is the value of that function at every point in that mesh imagine that the mesh is small enough so that the value of the function may be treated as constant in that region sum it up right integration was defined as a limit of sum and based on that you explained to the your students what is actually integration now I do exactly the same. So what we do is we said that all right the difference is my vector field not only has a magnitude but also has a direction. So when I choose my section when I have a line which I choose its I split it into various small sections I must make it so small that over every such infinitesimal section not only the magnitude of the function but the direction of the function also remains the same. And so therefore if you have an arbitrary curve the direction of the field there is simply along the tangent to that curve and what I do is I take the direction of the force field dot it with the tangent to the curve at that point and say that sum it up. Now this is then my definition of the line integral of a vector field. So this is this is what I was trying to tell you that take an arbitrary curve I am I am plotting over this and at every point I make it so small that supposing the field direction is this and the direction of the dL is along the tangent to the curve and this is just a normal field plotted and this is what you do and you define the line integral that way. The second thing that I need very badly is what is known as a surface integral. The picture that I have given you is that of a fluid flow v stands for the velocity direction there. You see the point is this that if you have a fluid which is flowing along that direction and if in the fluid I put a an area which is this. Now let us suppose the normal to that area is along the direction of the fluid flow. Now that would be the situation where the maximum amount of fluid would pass through this surface. On the other hand if you were to incline it if you are to incline it then the amount of fluid that will pass in is related to the projection of this surface which is normal is along this along the velocity direction. One of the things that you have to convince your students that if I consider a surface it is not a vector. A surface is not a vector but suppose I take make the surface very small make it a surface element. Now if I make it a surface element it can be considered as flat enough. Now if I have an actual optically flat surface like reasonably shown for example this blackboard. Then I can look at surface as a vector. Now how is surface as a vector? Go back to definition of what is a vector. School definition of a vector. School definition of a vector is it is a quantity which has a magnitude and a direction. The only thing that I change now is to say that it is a quantity which has a magnitude and a unique direction can be associated. Now so if I have a surface let us take the well this is not this is a surface. So I define or associate a direction with this surface which is perpendicular to it. Now notice that is the reason I wanted it to be as small as possible because if it is a surface like this then the direction of the perpendicular at this point is not the same as the direction of the perpendicular at that point. But on the other hand if I have an optically flat surface the and it is very small the directions are the same. Now so therefore if I have a surface element a unique direction can be associated with it and since my traditional definition of a vector is a quantity which has a magnitude in this case some millimeter square and a unique direction then I also call that surface element as a vector. So with this the I define the flux which is the same as the surface integral of this vector field over the surface. So the entire surface I split into small surfaces and ds is the direction of each surface element. So surface element can be regarded as a vector and this defines the concept of a surface integral. Now notice one thing here there is a picture there of a some sort of a let us take something like a hemispherical curve that is not badly drawn but that is not important. Now you notice one thing that I have a surface here but this is an open system so that the surface has a boundary which is a curve. Now on this surface the direction of the normal at every point is the outward normal that is take a small area and go ahead. There are some points that you should make it clear to the students not every surface has a surface integral. This surface the picture that I am showing you ok this is a Fisherman's net. Now it has a surface area you can see it it has a well defined inside surface and a well defined outside surface. So you take a piece of paper it has a one surface which is above and surface which is below. Now suppose you want to go from this surf of a piece of paper from above the surface to below it for instance take this net supposing in this net I want to move from the top of the net to under the rim. There is no way I can do that without crossing a boundary the same with your piece of paper you are writing something supposing now you want to go to the underside of that paper then you have to cross the edge of that paper. This is the property of a true surface that is in order to move from one surface to another to the other side you need to move through. But there are surfaces which have only one side I could have actually if someone gives me a piece of paper I can even show you yes he is giving it yes clear it off. See you can actually so this is two surface two sided surface this side that side if I want to go from here to there I have to cross the edge. But let us look at slightly different situations you can actually make it if you just had this have a cellotype with you. So, I take this now what I do is I fold it and instead of joining like this I join it like that put a put a cellotype there this is what I have drawn here you see that was a ribbon and what I have done is that while putting a cellotype there instead of doing it like this I have done it like that. Now you see the point is this as this is a surface but I can go from any point to any point on this right supposing I want I am on the top side I want to come to the bottom side I am not crossing any edge at all I can continuously come. These are called one sided surfaces a typical example is what is given here it is called a Mobius strip. The theorems that we talk about they are not applicable for one sided surface they are only applicable for proper true surfaces. So, that was about summation I am already as you can see running short of time because I said half an hour course but that is always difficult. Let us come back to scalar field let us come back to differentiation. So, summation or integration of scalar you know integration of vector field I told you let us look at differentiation. Now in order to do that recall what is meant by an ordinary derivative I just now told you that supposing I have a function f x what is df by dx? We know that df by dx essentially gives you the slope of the curve. So, if you take x going to x plus delta x then f of x plus delta x is given by this quantity but this is one dimension much easier and that is the reason why we teach them in school. The question is that when you are in more than one dimension how do you define a derivative. So, in more than one dimension your concept of derivative is also dependent on in which direction you are changing for example, I said let x go to x plus delta x. Now supposing I want to go from let us say this point. So, this is a this is a picture in this blackboard is two dimension. Now suppose I want to find out what is the value of the derivative of some function at this point. Now the question is that had it been in one dimension I would have said all right this is x go to x plus delta x. But now I can go from that point to a nearby point in any direction around 360 degrees and this concept is what is known as a directional derivative. In at every point there is a derivative, but you can define it in any particular direction that you want. And so let us suppose I am talking about derivative here, but going in this way. So, I go to a specific direction delta s from the point to x plus delta x y plus delta y z plus delta z. So, the directional derivative of the function phi along this direction delta s is given by the standard formula that is d phi by d x this is something which your students would have done that this is partial f with respect to x d x by d s and since phi depends upon x y z you take all the three. Just to give you a small exercise you say find the directional derivative of this point at the point 1, 1, 2. I have given the directional derivative of let us say i plus 2 j that is the field that I plotted in the beginning. Do not worry about that I will come back to that point. So, what do I do? I say all right this is a two dimensional picture. So, I want d f by d s. So, write down your function was given find out what is the partial of the function with respect to x which is 2 x and so this is your d f by d x definition. So, along i plus 2 j notice y is 2 x and d y by d x is 2 and this you can very trivia the calculate. So, this is the concept of a directional derivative. So, let me define the gradient the mathematical definition you all know, but what I am going to be talking about is this that what is the relationship of the gradient why is gradient so called. You remember that when I talked about directional derivative I had a partial derivative dotted with d f by d x. So, in other words if you define u unit vector u along the x y and z direction then your directional derivative in that direction is given by gradient of phi. Gradient of phi as you know is defined as i d phi by d x plus j d phi by d y plus k d phi by d z and that is given by the magnitude of the gradient times the angle between the gradient and the unit vector along that direction which is grad phi constant. So, if you look at that you find that the magnitude of the gradient is the maximum magnitude of the directional derivative. So, the maximum magnitude of the directional derivative happens when cos theta is equal to 1. And the direction of the gradient is along the direction in which the directional derivative is maximum. So, just to explain to you what it actually means let me show you this picture. See supposing I am at this point this is a hill I mean you will find lots of hills around i d. Now, I am at hill supposing you want to come from this point to that point. Now, there are many ways of coming from this point to that point. One of the possibilities is you just decide go like this or come like that, but there is a direction in which the descent is steepest. So, it is that direction which is the direction of the gradient. So, gradient direction as I told you is the direction in which the directional derivative has maximum value. So, in this picture your this is the direction of the gradient. Then what is the direction of the gradient? You all know what is meant by a level curve. The level curve if you have forgotten is the curve along which a function has a constant value. And since we are talking about the direction in which its change is maximum. So, the magnitude of the gradient is directed along the normal to this level curve. Now, why am I talking about that? We are going to be talking about electrostatic field. In electrostatic field you find its convenient to discuss things in terms of potential. Now, the point is this that take for example, a curve like function of x y which is x square plus y square. You know that level surfaces are circles because x square plus y square equal to constant is equation to a circle. Now, look at what is its gradient calculate the gradient which is partial x square by x which is 2 x i plus 2 y j which is nothing but vector 2 r. This is in the radial direction because it is along vector r and is normal to the level curve. As you already know that if I have an equi-potential surface what is an equi-potential? The surface or the curve along which the potential is constant. So, equi-potential surfaces are level curves of mathematics and the gradient is the direction of the electric field. Why? Because you want to go from one level curve to another the steepest would be if you go perpendicular to that. So, this is the idea that you find out where the potential remains constant and you know the gradient of the potential is the direction of the electric field. Having done that, let me go to the next one. There is another property of vector field namely the divergence of a vector field. Now, divergence is formally defined by this relation. I will show you a picture of the vector field. See as the name suggests the divergence tells you how much does a vector field spread out. I told you how to draw the pictures. I will come back to the pictures. So, it is possible that a vector field supposing I am looking at this point and I am looking at another point it spreads out like this. On the other hand it might remain much closed in which case the divergence is smaller and that is the most of the names are names with commonsense. Now, so therefore, we define the divergence of a vector field. I take a small volume element and you know that associated with every volume is a small surface. I calculate how much is the surface integral of that divided by the volume and it will take the volume go to 0. This is the formal definition of a divergence. Now, look at what is happening. See here I have given supposing this is my whole system the big one this plus this. So, there is a surface here the bottom surface there is a surface at the top. Now, suppose this volume I am to split. So, look at I have done the same thing here that I have now split this and that. So, you notice now that at the place where I sliced the remember the direction of a surface is always along the outward that is convention. So, what I say is this that where you split there was no surface. Now, a pair of surface was created one with its outward normal like this and another surface with its outward normal like that and since the magnitude of both these surfaces which you cut must be the same. It tells you that when you add them up I still am left with only the outside ones. Look at what is this divergence like. So, the question as I told you is how much does the vector field spread out? Look at this picture this is a picture of a vector field plotted in Mathematica. Now, look at this that now how do you find out you take certain region remember the best picture of divergence is to think of a liquid flow. Now supposing I consider a some area here you notice that if I take an area here the vectors which are going out seem to be much bigger than the vectors which are coming in I take a small circle here which means more liquid is going out than coming in as to whether that is a permitted in liquid theory or not that is unimportant ok. But this tells you my divergence is positive my divergence is positive ok. On the other hand here you look at it here what happens is if you take again a circle you see much bigger vectors are coming in and smaller things are going out. There is an example of a negative velocity negative divergence situation. On the other hand I hope I have that picture. So, I do not have it does not matter you could have situations where the amount of liquid that is going in is equal to the amount of liquid that is coming going coming out the vectors have the same magnitudes coming in going out there the usual 0 divergence. So, look at this the same picture now. So, what we do is this that I take a volume and I say that look how much of fluid is flowing through this space I am going to accelerate it a little bit because I have already taken half of my time and then I say that all right supposing rho is the density and let us let us call this this is this is the way we have done it x axis y axis z axis and you can sort of see that this area is along the x z direction. So, therefore, the mass of the fluid flowing and this phase you notice this that is this is going into the thing and direction is minus y this is the positive y this is the both of them are positive y here the fluid is going in here it is coming out. So, therefore, the net increase in the volume mass of the fluid is given by this. So, now this is something which is really not directly connected with us immediately and I can add up from six phases and I find out from six phases my total increase in mass is given by this which if you recall is the definition of del dot that is divergence of v times this ok. Now, since the volume is fixed rate of increase of mass can only happen if there is a change in the density. So, therefore, I get by equating this to d rho by dt times dx dy dz what is known as continuity equation which is true in fluid flow and later on when we talk about correct. So, this is a picture which I was talking to you about you can see it here that this has been drawn on Mathematica the this is a field which is x square y i plus x y square j divergence you can immediately calculate is 4 x y. If it is 4 x y if x and y are positive that is in even in this quadrant then you can see my divergence is positive. Now, how do I know divergence is positive you can see much bigger arrows are going out. On the other hand if I go to a fourth quadrant or a second quadrant where x and y have opposite sign I will have the divergence to be negative which is again shown by bigger arrows going in smaller arrows coming out. And as I was looking here this is the field which is x i minus y j divergence of x is equal to 0 this is the field. You might like it to ask your students particularly because if you do that they have a much better appreciation of what divergence well my next picture will be on curl. Now remember I talked about the divergence definition was this. So, therefore, if you write down what is f dot n d s you can simply write it as divergence of f d v. Now, if you take the limit and do the sum what you find is what is known as divergence theorem. It is a very important theorem I mean which simply says if you take divergence of a vector and take a volume integral there of what you get is this surface integral of that vector itself. Now you know sometimes the students sort of say sir are we supposed to memorize it you know he might suddenly say f dot n d tau equal to d v d s this routine. But then you could sort of tell them that look at there is a simple way of knowing whether d s should come here or there and the reason is this that when you take a divergence you are essentially doing a differentiation right i d y d x etcetera right order d y d x of that. Now when you do a differentiation your length magnitude or length scale reduces by 1 d f by d x since you are dividing it by a distance. So, therefore, divergence of f ok multiplied with d tau here one length scale was reduced because you have taken divergence and you have got a tau which is the volume. So, therefore, the the this dimension of this side is dimension of a surface. So, this is an important theorem and likewise I now go to talk about what is known as a curl. Now the definition of the curl the word came from circulation of a vector is very similar to what I gave as the definition of a divergence. Remember in the divergence I said integral f dot d s divided by volume. Now I say similarly define a line integral divided by a surface and this quantity ok. The see in the other case I had a volume integral with a dot. So, it is and and now with the curl I associate a directional. Now you can see what is happening here the same type of thing that if you have a surface divide it into many the curls are given like this. So, if you look at an area in between you find one of the circulations is going in the clockwise direction the other one is going in the anticlockwise direction. So, therefore, there is again no contribution from the inside the only thing that I am left with will be the outside fix. So, the line integral of the vector is the surface integral of the curl of the vector again the dimensional question comes out. Curl is a differentiation. So, therefore, the dimension reduces by one length. So, therefore, if I am multiplying with a area. So, the net I have is f times a length area. So, this is all the basic mathematics that I wanted to do this is nothing you are all familiar with the expression for the curl. So, let me give you a physical reason for what is a curl. See the lame curl came from again fluid dynamics. See the entire vector calculus the people had worked out when they used to do fluid dynamics. Now, notice what we are trying to say is this that consider a pedal wheel inside a liquid. Now, the point is that supposing you have a pedal wheel like this under what condition will this wheel rotate? So, here I have given a profile of the liquid velocities. You can see that this is the x direction. So, in this my dVy by dx is greater than 0. On the other hand this is the y direction. So, Vy is increasing with x. On the other hand here my velocities are in the x direction and my Vx magnitude is decreasing with y because it is in the negative direction. Now, only you can now see by common sense only if such a thing takes place that there is proper sense then only this wheel will rotate. Now, this is the picturization of a vector field which has a constant curl in the z direction. So, if you ask your students that try to work these out using Mathematica, Maple or whatever they use you will find them their experience of dealing with vector fields become much better ok couple of languages. So, you express a vector field remember curl is a vector. So, divergence of a vector field is a scalar field curl of a vector field is a vector field. Since curl itself is a vector supposing I have a field f which I use it to express it as a curl of another vector field. So, if f is equal to del cross a then you can immediately show that the divergence of f is equal to 0 because divergence of a curl is always equal to 0. This is trivial the arithmetic you can work it out. Such a field is known as a solenoidal field a divergence less field. So, you can either call it divergence less field or say that field is expressible as a curl of a. In electromagnetic theory which is the field which is a solenoidal field anybody. Sorry. The magnetic field. The magnetic field. Why is magnetic field solenoidal? That is simply restating my question, but why did it does it happen? Because. No the answers are coming in. So, the if you have seen the lines of forces due to an electric charge. For example, it is a positive charge you see the lines of forces are all diverging from there negative charge they are coming in. You do not have a corresponding situation in magnetism. You take a magnet which has a north-south pole cut it you again get north south pole. In other words it has not been possible. Incidentally I must also tell you there is no theory which says magnetic monopoles do not exist. In practice nobody has been able to isolate a magnetic monopole. Experimentally magnetic monopoles have never been found. Theoretically there is no reason why they should not exist, but since we deal with practice magnetic monopoles do not exist. And this comes out as that the magnetic field is a solenoidal field. It is divergence is 0. Divergence of B is equal to 0 is one of the Maxwell's equations that we have. Yeah, sit down and shout. Sir you said theoretically it is still not there that magnetic monopole cannot exist. Yes. But what about Maxwell's second equation it says del dot B is equal to 0 that means there is no source for. No, no, no. So the point is this that if you recall I will be coming to that equation his he is asking maybe you should always say who you are. Sir I am Dr. Samrat. They are from Assam. Ok good. So he is asking that if magnetic monopoles there is no theoretical reason why they do not exist how do you explain divergence of B equal to 0. The point I am trying to make is if magnetic monopoles were to be discovered tomorrow then your divergence B equation will change not only that something else will also change which other equation will change. See remember we are calling these laws right. What are laws? Laws are things which on which your entire edifice of physics is based. You do not try to explain law as to why it happens. The matter of fact is magnetic monopoles do not exist. As a result we find that the divergence of B is equal to 0. Where did it come from? Experimentally it was found that the Coulomb's law is the basic for the electrostatics. What is the corresponding law for the magnetostatics? The Biot-Savart's law. But so the fact that divergence of B is equal to 0 is contained in the Biot-Savart's law. That is a law. If tomorrow the magnetic poles existed I will have an equation which is very similar to del dot e equal to rho by epsilon 0. It will not be of course that because of magnitude problem, dimension problems. But then Biot-Savart's law has to be changed. Remember nobody derived Biot-Savart's law for you. It was given to you just as Newton's laws were given to you. Am I clear? Okay. So now if the field is expressible as a gradient of a scalar field. We have just now said its curl is 0. What is such a field is called irrotational field? Which field that you know which is an irrotational field? Yes somebody? Partially correct? He says electric field. Louder electrostatic field is the correct situation. Okay? Gravitational field is of course correct. There is a fundamental theorem of vector calculus which says if you have any vector field you can express it as a sum of two components. One component which will have divergence equal to 0, another part whose currently good. So this is true of any vector field. So we were just now talking about Coulomb's law. This is a law. What is found is this? That Coulomb's law basically stated what is the force between two charges. I will not go through it because we have done this several times. But and the supplemented with our favorite Newton's third law whatever force 1 exerts on 2, 2 exerts the same force on 1. So you know that the force on particle charge number 1 due to 2 is there is a constant there which is 1 over 4 pi epsilon 0. Some books you will find that 1 by 4 pi epsilon 0 is missing because this is all related to in which units you want to do your method. This is standard in all engineering institutions to do what is known as SI units. Which units physicists use? You know we are very very confused people. We will teach our electricity magnetism in SI units. Okay? Maybe MKSA unit. But when you go to quantum mechanics you do not like it. Have you seen a quantum mechanics book in SI units? No. In fact the even the electricity magnetism books about 10-15 years back used to be in Gaussian units, MKSA units and things like that. But today I think all of us have learned that engineers prefer this unit. So therefore let us keep it SI units. Even most of the advanced books also like even Jackson's book which used to be till recently in MKSA unit has changed over and now it is available in SI units. So therefore don't worry about this factor. This is 1 over 4 pi epsilon 0. It is permittivity of free space and things like that. Alright? So let us summarize what we know about the electrostatic forces. So firstly this is very interesting. It is an inverse square law. Inverse square law is a very interesting law. Gravitational field also is inverse square. In fact the you know very interesting thing. I do not know how many of you have realized it. See planets are bound to the earth by gravitational force. The electrons are bound to the nucleus by electrostatic force which are both inverse square law. The inverse square law forces are central forces. What is the central force? Sorry? That is one part of its definition. Is that one part of the definition? Something else is very important. No, no, no. Let us not worry about centripetal, centrifugal. Both of them are central force. See that talks about direction. That still talks about direction. There is something else there. See you are all talking about radial. That is correct. It is a central force. But that simply talks about the direction either outward or inward. But what else is important there? No, it is not inversely proportional. Anybody who knows? No, that is the inverse square law force. Sir, centripetal acceleration will be there. No, no. We are talking about force. I am sure all of you know it but you are not you know like happens to all our students when a teacher asks a question you sort of get confused. I am not trying to confuse you. You see central force has two characters. First part is what he pointed out and all of you agreed that from the source of the force it is either radially outward or radially inward. That is one part. But the magnitude of the force not inverse square necessarily depends only on the distance between the two. That is the it would be a central force if the force depends the magnitude of the force is only dependent on distance. One over r square is a special case. In fact, it is known. There are just two types of central forces which can give you a bound state. The first one obviously is the inverse square law which we all know because after all we would not survive if the inverse square law did not bind. The planets are bound to the sun. What is the other force? Anybody remember? Just two forces. No other force will bind it. Closed orbits are possible only under two forces. No, no, no. You see my question is different. Electrostatic or gravitational they have the same feature. Both of them are central forces, both of them are inverse square law. And we know that both of them bind because atoms are bound, the planets are bound. Okay. I am saying there is another force, again central force which also binds. Anybody have an idea? Where is the magnetic force? What binding force? Okay. So, let me, you know, I will be unnecessarily wasting time if I keep on asking. The only other force, central force, which is known to give closed orbits is Hooke's law force. You know what is the Hooke's law force? Okay. Three-dimensional Hooke's law force. Spring force. Okay. That does not go as one over r square. It is proportional to r. Right? That force also binds it. Supposing you took a r cube or one over r cube, it will not bind. Inverse square law is important, Hooke's law is important. You know that is the reason why these two have been investigated by us like nothing else. So we said it's a central force. Actually speaking when I asked you the question, the answer was already here. The magnitude depends only on the distance and the direction is along the line joining chances. Second thing is it's a long range force. A long range force is one which really never becomes zero no matter how far the distance is. One over r square force doesn't become zero. Which force becomes zero? Any force that you know in physics which becomes zero? Now listen to me. Just because I am in an electricity magnetism course always you don't have to think electricity magnetism. Yes, some people are talking about it here. Nuclear force. The short range nuclear force, the strong force. That is very strong inside the nucleus and soon becomes zero. That's a short range force. So Coulomb force is a long range force. These are things which you of course know. Okay? Look at it. The other force that we know, there are four types of forces in nature. Just four forces which can explain everything. The weakest is the gravitational force. Typical strength if you take strong nuclear force to be 1 is 10 to the power minus 38 or minus 39. It has an infinite range because gravitational force doesn't become zero. The next in line weak, it's actually called weak nuclear force. This is a force which has a range of about 10 to the power minus 18 meters. Its strength is 10 to the power minus 6 compared to the strong force and that is responsible for things like beta decay. A force with which we are very much interested is the electromagnetic force which typical strength is given here. Do you know what is this number called? Yeah? That number has a name, very well known number in physics. Fine structure constant. It's called the fine structure constant, right? Okay? And of course strong nuclear force whose range is typically the size of the nucleus which is about a Fermi. Now you know, let me go to slightly not part of our course. You know, the modern theory of forces is this. But how do particles interact? You know, there's a problem there. We don't believe in action at a distance. What is meant by action at a distance? There are two particles here. They attract. How does this particle know that there is a particle there? No, but how does it know? How does the information come? Now the question is this. That look, I know you are here because I am seeing you through a visible light. But what happens? That how does this mass suddenly know that there is a mass there? So the question is this. There has to be a mediator. In the example that I gave you, the mediator was light. Okay? So basically we were exchanging photons. Now so what happens is this that in the modern theory we say that particles interact with each other by exchanging things. Exchanging things I have said. It turns out these things which they exchange are bosons. Okay? As you know that all particles can be divided into two categories. All particles of the world, the fermions and the bosons. Okay? So the particles exchange bosons. And it is because they keep on exchanging, they are in some sense boundary. It is imagined that what we are doing is they, you know, we are continuously I am throwing you a ball, you are returning it back, I am throwing a ball, you are returning it back. Now it turns out the carriers of long range forces are massless. The electromagnetic force carrier is what is called photon. And photon doesn't carry any charge. So in other words, the two charges are interacting because they are continuously exchanging photons. It is just, you know, occasionally when you teach, you sort of stop for a while and teach something which is not necessarily related to the, you know, the coursework or a problem that they will have. Another important point that you need to impress your students with is the superposition principle. Superposition principle is very interesting. It says, let us come back to the Coulomb field. We say suppose that the electric field at a point P is E. What does it mean? It means that if there is a charge Q, then it will experience a force QE. So if I have a charge Q1, this picture is not visible. If I have charge Q1 at R1, the field at R is given by this. This is Coulomb's law. Now mind you, superposition principle is not a trivial principle because if superposition principle did not work, whole of electrodynamics has to change. Again, this is the way things have been found to be true. That is, if I have two sources of the electric field and I have a calculating the effect or the force experienced by a charge particle at the third point, the superposition principle tells me that you have to find out the force due to one as if the other one did not exist. Then consider the force due to the second one, imagining the first one did not exist and do a vector addition of that. Now mind you, things do not always add up. Effects do not necessarily have to add up. So the fact that they are adding up is helping us in simplifying problem. Addition of the field, so this is the field at P due to charges Q i located at R i is given by 1 over 4 pi epsilon 0 sum over i Q i R minus R i by this Q. The after you have gone through this, probably the next thing that you teach the students is the Gauss's law. One thing I want to tell you, Gauss's law and Coulomb's law are equivalent. There is no new physics in Gauss's law, absolutely no new physics in Gauss's law. Then why do we use Gauss's law? Why not Coulomb's law? Any idea? If Gauss's law is absolutely equivalent to Coulomb's law, why have a separate law? Why give a name to Gauss? Any idea? Why do you teach Gauss's law? Absolutely. The point is that in situations where the problem shows a large degree of symmetry, you see Coulomb's law is very difficult to work with. Superposition principle, you are, you have one or two is alright, have 10 charges you will have problem. Even good computers may not be able to help you. But if your problem has sufficient symmetry, then you find Gauss's law provides a simplification. So, remember again I talked about how surface integrals are calculated. Again, so the pictures is not, so there is a, so what we are trying to say is that remember I define surface integral of a vector field. Electric field is a special vector field, nothing shows. So, therefore I defined f dot ds, this is defined as e dot ds. So, flux has the dimension of electric field times an area, that is the dimension of flux. Look at why Gauss's law works. Remember what was Gauss's law? It says that the flux out of a surface gives me the charge enclosed divided by some constant like permittivity of these things. So, what happens is this, that if you look at, I have sort of tried to tell you that there is a relationship with a solid angle. Remember how solid angle is defined. Recall also how an ordinary angle is defined. See the way you define an ordinary angle is, I do not have a picture here, they have disabled that other one, that ok forget about that this is solid supposing I have one line and you have another line here. Then this angle that I have is basically that this length, length of the arc divided by this length radius right r theta that is the length of the arc. Also understand that an ordinary angle is dimensionless because you are dividing length of an arc by a distance, both of them are in meter. So, therefore it is dimensionless but it is traditional to measure it in degrees or radians or things like that. So, degree and radian are dimensionless quantity. Now suppose instead I have a area like this and I am looking at what is the angle that this makes at a point B. So, what I do is this from all parts of this area I draw tangents to that point. So, basically I will have a cone like thing generated there. Now it is this cone I define a solid angle as the perpendicular projection of this area divided by the square of this. Remember again I had earlier arc length by distance now I have a surface by distance square once again the solid angle does not have a dimension. So, this is take the projection of this in the perpendicular direction so that you have a right cone and you get this as the solid angle. Why did I bring in solid angle? See look at what is happening here. Supposing I have a volume here and let us suppose there is a charge cube there. Now I am looking at the flux of the electric field through this surface. Now if the charge cube is inside you notice that it subtends certain area there and I can go on adding this out and at every time my direction of the normal to this surface is outward. So, whether the surface is here or there the line if I join here then the surface will be this. If I join there the outward normal always it is outward normal. But supposing the charge is supposing the charge is outside the volume then you notice that if you draw these lines then it cuts the solid at two places here and there. The solid angle subtended by both are the same though the magnitude of the surface areas will be different. But more important than that the direction of the normal on this sector is outward here and this is there. So, therefore, what happens is if I am looking at how much is the flux? The flux can be written as that it is q d omega by 4 pi epsilon there. So, because of this reason that if you add up these angles then in one case the when the charge was outside the things cancel out because each point subtends two area on the surface and they will cancel out. So, therefore, we say the flux is q n close divided by epsilon 0 whatever charge is inside the outside one does not matter. So, I have got this the flux definition is e dot d s that is equal to according to divergence theorem it is the same as divergence of e d v which is equal to charge enclosed charge enclosed is density multiplied by d v by 1 over epsilon 0. Now, you compare these two expressions both of them are volumetric ok and valid for arbitrary volumes. So, therefore, I get del dot of e equal to rho by epsilon there that is the first Maxwell's equation that we have derived in differential form. So, you notice now let us come back to what happened to divergence of b because I could not close I could not get rid of a north pole if I have a south pole I could not get rid of a south pole if I have a north pole. So, if here I had to worry about a positive charge the moment I have a negative charge then also I will always get this to be equal to 0 because q n close will always be 0 in that place. But fortunately we have charges there ok. So, therefore, del dot of e equal to rho by epsilon 0 is our first Maxwell's equations for electrostatics ok alright. Now, you will find sometimes I will write down big expressions, but do not worry about it I am not going to be deriving it here it might be found for your in the Moodle in case you are interested. But for example, I made a statement that Gauss's law can be derived from Coulomb's law. Remember Gauss's law is del dot of e ok. So, del dot of e equal to rho by epsilon 0. Now, electric field you write down in the form of Coulomb's law. Now, if you do that then I can do a little bit of an algebra which I am not going to do ok, but I will put it in my Moodles and there is a pi missing there which is because of some reason I do not know why that square there is a pi ok. And you can show that from Coulomb's law I can get the differential form of Gauss's law they are absolutely coherent ok. Let us look at a very standard problem. The standard problem is this supposing there is a charge which is the at the centre which is at the centre of a cube of dimension a by a by a. How much is the flux through each surface 1 6 absolutely correct because they must be the same. The problem is not that the problem is put that charge at one of the corners the symmetry is not there it is in one corner a q by 24 is written here, but you have to explain why he is saying 1 by 8. So, basically the concept of symmetry it is not necessary concept of symmetry does not have to be real it could be an imagined symmetry also these are called Gaussian surfaces. So, what you do is that this is what you have here. Now, I know how to solve the problem of if the charge was at the centre you all immediately said 1 6. Can I put that charge at the centre? I can do that if I stack it up with more cubes supposing I stacked it up with more cubes. So, that it becomes like this and my charge becomes at the centre. Now, what has happened is that if I had a bigger cube side 2 a by 2 a and this point was at the centre you would agree that is 1 6 right q by epsilon 0 q by 6 epsilon 0, but then this side is one fourth of the side of this this this size is one fourth of this size. So, from q by 6 I get another one fourth. So, what is important is you also realize how the symmetry can be used for your benefit ok. I will begin my next session with an example from here standard problems which you people do as I have announced earlier when you go back because here I have very little time to do problems. Now, suppose you feel that some problems which you would like the students to understand better and you would like us to put it at that time you can send it send it to us by email the emails will be provided. Point out that look it will be nice if this thing is done and we will try to collect all the such requests and try to see what is the best we can do any question any doubts yeah. Sir, when we teach the students they just pass their senior secondary examinations right and whenever we ask them about flux of a vector field they just answer that these are the lines of force that pass through a surface normally. Absolutely. So, the point is you are absolutely right that is what they do because that is the way they have been taught. Yeah, but my question is that without doing the mathematics how can we make them the concept of flux clear. So, you could do that that is the reason I brought in the question of a liquid flow ok. So, the thing is it is very easy for you to talk to them supposing I have a pipe which is going through if they have seen it right. Now, you put a along the line you put a surface how much you know I mean let us suppose that surface is like a sieve so that the water can flow through. Now, you ask them that look you see that if I keep it perpendicular the amount of water that flows in and then you ask them supposing you tilt it what happens is this correct. Now, this is an experiment which the students would be able to think up in mentally also alright. So, see the lines of forces came in because of that that is because no there is nothing like you know what are these lines of forces and all that nothing wrong with it their representations. But when your students are going to first year after having done the school you have to also do a little bit of unlearning for them right. Now, this is the problem all of us have. Now, maybe we get much better students here, but nevertheless they have also been spoiled by the coaching classes. They tell them look don't bother about understanding how fast you can crack the examination is the most important. So, don't think that what we have maybe they are on an average better student, but in terms of their training they are exactly the same. What they will do is they will not talk in terms of lines of forces, but they will say well memorize it is integral so making student unlearn something is one of the professional hazards all of us have. Try to see wherever possible whether you can ask them that can you think of such a thing. You see flux is very easily explained with water jets just get a pipe in the class if you like I mean I know that it is not possible, but if they go to a lab you can do that right. Thank you. Okay, thank you very much. I will see you again in the second half of the today.