 a very good morning to all of you, I am Professor Dipan Ghosh and along with my colleagues Professor Shiv Prashad and Professor KG Suresh, we will be conducting this workshop for the next 10 days. But we have another couple of colleagues, I would now introduce them because they will be engaging some of the tutorial sessions and will be also helping us to you know formulate quizzes etc. So to my left is Varun Gandhi and he will be taking up some tutorial sessions on various subjects and to next to him is Atul Kedya. So these two friends will be with me and with other colleagues of mine to take us through this course. Let me go to the subject of the, I will be teaching taking a few lectures on electricity magnetism. So let me quickly tell you what am I going to be covering in these next few days on the electricity magnetism. The first thing which most students find it little confusing is what is this field. So I will be talking today about what the concept of the field and you will see that there are fields which have different characteristics, only about two of them we will be interested in talking about the scalar field and the vector field, we will be talking about its mathematical representation and I will attempt to give you within this lecture itself, namely within the next one hour a crash course on vector calculus, I have written half an hour but on the other hand it will probably take the whole lecture. And this is very important for your students that irrespective of the course that they might have done in mathematics. You have an introductory class on the vector calculus itself because having together all the concepts of vector calculus that a physicist needs for this course is an extremely important point. Now once I have done the course on vector calculus, I will be going over and reviewing for you the electric field, the concept of electric field and the potential. And of course I will spend quite a bit of a time on electrostatics, talking about electric flux, Gauss's law, properties of conductors and electrics. Now that would complete our coverage of electrostatics. Having done electrostatics, I go over to the discussion of magnetostatics, the magnetic field as you will see is in its character quite different from the electric field. We will see that while an electric field source has to be a charge, a magnetic field can be produced by a charge only when the charge is in motion. One of the very important applications of the electromagnetic field is the charged particle behavior in electric and the magnetic field which we will be talking about in detail. I will be talking then go over to steady currents and talk about what is the force on a current carrying conductor. The concept of potential energy of a magnetic dipole, now having done this, we talk about go over to a discussion of Faraday's law and electromagnetic induction and briefly talk about displacement current, bring in a concept which students find difficult to understand namely vector potential and finally ending this program with a discussion of electromagnetic waves. A natural extension of this is to take us to applications to optics which I will be covering in another four lectures in the following week. So with that introduction, let me talk about the concept of a field. So my question is what is a field? So let me try to begin a difference between an ordinary scalar or an ordinary vector with the concept of a field. So look at this room, you know that in principle this room for instance at every point in this room I can define a temperature, now of course very loosely speaking we say that the temperature of this room let us say at this moment in Bombay could be something like 31, 32 degrees but when we make that statement that the temperature of the room is let us say 32 degrees, what I mean is that is an average sense, the temperature that we talk about are in an average sense. However, that technically every point in this room has a different temperature and this is very much apparent to you instead of looking at an air conditioned room like this if you are to look at for instance the kitchen in your house. Now in the kitchen when you are closer to the oven or stove you know that the temperature is much more when you go towards the window the temperature decreases. So in other words kitchen is a place where you can see that various parts of the room have different temperatures. Now so the temperature as we know is a scalar, so temperature being a scalar, so this is a scalar and when I am talking about the defining the temperature at every region in a given space as I said this room for instance. So if you define a temperature at the function of let us say the point x, y, z in a certain region of space I can call this though this language is not very much used that my room then is a place in which a temperature field exists. Now this is temperature but on the other hand I could think of other scalar variables which are defined in a certain region of space. So the it is a scalar which has definite value at every point now this is an example of what I can call as a scalar field, so temperature for example is a scalar field. Now let us go to the what we are going to be using more often we would be more interested in talking about a vector field because the electric field or the magnetic field they are all vector fields. So now what is the difference between the two fields first let us try to understand the difference between a scalar and a vector as we know that a vector differs from scalar in the sense that the a vector is characterized by both a magnitude and a direction. Now so therefore when I am talking about a vector field I am still doing the same thing as I did in case of a temperature that at every point in a region of space I have a definite value for a quantity accepting that unlike temperature this quantity has both a magnitude and a direction. So typical examples of course we will be discussing a lot about electric field or a magnetic field but let us say the more familiar one to everybody is the gravitational field. You look at again a certain region of field let us say this room now we say for example gravity which arises due to the gravitational pull of the earth we very frequently make a statement that the acceleration due to gravity let us say in this room is 9.8 meter per second square but that statement is again an average approximate statement. Now technically speaking before or any different points in this room are at different distances from the earth's mass distribution so though it may not be measurable but every point in this room or let us say in your campus would have a different value of the gravity gravitational field due to the earth both in magnitude and in direction but of course because the in the scale of the earth these things are not all that big you will not be in a position to measure the difference but nevertheless the gravity is indeed defined at every point and it has in principle a different value at different point either in magnitude or in direction or both. So gravitational field is an example of a vector field I am of course stating at this moment that the electric field and the magnetic field are also examples of vector fields but of course this would be the content of my course here I will be carrying on with them later. So therefore to summarize I said field is a scalar or a vector quantity whose value is defined at every point in a certain region of space the room for example the example I gave you is a seat of temperature field in which a scalar is defined at every point in the room there is an example of a scalar field. Now so likewise for a vector field we assign to each point a vector quantity which supposing I am looking at in two dimensions it is a function of xy but it is a directed quantity and in three dimension it is a function of xyz but again it is a directed quantity examples as I pointed out to you are the gravitational field and the electric field. So let us look at how does one visualize this I will try to first the screen is of course in front of you but let me illustrate this by an example. Now let us suppose I have a vector field in two dimension I am doing it in two dimension because I am in a position to sketch it so let me take that vector field to be given by so vector f to be given by let us say y times i minus 2x times j so this is a vector field and so let me see how does one visually sketch this. So the first thing to do is this since I am plotting a vector I will require to fix some units so we could decide for example that this would be a nice unit to have for one unit and so let us try to draw a two dimensional graph so this is my x and this is my y. So for instance if I look at the point let us say 0 1 now at the point 0 1 which is of course here the if you plot x is equal to 0 y is equal to 1 you notice that f is simply equal to i so therefore since it is 1 i I plot it with an arrow and I said ok this is the length here similarly supposing you went to 0 2 at the point 0 2 this is 2 i so I just go one step further and draw another line along the x direction which is twice the length and like this we could go likewise if I want to plot it at 0 minus 1 it is equal to minus i so therefore it would be something like this and like this and so on and so on. Now when I come to the point let us say a different points let us say point 1 2 so the point 1 2 is if this is 1 this is 2 so this is the point 1 2 there and at the point 1 2 this is 2 i minus 2 j so look at that so you try to draw a arrow directed arrow of that length in the direction of 2 i minus 2 j. So this screen which you could see I have listed a few the values of the field y i minus 2 x j at certain points I mean these are representative points but of course you could take a graph paper and draw this whole thing out but this is the drawing that I made with my hand taking some representative values that you have taken and this is something which you can ask your students to be able to do. But since today our students are much more savvy at computers you will be able to ask them to take up a mathematical computer like for example a Mathematica or a Maple or things like that and draw sketch. Now this is the mathematical plot of the same vector field that I showed you and so therefore this sort of tells you about this is of course a two dimensional picture you can get such pictures in three dimensions also using a Mathematica they look really beautiful and you can sort of get a feel of what the field is alright. So let me now go a little more to the properties of the now we have said that a vector field is defined at every point in certain region of space and like a scalar function and the only difference is it also has in addition to a magnitude it also has a direction. Now since it is a function which like an ordinary scalar function has different values at different points or basically saying that it is a function which is defined at various points. Now I can treat it like and I can subject it to processes such as integration and differentiations and things like that and in fact what I am going to do now is to do both and firstly because the vector field has both magnitude and direction at every point I while defining an integration or a differentiation remember integration is nothing but a sum. So therefore if I am looking at an integration of vector fields then the result that I get will be essentially a directed quantity unless I happen to do some operation with it to make it a scalar and so therefore it is not just like a scalar field but its differentiation or integration has to take care of due to the fact that these are directed quantities. So let us continue with that so let me first define what is known as the line integral of a vector field. So now what we said is this that here I have a let me do it this way supposing this is the way let us say my vector field is there I will show you a better picture when I go over to the monitor and so this is the direction of the vector field let us suppose this is f. Now I want to integrate it now I want to integrate it along a path supposing this is my path so this is the path along which I want to do the integration. So what I do is this now notice supposing I want to do remember that the vector field is defined at every point so therefore it is defined all along this path. So let me take this point for instance the now at this point my path direction is along the tangent let me call it dL direction and the vector direction is of course as you notice that field direction is this. So there is an angle which f makes with dL which of course I will call it as theta. So the point is this that I define a line integral a line integral along this path by first looking at a small length element dL which is as I said is directed like this and every point along this path the direction of the line element is along the tangent. So therefore at every point I can define a quantity which is dot product of f with this dL and if I take it along this path which is saying that take it along this contour then this is my definition of the line integral of a vector field along the contour right. So this is a better picture of what I tried to do on my paper you can see that green path that you see is the path along which I am trying to integrate and the blue direction are the directions of the field there the direction of dL at a point where I have shown the direction of the field as well as the direction of the length element dL separately they make an angle theta and if you do that at every point in space add it up because integral is nothing but an addition you get the definition of a line integral. Having done the line integral let me now go over to the concept of a surface integral. Now the surface integral but before I go to the concept of surface integral let me sort of recollect for you and which you should always do that when you teach in a class is that an area can be under certain circumstances regarded as a vector. Now the reason why area can be regarded as a vector is the following I will come back to the screen but first let me explain the reason why area can be regarded as a vector is because you remember that the definition of a vector is that it is a quantity which is both magnitude and direction. Now area certainly has a magnitude but the question is does it have a direction that you would normally say that area does not seem to have a direction but this is not true if you look at a small enough area so that it can be regarded as essentially flat. Now if you look at an area element ds it is essentially a flat thing and so if you have a perfectly two dimensional flat quantity you can uniquely associate a direction with it and that direction is the direction which is perpendicular to that surface area. So therefore to this surface element with this surface element I have associated a magnitude which is just nothing but the area in some units like meter square centimeter square or whatever and a direction which is along the direction which is perpendicular to that surface area. Now there is a small problem there the problem is that if you have a flat surface the direction of the normal is not unique because the normal could be coming out of that surface or it could be going into that surface. So therefore what we do is take a make a convention we say that the direction of an area element is given by the outward normal that is the normal which is perpendicular to it of course but is going out of that not into that surface. So this is a convention nothing would go around if you are to take the other convention but the entire physics uses this convention so therefore we also use it. Now let us look at the concept of a surface integral. Now that we have talked about the principle that is an area element or a surface element can be regarded as a vector. Now I can now talk about a surface integral that is the best understood by this picture that I am showing on the screen and this is you imagine that through this surface in the first case I have put that surface perpendicular to the direction of the left picture I put it as perpendicular to the direction of the vector field imagine this vector field to be a the fluid flowing through that surface. So what I have done is I have water or some other fluid flowing and take it as a streamline motion and put a surface area perpendicular to that. So notice that in this case that I have shown that direction of the surface element and the direction of the vector field which is velocity field in this case are parallel to each other. Now suppose I take the second picture the right hand side picture there the surface area is inclined to the direction of the vector and I define the surf flux that is the name that physics people will be using as the dot product of the vector field with the surface element where I am doing the calculation. So this is the flux is f dotted with ds which is the surface element times the field magnitude times the angle cosine of the angle between this. So we have been able to do it because we regarded surface element vector for an arbitrary surface the third picture is showing you the direction of an outward normal which of course for this type of a surface will vary from point to point. Another thing that we need to be very careful about when we talk about surface integral see the there are different types of surfaces. Now this surface I am talking about must be two sided and not one sided. Now you would say what is a one sided surface I will give you an example of a one sided surface for example Mobius strip is a one sided surface on the left figure here I am showing a normal Fisserman's net which is obviously a two sided surface this is because there is an inside and there is an outside and there is a rim which sort of defines the boundary of that surface. Now let us look at another type of surface but realize what is meant by a two sided surface supposing I am on the outside surface and I want to go to the directly opposite point on the inside surface. Now on a two sided surface there is no way of doing that unless I cross the boundary that I have to cross this rim get underneath and then come back to a point below. Now so therefore this is a characteristic of a two sided surface that is that in order to go from the a point on the surface to a point directly below that surface for example if you take a piece of paper then it is from the top of the paper to the bottom of the paper at the same point. Now if you want to do that then you have to cross the edges of the paper in this case the edges of the rim. So this is an example of a two sided surface. Now you can easily construct a one sided surface to be shown to your students and this is what is known as a mobius strip. So in order to construct a mobius strip I will try to tell you how to how to construct a one sided surface. So what you do is you cut out a ribbon first. So I have a paper ribbon now you try to fold it. Now you see if you just fold it like this what you have got is a two sided surface. If I am here I want to come to the point below I will have to cross an edge but supposing I did not do that what I did is I did this and folded it back like that making something like a figure of 8 this you can easily do. Now if I do that and you are on one side of the surface and you want to come to the bottom now you can seamlessly do it without passing through any edge and directly come to the bottom of it. Now so in other words the there are surfaces which you will be able to go from its top to the bottom side without ever crossing an edge. This example for example is a mobius strip which is very trivial to do all that you need is a piece of paper and a cello tape in order to do that. Now the surface integrals of the type I am talking about they are applicable only to two sided surfaces and not to the one sided surface. So the flux which is basically a surface integral of the vector field is defined like that it is only for two sided surfaces. Having done these two different integrations namely the your surface and the this integral. So let me then come over to the concept of differentiation but before I do that let us look at what is the concept of differentiation that you have in a scalar field and we will try to suitably generalize. So if you recall that the way we define the differentiation of a single variable when we are in school is to say that I take the value of the function at some point x plus h subtract from it the value of the function at x divide it by h and then take the limit of h going to 0. This is the class tenth or ninth definition of a differentiation but basically what does it do? What it does is to say that take a point fx x value of f there is x fx and take a close enough point and so value is x plus h there. So if I do that then this tells me that if I want to write what is for example f of x plus delta x delta x is a neighboring point then this is nothing but f of x plus the derivative df by dx times delta x. This will be of course 2 rigorously if delta x is a small distance. Now so this is our idea of a one-dimensional differentiation. Now what happens in three-dimension? Now notice that the basic difference between the one-dimensional differentiation and three-dimensional differentiation arises because in one dimension when I say take a neighboring point then basically I take the neighboring point either to the left or to the right that is along the same line. But in three dimension when we talk about a neighboring point we can go in different direction and so therefore what we will do is to generalize it to the concept of a directional derivative. So for example now I have df by ds, s is a length element if you like along a particular direction. So the point is this by similar argument that what we have just now given for one-dimensional differentiation the only thing now is that when you want to go to a neighboring point which is separated by a length element ds. Now you could go to that point by traveling along x keeping y and z constant then traveling along y keeping x and z constant and finally traveling along z keeping x and y constant. So therefore my df by dx df by ds is now this is traveling along x keeping y and z constant and that is why I put it as a partial differentiation times of course dx by ds and similarly df by dy dy by ds d5 by dz dz by dz. Now to illustrate let me work out the directional derivative of a function f of xy let me take a very simple function with which you are all familiar let us say it is equal to x square plus y square. So now this is what I just now told you that I have a I am going along a direction s that is done by going first along by delta x along the x direction delta y along the y direction keeping x and z constant and delta z along the z direction keeping x and y constant and where d5 by ds is of course given by this. So let us look at what does the figure z equal to z is the this is a mathematical figure or a genu plot figure basically I am trying to plot that function the function has two variables xy which is equal to x square plus y square is my function. So the value of the function is plotted along the z direction and this is the type of picture which you can generate from any of the standard packages. Now it is a couple like structure the it has a couple like structure and so people can get a very good view of that yeah let me show you how to calculate directional derivative of this function in two different directions for example I will show it first along a direction like i plus 2j. Now first thing is that we have said already that look at x square plus y square equal to z that is a couple like structure which I have shown you. So my df by ds is partial f by partial x dx by ds plus partial f by partial y dy by ds. Now these I can calculate easily because this is a function f is being differentiated with respect to x treating y as constant so this is nothing but 2x times dx by ds and likewise this is 2y times dy by ds. Now I need to calculate this dx by ds and dy by ds along this direction for example. Now notice along the direction i plus 2j my x and y are related by so I have y is equal to 2x is what is there which means my dy by dx is simply equal to 2. So what is ds? ds if you remember is an element of length which since I have in two dimension is given by dx square plus dy square which you can rewrite by pulling out a dx as dx times square root of 1 plus dy by dx whole square so that ds by dx becomes equal to root of 1 plus dy by dx whole square but we just now showed that dy by dx is equal to 2 so this is equal to 1 plus 4 square root which is equal to square root of 5 and what you could do is of course to rewrite by instead of pulling out a dx you could pull out a dy and write it as 1 plus dx by dy whole square so that my ds by dy can also be shown to be equal to square root of 5 by 2. So therefore the if I substitute these quantities I find my d5 by ds at the point 1 2 is given by your 2x times 1 by square root of 5 which we just now calculated plus 2y times 2 by square root of 5 and I am interested in the point 1 2 so therefore put x is equal to 1 y is equal to 2 if you do that this quantity works out to 2 times square root of 5. So this is the directional derivative at that point we will do more calculation with different things later. So let me then say what is this gradient? So in defining the gradient what I did is to say that look the sense the gradient is basically connected with a directional derivative what I am trying to say is that look you go along the x direction the unit vector is i the variation of the function scalar function 5 is dou 5 by dou x that is vary with respect to x keeping yz constant plus unit vector j times dou 5 by dy plus unit vector k times dou 5 by dz. Supposing a unit vector u is given by some a plus a i plus b j plus ck obviously it means square root of a square plus b square plus c square is equal to 1 supposing this is the unit vector and the vector s is given by some s times the unit vector u. So therefore I know that this tells me that if I am going from the point x0 y0 z0 to a point which is given by x0 plus a s y0 plus b s z0 plus c s then my directional derivative d5 by ds is given by dou 5 by dou x times a plus dou 5 by dou y times b plus dou 5 by dou z times c which if you look at the way I define my gradient operator which is i d by dx actually the gradient of phi I have defined there. So it is grad phi del phi dotted with the unit vector u. So this tells you since del phi dot u is by definition remember u is a unit vector so it is the magnitude of del phi times magnitude of u which is equal to 1 times the cosine of the angle between the two. So the magnitude of the gradient is nothing but the maximum magnitude of the left hand side that is the maximum magnitude of the directional derivative and secondly the direction of the gradient is along the direction in which the directional derivative is maximum. So this is a very good picture which all of you can use to illustrate what is a gradient. Well you do not have to be at the top I have shown that suppose you are at this point on the hill. Now suppose I want to come down from the hill from this point to this point now notice there are many ways of doing it I could of course be a little crazy go up then come like this I could come like this but look at that there is a direction along which the slope is the steepest that is normally the path that you take can do it to come down from the hill that is come in the direction of the maximum slope and that is your the picture of the gradient. The magnitude of the gradient is well we have said that the gradient is now directed normal to a level surface. Now level surface of a function means the surface on which the function is constant of course in this case I have given you a two dimensional function. So therefore I will be talking about the my level surfaces are actually curves and since x square plus y square is going to is my function x square plus y square equal to constant is in this case the level line if you like or level curve if you like level curves this is nothing but equation to a circle because I am looking at the surfaces or in this case the curve on which the function remains constant. Now the gradient is normal to the level curve which is nothing but in the radial direction you can see it why because grad phi from for this function is dou phi by dou x which is 2x times i plus dou phi by dou i which is 2 y times i which is nothing but 2 times the vector r. So this illustrates the point you see the idea is this that the I simply my picture as I showed you was a couple like structure and if I want to put a level surface that is curve along which my value of the function remains constant then this is the type of intersection that I have. So the question is that why are we studying the gradients the idea is this that we will be very much interested in finding out you are all familiar what are known as the equipotential surface that is surfaces on which the potential function remains constant that is my equivalent to the level surface. And the direction of the gradient that is which is normal to such equipotential surface I am anticipating what I will be doing in my next class and that is obviously the direction of the gradient. Now which if I take the level surface to be equipotential surface then my direction of the gradient is the direction of the electric field. In this particular example I have shown that phi of x y is x square plus y square. So gradient of phi is 2 times the unit vector r and that is what is expected in the radial direction because that is normal to the level curve which is a circle. Now I am left with one last thing in my discussion of the vector and that is to talk about the divergence of a vector field. Now let me first explain the concept of a divergence. The top thing here defines divergence though this is not a definition which you will be requiring very often. So what we say is this we have already defined what is a surface interval. So what I do is this that I take I have a volume and I cut that volume into many small volume as you can see I have done here. This is a big volume here and I have split it by cut it into two different volumes. Now when I have cut it into two different volumes and I can do that many many. So volume surfaces of each constituent volume. Now if I take a small enum volume and then calculate the surface integral over each surfaces what I mean by that is this supposing this is my volume in this case I have shown you that I have cut it into two. Now I am saying that I am supposing this is my smallest unit then I take the surface integral over the six faces that I have in this cube. Now then if I divide it by the volume of this element and if I make this volume small enough then this quantity it is limit when delta v is very small is the definition of divergence. Now let us see why the first thing that you notice is this that when I have the bigger surface then I am illustrating with two surfaces but you can imagine this should be true with all the six surfaces. So for example on the top surface the outward normal is this and on the bottom surface the outward normal is that and similarly on the right surface it will be towards this on the left surface it will be towards this on the front surface it will be towards me on the back surface it will be away from me. Now when I have spliced it into two then the my concept changes because on the top surface here it is there it is this way but now the intermediate surface their directions are opposite because this is the outward normal to this surface this is the outward normal to this surface and of course the bottom surface is as same as before. Now and this I could do it for all the stacked volumes with respect to which I have done the splitting. So the point is this so if I take a large enough volume and keep on cutting it back make it into smaller and smaller volumes like this you notice that because the interfacial surfaces have their outward normals opposite to each other the surface integrals from these will cancel out and in the aggregate volume what I will be left with will only be the surfaces which are not in the interfacial surfaces but the outside surface of the main volume. So what is this divergence what does it actually tell me basically a divergence tells you what is by how much does a vector field spread out. Now let me give an example to you you look at this picture you notice that the vector field again here I have drawn the vector fields plotted the vector fields using a Mathematica and I have taken a convenient two dimensional plot. So you can see that the vector fields are spreading out in each of the quadrants. So here this is an example of where my divergence of the vector field is positive and the if you look at the picture on the right you notice that the fields are being directed inwards. Now this is an example where my divergence is negative. Now let me finally try to complete this by the best way of understanding divergence is to assume that there is a fluid flowing through a volume. So suppose I have a fluid which is passing through and inside a fluid I have put in a volume and so you notice that I am asking the question how much fluid is flowing through the surface. Now notice take the left hand surface in this case my direction of the surface is towards my left towards my left. In other words the direction of the in this is the y axis this is the x axis the z axis. So direction of the surface is along minus j. Now if my density of the liquid is rho the velocity y component of the velocity is v y then the mass of the fluid which is flowing through this surface is given by v y dx v z and the surface is in the negative direction. And similarly that since the velocity could have changed in going from here to there I have if the velocity in this surface is v y the velocity in that surface is v y plus dou v y by d y d y times dx d z. And so therefore what is the net increase in the amount of fluid which is inside this volume. So notice that this is coming in this is going out. So therefore decrease or increase will depend upon what is the sign of this but minus d v y by d y dx d y d z. Now the question is this that where did this come from. So let me again use the this quantity is my net increase in this and supposing I take now my three dimension then I will have since notice this is d x d y d z. So if I considered fluid coming from all directions I can write this as minus d v x by d x plus minus d v y by d y minus d v y by d z. So therefore this would be given by minus of this quantity times d x d y d z which by definition of the divergence in minus del dot v d x d y d z. Where did it come from? Now since the volume is fixed the rate of increase in the mass could have come in because there is a change in the density. So which is nothing but d rho by d t d x d y d z which if you equate the two you get the equation to the continuity equation. Now here is a more pictorial representation of what the divergence of the field looks like. Here I have taken a field which is x square y i plus x y square j. You can easily calculate the divergence which is 4 x y here because del dot f x the divergence is d f x by d x. So this is your f x and since I am doing only a partial with respect to x I get 2 x y and another 2 x y from here which is 4 x y. On the other hand if I take the our old example f equal to x i minus y j the divergence obviously 0. Now you can see it what is happening here. See when I am in the first quadrant where x is positive y is positive 4 x y is positive you can see that the divergence is positive everywhere. Now come to for example the fourth quadrant. In the fourth quadrant x is negative y is negative ok. Again the divergence is positive. You can look at it that by taking a small area and you can realize whether the divergence is positive or negative. On the other hand look at this here. Here the divergence is 0. If you take a circle like this you notice the amount of things coming in is equal to the amount of things going out and that gives me the divergence to be equal to 0. In the next session I will derive the necessary equation connected with the divergence of the vector field. Thank you.