 Let us begin our electrostatics. So, now that we have had a reasonable amount of discussion on the vector calculus required for it, let me now go over to what we actually want to do. We are interested in talking about electric field. So, I will not waste a lot of time on some of these things because this is all very familiar to you, but it is a good idea to realize that physics is not in isolation. No matter what you are doing, whichever part of physics you are doing, you need a set of building blocks and the fundamental law of electrostatics is what is known as Coulomb's law and we all know that the Coulomb's law, I will not state the way you are doing it in school. They like charges, repel, unlike charges attract all those you know, but basically this is the idea that I will summarize the essential point of this. Supposing I have a charge q1 here at a position vector r1 and a charge q2 there at a position vector r2, so that the vector r is the relative position vector of q2 with respect to q1, then the force on the second particle due to 1 is given by 1 over 4 pi epsilon 0 which is a constant that comes q1 q2 the product of the charges divided by square of the distance between them and a vector r is there. Depending upon whether q1 q2 is positive or negative the product, I have either a repulsive force or an attractive force. So what one does very frequently is to write this instead of a unit vector and r square make this an r cube and a vector r this is the same thing really and that is the inverse square law. And since by Newton's third law the force due on 2 due to 1 is the same as force due on 1 due to 2, but with a negative sign I have f12 equal to f21. So that is my Coulomb's law and the Newton's third law, let us summarize this what is so special about Newton's law what are the points that one has to know first inverse square law it is a inverse square law force of course we already know that that the force varies inversely as the square of the distance like charges repel unlike charges attract is also something which you know. The third point which is an important point which should be made very clear it is a long range force. You see a long range force means it is a force which really never becomes 0 never becomes 0 no matter how far you keep one from the other it may become weaker and weaker, but never becomes 0. So inverse square law is a long range force another important point is the force is a central force meaning thereby that the magnitude of the force only depends on the distance between the two and the direction is along the line joining charges. So this is the central force and well it we have defined something called a permittivity of the free space which is an epsilon 0. Now I want to summarize the Coulomb's law force with a bit of a comparison of various types of forces. Now you see in the nature we only have four types of basic forces does any force that you think of can belong to one or the other of this category. Now in terms from weak to strong from weak force to strong force the weakest force is the gravitational force and taking some scale of one which is the strength of the strong nuclear force between nucleons attractive nuclear force between the nucleons. The gravitational force has a strength which is 10 to the power minus 39 or 38 times the strength of a nuclear force which is essentially a negligible force and its range is infinite as I told you it is a long range force. The next in order of strength increasing order of strength comes a weak nuclear force and that has a relative strength of 10 to the power minus 6 with respect to the strong force and it has a range of 10 to the power minus 18 meters. Remember gravitational force is infinite only ranged that is long range the nuclear force is very short range but not that short but it has a 10 to the power minus 18 meter. Now the real strong force with which we deal with leaving aside the nucleus is the electromagnetic force and the relative strength of it is about 1 by 100 actually a typical characteristic number is E square by C H cross which is 1 over 137. So that is the fine structure constant which is 1 over 137 but roughly less than 1 percent of the strength of the nuclear force and the strong nuclear force with the range of a water farming which binds the nucleons namely neutrons and protons together that has a strength of 1. So since we are talking about the fundamental four fundamental forces there is also a question of how do these things act. Now remember that I have supposing I put two charges now they have a force even when they do not have to be in contact in order that there is a force they have a force even when they are separated by a distance the question is how does one particle know that the other particle is around. So, as to either attract or repair. So, this is a question that has bothered people for a very long time that you know I mean. So, what actually happens now this is the reason why we are learning about the field what we say is this that let me give all my explanation with respect to the electric magnetic forces. So, what we say is that when you have a charge now the all the field all the space around that charge is a seat of electric field the region carries the information that there is a charge there. Now, so when another charge comes to that region it has it is of course, own region, but when another charge comes it gets influenced by this field. Now this is let me give you a very bad example similarity that is supposing you are in the in the presence of a person of very strong personality the fact of his presence is sometimes felt by you. Now as I said it is a bad example, but this is precisely what we are talking about we are saying the fact that one charge is there it creates its field and since it is the range of electromagnetic forces infinite it essentially means that this information is there all over the space. In the modern language what we believe is that these charges are or any objects where we are looking for forces they continuously exchange certain things between them and these exchange particles are bosons since you are all teachers I do not have to explain to you. So, the carriers of long range forces which are the electromagnetic forces they are massless and in this case it is photon and as you know photon does not carry any charge. So, that is all about the Coulomb's law now I want you to realize another very important concept. So, I start with the building blocks the first building block is the existence of Coulomb's law which is of course, an axiom this second axiom is that supposing you are looking for calculating the electric field at a point due to let us say multiple charges. Now the point is that how do what is the effect at a point due to the presence of multiple charges they remember I told you that each one of them has their own electric field. Now this is where we talk about what is called superposition principle what we are saying is this that suppose I have a charge q at let us say some point and I said already that all region around it is a seat of electric field. Now if I have an electric field whatever is the reason for of its creation supposing there is an electric field at a point p and that field is e if a charge q finds itself at the point p then it experiences a force q times e. So, basically the electric field is the force exerted on a unit charge at a particular point. So, what we are now saying is this that supposing there are charges q i's at different point let us suppose R i's are the positions at which the charge q i's are there then the electric field at a point p is simply given by a sum remember it is very important it is not given by product or division or by anything it is given by a vector sum of the forces exerted on that a part charge at the point p due to all the charges q i's. Now since it is an electric field at the point p I put a unit charge. Now this unit charge due to the presence of the charge q i at the point R i experiences a Coulomb's law force 1 over 4 pi epsilon 0 q i this is the relative position vector of the point p with respect to the position R i of the point where q i is there divided by of course R cube as I told you this is Coulomb's law. Now this is a very important concept we are trying to say that physical effects add up in other words if there is a force exerted on a charge due to one field is E 1 due to another field is E 2 then when they are simultaneously there it will be the vector sum of E 1 plus E 2. Now notice this is not something which you can prove it is a superposition principle which is axiomatic that I take Coulomb's law and the superposition principle as the basic building block of my electromagnetism. So having defined what is an electric field and having talked about the principle of superposition I need to know there is a question raised which I said I will take up this is almost now time come for discussing the electric flux and the Gauss's law. So remember we defined a flux of a vector field we said that if you have a surface then you define the flux as the surface integral of the vector field over it which we defined in a particular way. Now supposing this field now is my electric field now that this tells me that I can define the flux through any surface as given by this is the definition of flux the surface integral of the electric field over that surface and I have told you that the definition of ds vector is the outward normal. So let me now come to this question which came up the question was that why is it that when we put a charge inside I have a flux coming out positive or negative will depend upon the sign of the charge let us not worry about that then but on the other hand if the charge is outside I do not have a flux now let us look at it in a slightly different way. But before I do that let me explain the concept of what is known as a solid angle look at this supposing I have a surface here this is a surface and I am looking at a point p. Now I would say I will define that a surface makes a solid angle at a point p I will sort of explain it to you by comparison with something else. So let me come back and try to explain this. So let us first go to the definition of an angle ordinary angle how do I define an ordinary angle remember that this is the way we looked at it supposing this angle is theta. Now if this angle is theta how do I define this it is an ordinary angle. Now what we did is to say that look if this remember angle is a dimensionless quantity the you may sometimes measure it in degrees radians or whatever but angle does not have a dimension and the reason is this supposing I look at this situation where this length is L and this is let us say R then my L is given by R theta this was your definition of ordinary angle theta of course is in radians this is my definition of ordinary angle. So what did I do I said that look take a curve which is remember that this curve is such that these are two radial lines. So therefore these are perpendicular to it take a curve and then divide the length that you make between these two points by R and that will give you angle theta but supposing you had this a different length then also the definition is true accepting that you have to take the projection of this along the normal to this one. So this is basically and this is also the definition tells you that theta is dimensionless because L is a length unit R is a length unit I may have a unit but I may not have a dimension. Now the so far as this thing is concerned now the solid angle is concerned now it is a straight forward generalization of whatever I told you just now. So accepting that we had said that in defining a an angle we look at a curve and ask the what is the angle that is subtended between the two radial lines which intersect that arc. Now here what we do is this we go one step and we say supposing I have a surface. Now this surface at a point P makes a cone like structure remember the process is in three dimension earlier we are in two dimension. Now so what we said is in the earlier case an arc made an angle ordinary angle but in this case a surface is making a type of a cone angle and that is what I will call as the solid angle look at this picture. So what we are trying to say is this that here is a surface I have taken it to be small because most of our definitions are defined like this. So what we are saying is imagine not keep on drawing tangents from point P to various points on that surface and then you will get a cone angle here. Now remember I define the angle theta as the ratio of the length to the distance. Now in this case I define a solid angle as the area like in the other case I also take the perpendicular area that is the area which is normal to those and divide it by because of dimensional reason I do not divide by a length but I divide it by a square of a length. So once again if this surface is not directly perpendicular then you take its projection. In other words if the outward normal to this surface is making an angle alpha with the radially outward direction then you have ds perpendicular as ds cosine alpha divided by r square. So that is the solid angle that this surface element subtends at the point P. Tomorrow I will take up the questions that have come in because we will collect all the similar questions together but what I will do tomorrow is this. I will go repeat the idea of solid angle and then tell you that how the Gauss's law follows from our idea of what is an electric flux and the idea of a solid angle. The question that was asked is why is it that when we have a charge inside a volume we have an outward flux whereas why is it that if the charge is outside the flux through that surface is equal to 0. So since I have run out of time I will take over from there. Thank you very much.