 So, now we go to the fourth lecture of this course, we will talk about more of vector calculus in this lecture. So, first is you know gradient concept, the gradient is has got both magnitude and direction, magnitude is describing the maximum rate of change of a scalar field and direction is the direction of the maximum, right. So, we will see you know example of this in the next slide and more explanation. So, again the gradient in Cartesian coordinates you have this expression, in cylindrical coordinates you have this expression and in electrostatics you have E vector given by minus del V. Now here you know of course units match because this is volts per meter because del is 1 by meter. So, volts per meter and E of course is volts per meter. Now, other thing to note here this minus sign you know let us understand intuitively why there should be minus sign. So, E is always from high to low, electric field intensity is say from positive to negative charge, by convention we take positive things at higher negative things as lower. So, E is from high to low, gradient is always from low to high, when we say gradient it is always low to high. So, that is you know very logical that is why that E and del V are separated by minus sign, okay. So, if you take an equipotential surface, right any equipotential surface if suppose this is an equipotential surface and suppose there are two points with you know x x x 1 y 1 and x 2 y 2 then you can write the distance vector P joining these two points as dx ax hat plus dy ay hat plus dz az hat whereas dx is x 2 minus x 1 and similarly, right. So, now since it is an equipotential surface the potential difference between any two points on this surface is 0. So, that means dV is equal to 0 between any two points. So, dV equal to 0 now dV can be written from you know total differential formula as this is it not this is dV can be written as this expression. Now, this expression can be written as del V dot del P because you have both the expressions del V is here and dP as a vector expression is here if you take the dot product of these two vectors you will get this, right. So, that means you have dV is equal to 0 is the gradient of V dot dP and now it is 0 because it is equipotential surface, right. That means del V and dP are orthogonal because we have just you know in the previous lecture we saw dot product is 0 means the cos of angle is 0. So, that means angle has to be 90 degrees. So, that means this del V and dP distance vector they are orthogonal but E and del V they are in the same direction in the sense they are oriented along the same direction of course their directions are opposite, right. But that means what it means is del V as well as E are normal to the equipotential surface, right. Next comes the divergence. Divergence gives the net flux outflow from a close surface per unit volume with volume tending to 0. By definition divergence A is given by this limit del the volume small unit volume tending to 0 close surface integral A dot dS by del of volume or delta volume. So, now this divergence A in Cartesian coordinates is this and in cylindrical coordinates is given by this expression. Now remember divergence is always flow source. If divergence is non-zero that means there is some source there which is responsible for flow of fields, right. Now let us again take the example that we have been seeing quite often since beginning of this course this lead to ground structure. So, when now first you notice that I have purposely drawn this positive charges here little bit closer to each other on this side and then little bit sparsely spaced things on this. Why that happens because because of presence of this other electron electrode and which is at ground potential those you know charges negative charges on that will basically attract positive charges. So, if and if it was an isolated conductor cylindrical conductor then the charges would be uniformly distributed, right. So, this why I am telling you this is whenever you see or draw anything you should draw with the background physics taken into account. So, you should not just draw positive charges moment you know you have this another conductor coming the charges get redistributed. Similarly, on this ground plane although I have not shown these three charges will be more closely spaced as compared to the charges on the at the far ends is it not. So, now you know it is easy to understand that divergence will be positive here because field vector is originating, right. So, nothing is coming in and something is going out. So, divergence net divergence is positive whereas on this negative charges something is coming in, but nothing is going out from those points. So, the divergence is negative, right. Whereas point P if you take some point P in between the divergence is 0. Now, how do we understand that now this this line middle line field line you assume that it is impinging on this some kind of you know cube and this at this point you know you have that at the center of the cube you have point P at the center of the cube you have point P and now by definition if you calculate your closed surface integral a.ds. So, now there are six sides here is it not for this cube the four these four sides will not contribute to the you know dot product because of the reason that you know you have this the field is perpendicular to the area vector is it not field is perpendicular for because for this phase for example the area vector is going to be like this that is going to be perpendicular to this field. So, the dot product is going to be 0. So, the only two sides that will contribute to the dot product are the front and backside because in that case you have you know field as well as you know normals along the same direction the difference being the unit normal on the front phase is going to be outside. See concept of unit normal we need to understand with reference to open surfaces and closed surfaces. Suppose you know you take this open surface now these are open surface now these suppose you take this surface for this surface you have two choices for unit normal either it could be like this or it could be on the opposite side going in the opposite direction. So, but this choice does not remain with you if you consider if you are considering the corresponding counter integral if the corresponding counter integral you are considering like this then counter integral is in this direction. So, the unit normal will be in this direction whereas if the counter integral is being taken like this in anticlockwise direction your unit normal to this open surface will be in this direction. But if you take closed surface something like this there is a closed surface all these know for all sides of surfaces of this closed surface the unit normal is always outward normal. So, for this surface it will be like this for this surface it will be like this like this and like this. So, for closed surface there is no problem it is always outward normal for open surface if you are considering the corresponding counter integral then again you are unit normal direction gets fixed if counter integral is not being considered with reference to the open surface then you can take the unit normal in either direction. So, the unit normal in this case is opposite to the direction of the field for this front face whereas for the back face the unit normal is along the direction of the field and both those that is why those dot products will also cancel each other that is why overall dot product for the entire closed surface is 0 and that proves that you know since this is 0 divergence is 0 that is why you know it is the case. So, the divergence theorem is divergence A dv evaluated over volume is equal to closed surface integral A dot ds. Now effectively why this theorem is useful because you are converting 3D into 2D integral this is very very useful particularly you know in you know integral equation numerical techniques where the dimension of integral reduced by 1 similarly if suppose you know you are doing a 2D analysis mostly you know in this course also we are doing a 2 dimension analysis. So, when we are doing 2 dimension analysis what we are typically do we take the cross section which is perpendicular to the current always you know whenever you show a cross section we will find we either show dot or cross that is the current direction that means you are taking always cross section which is perpendicular to the current and then we do what is the approximation that we do the infinite you know the same 2D condition that you see on the paper it continues infinitely in z direction right. So, that is the approximation we do. So, when you do that approximation there also if you apply you know this divergence theorem effectively what you are doing is this divergence A dv is the same thing here it is effectively ds into 1. So, this is volume ds into 1 is volume that means you are taking per unit depth in the z direction is it clear what I am saying. So, per unit depth in z because it is a two dimensional approximation and although z is infinite we take for calculation we take we do calculation per per unit depth. So, effectively it is ds into 1 and this is dl into 1. So, this is also ds this ds and this is same dl into 1 is this ds. So, both the equations are same, but since we are actually doing a 2D approximation already already we have reduced the dimension by 1 by further you know invoking divergence theorem we have further reduced the dimension of the integral by 1 again this is very useful when you do two dimensional analysis. We again go to that virtual electromagnetic laboratory where in divergence is explained through some field plots. Now, you can see here divergence of now what is this equation it is x ax hat y y a y hat so divergence by formula is 0 is it not. But then how do we understand this. So, here for example you can see take any point here and you will understand that the rate of change of field with x and rate of change of field with y they are exactly cancelling out. If x the direction the field slope in x direction if it is increasing the field slope with y direction is reducing by the same amount. So, that makes you know the divergence equal to 0. The vector with positive divergence it is very easy. At every point for example in electrostatic field there is positive volume charge density that is positive flow source. Similarly, for negative divergence at every point there will be negative volume charge density that is negative flow source. So, now we will go to the next which is curl. Curl is measure of circulation of a vector field and it is circulation per unit area and it is a vortex source and by definition del cross a is given by this formula and again earlier in case of divergence volume was tending to 0 here the corresponding surface is tending to 0. So, here it is what is in question a surface which is bounded by close counter. And you know you have this the vortex source the most common example that we can think of is a current source. The current here is coming out of this paper and the corresponding field again given by the right hand rule. So, current the field direction is in anticlockwise way. So, here the loop this area is normal to the vortex source and that will give the maximum circulation that is the meaning of this max. So, your loop or the area vector is arranged such that it basically that area vector because area vector will be normal to the plane and that area vector is along the same direction of the vortex source and that will give the maximum circulation. So, for example if you you know current source is like this and if you your loop is not along this but it is like this then you will not get the maximum circulation calculated by this formula. Then the formula for curl in Cartesian and cylindrical coordinates is as given here. So, just to you know sort of quickly understand this just see the direction of this. So, it is yz zx xy better to write in this form rather than putting a minus sign and just interchanging these two terms because this makes it simple yz zx xy. So, it is in the same sequence and then finally you know you have the Ampere's law in point form is del cross h is equal to j. Remember probably I did not mention when I was talking about Maxwell's equations. So, Maxwell's equations that we have seen they are like point form of differential form of equations because they are involving all curl divergence and all that and they they involve derivatives partial derivatives. So, that means Maxwell's equations that we have seen till now they are basically point form of equations they are telling you what is happening at a point because derivatives tell you because derivatives change from point to point is it not. So, this Maxwell's equation that we saw in the till now involving divergence and curl operations they are basically point form. So, this is you know point form of Ampere's law and when you actually take the surface integral from both sides on both sides then you have integral j dot ds is equal to i and apply Stokes theorem which converts surface integral open surface integral into closed line integral l. So, you will get h dot dl is equal to i. So, again here Stokes theorem is useful in converting 2D integral into 1D integral. Similarly, you can actually see the there is some excrement in virtual lab on curl also that actually you can see in your leisure time. If you go to electrostatic fields we start with Coulomb's law which is f is equal to q 1 q 2 upon 4 pi epsilon 0 r square wherein this q 2 upon 4 pi epsilon 0 r square is the electric field intensity due to charge q 2 and if that electric field intensity is acting on q 1 then you will get force on charge q 1 as q 1 times e. So, e is q 2 upon 4 pi epsilon 0 r square. Now, in case of you know electric fields you can either have line charge surface charge or volume charge. So, line charge is rho l dl and the corresponding electric field intensity is this. Again remember both rho l is the Coulomb per meter line charge per unit length whereas this rho is the distance variable in cylindrical coordinates distance from the axis. When it comes to surface charge you have dq as rho s ds and corresponding electric field intensity is given by rho s upon twice epsilon naught a n vector. Now, here how do we understand this? Now, you know you have this capacitor problem we know the field distribution like this. Now, understand this field distribution using this formula. If you take only the positive plate of the capacitor then you have the field lines like this as well as field lines going up. Remember because positive charge the field will emanate from below as well as on the top but I have shown only the field lines on the bottom. Similarly, the negative plate you have the field lines terminating on that. Similarly, there will be field lines terminating on this side. So, now if you superpose because linearity is there there are no non-linearities in this problem you will get the resultant field distribution as given by this. So, only in this region between the two plates you will get addition because both the fields are in the same direction whereas above and below you will have cancellation of fields because of the two fields. And that is how you get this and the corresponding formula of E resultant as rho s surface charge upon epsilon naught a n hat. And remember what we have seen d vector is nothing but really rho s is it not because d upon epsilon is E and unit of rho s is also coulomb per meter square unit of d is also coulomb per meter square. D can be related to surface charge density on the conductor surface or to volume charge density in non-conductors through the equation divergence d is equal to rho V. So, as mentioned here in free space d is equal to epsilon 0 E and only one of the two vectors d and a is sufficient both vectors are not required only when you have material and corresponding polarization which we will see later you need both the vector that means when p vector is also there polarization you have to actually you can differentiate between d and E and you have to have a separate treatment given to both vectors. Now the Gauss law is basically total electric plus 2 closed surface is equal to charge enclosed within it. So, that actually can be expressed as q is equal to closed surface integral d dot d s which is also but q is also integral rho V d V volume charge integrated over the volume charge density integrated over the volume. So, again now if you apply divergence theorem you will get divergence d d V and that is why now if you correlate this and this expression you will get divergence d is equal to rho V. So, in a way we have derived the first Maxwell's equations from Gauss law. Now the next point is potential work and energy. Potential difference V AB by definition is minus integral A to B E dot dl and that is also potential also is defined as work done per unit charge. So, now actually if you see here if V AB is this V B A is minus B to A because I am just changing this interchanging this as well as the integral I am interchanging. So, it is minus B to A and E is minus del V. So, this minus sign and this minus sign goes and you get something positive. So, when you go from B to A you will get positive voltage and that is what it should be because it is a gradient. So, that way we have just verified the sign convention is correct. The other thing is important thing is electrostatic field is a conservative field. That means if you take two points C and D here the E dot dl along this path and along this path is same because the potential difference is same. Is it not? Both will give you the same answer. And now since both these are in the same you know they are their magnitudes are same. Now you take the total integral like this circular total integral will be 0. Is it not? And now you apply Stokes theorem you will get this and then finally del cross E equal to 0. Now, this becomes in case of time varying fields it will become del cross will become minus daba V by daba T which we will see later. But for electrostatics this is 0. Now this one important thing what is this conservative field? See here it is written conservative fields cannot do any work. So, what is the meaning? Suppose you have a capacitor which is you know connected to a battery and you basically you know charge this capacitor by closing this switch and this is plus minus. So, in certain time constant you know this will get charged and the corresponding energy will get stored. We call this as potential energy but this potential energy got stored only when the some charges move. So, remember always the work will be done only when charges move. The static charges will not do any work. Similarly, now this charge capacitor with this potential energy is not going to do any useful work unless you make those charges move. For example, now this capacitor you short by connecting in parallel with resistor this capacitor will get discharged in no time and then it will produce heat in this resistor. Now that heat of course is a work it may be useful or non-useful but it is a work. In some cases it will be useful work for heating applications. So, the point to be noted here is when it was only capacitor was charged and you know the charges were static it was only potential energy that is not capable of doing any useful work because charges are static and it is a conservative field. Only when those charges were allowed to move by shorting that capacitor what you have is basically the charges moving and the corresponding kinetic energy and the work. So, kinetic energy you can associate with work always that is why the charges have to move in electromagnetic charges have to be moving to effectively do some work. These are very important concept to be understood. Now the next point here is the energy density that is energy per unit volume is half epsilon E square. This expression we will use you know quite often in the FEM theory to start with when we talk of energy minimization and energy total energy is the volume integral of this. The last topic of this lecture is here we will talk about current densities. Three types of current densities are discussed here conduction current density, convection current density and displacement current density. For all these current densities this expression of I is same. I is dq by dt and it is integral j dot ds that remains same but j conduction is given by sigma E it is governed by Ohm's law potential difference and again this is j equal to sigma E is called as point form of Ohm's law which is V equal to I R. So, j equal to sigma E is more universally applicable as compared to V equal to I R because for application of V equal to I R you need to have you know the conductor of certain you know uniform dimensions so that you can calculate resistance as O L by A and all that is it not. Then you can easily apply that Ohm's law but j equal to sigma E you can apply for any arbitrary shape conductor at every point. Whereas V equal to I R is like a integral form because we are actually finding the potential difference across the end of a conductor. So that is a integral form. So, in finite element method we will use this when we calculate for example losses, eddy current losses or losses in any conducting part. The loss is given by j square upon sigma dV integrated over volume. This also can be represented as rho j square dV because one upon sigma is the resistivity. Note that watts per meter cube is unit of j square by sigma or rho j square. So, again I think rho in today's lecture we have used third time for a different, is it not? So, now this rho is resistivity. So, rho was earlier used for volume charge density then cylindrical axis distance and now this is the resistivity. So, please bear in mind the differences. So, this is resistivity. So, convection current density is given by this rho V there is a volume charge density into its velocity. So, now this is not governed by Ohm's law. That means potential difference is not causing this convection current to flow. It is like you can imagine suppose there is some liquid insulation and you have by some means generated charges you have deposited those charges there and you make that liquid to flow by some mechanical means like pump. So, now those charges which you have introduced they will flow and constitute current and that will be convection current. So, that is convection current density. So, needless to say in this course on machines and equipment we will not have this convection current density except if you are doing some advance analysis for example in case of transomers wherein you are trying to understand static electrification phenomena for example whereas wherein static charges get developed due to you know friction between oil and solid insulation and because of pump action those charges move there this convection current would be applicable. But otherwise in normal circumstances this conduction current density would be applicable. In Maxwell's equation what we actually you know C, J right is basically combination of both conduction and convection if both are applicable. Remember all these equations generally are they are general in our this course also we will be writing most general form of equations but then depending upon the problem solved depending upon the variables applicable some variables will get eliminated in the formulation. Then finally the displacement current density which is you know sometimes you know not very easy to understand. So, this was introduced by Maxwell for bring in consistency with the continuity equation. So, continuity equation is this divergence J is equal to minus dou by dou by t. So, what is you know this first let us understand continuity equation. Suppose you have some volume and if you evaluate divergence J for that volume and if the divergence J is positive that means some charges would get depleted from that volume is it not. So, that is the meaning. So, some charges getting depleted means dou by rho v by dou by t, dou by t is becoming negative and this negative of negative is positive and that is why divergence J is positive right. So, this is the first explanation. In the you know starting of the next lecture I will explain the other view point with respect to circuit. But what is the inconsistency here? So, del cross H is equal to J if we you know consider the original Ampere's law in point form and divergence if we take the divergence of this. So, we know divergence of curl is identically 0 right and that means divergence J will be equal to 0 because we are taking divergence of both sides. So, divergence J here is given by this that means we are forcing dou by rho v by dou by t equal to 0 which may not be the case particularly at the high frequencies because if you convert this into time harmonic form then it will become minus j omega rho v at very high frequencies particularly this is not going to be insignificant. So, this is the inconsistency right. So, whenever you have you know for example capacitor currents becoming appreciable in at high frequencies stray capacitance currents becoming high at you know significant at high frequencies you cannot actually you know have divergence J equal to 0 because then we are actually not allowing this term to be finite in some cases right. So, more about this point in the next lecture. Thank you.