 So, welcome to 7th lecture. So, till now we have covered vector calculus and electrostatic fields. So, from now onwards we will see for next 2, 2 and half lectures magnetic fields and time varying fields. So, which are you know very important for this course because most of the time we are interested in calculation of magnetic fields in electromagnetic devices that we deal with. So, when we talk of magnetostatics we start with Biot-Severt law. So, for you know line current carrying current i the value of magnetic field intensity dH at point P at a distance r capital R is given by this formula right with alpha as the angle as shown there. So, in the vector notation it becomes this formula right. So, this is well known Biot-Severt law. Now, thing to note here is that either you have line current, you can have surface current or you can have volume current. All 3 cases that overall unit. So, is ampere meter ideal is ampere meter k ds k is the surface current. When, when does surface current come into picture? For example, when we will see later on when a high frequency excitation is there and skin depth is very small current tries to remain at the surface and then you have what is known as surface current density. We will see talk little bit more about later on this. So, k ds, so k since it is surface current density as shown here this is the surface here and the corresponding k is the unit is ampere per meter because it is only surface. So, one dimension is not there because it is just a surface very thin surface. So, ampere per meter into meter square is again unit is ampere meter and the third well known thing which we commonly use in field calculations is J. J is volume current density ampere per meter square into volume meter cube again the overall unit is ampere meter. So, all these 3 the unit is overall unit of current multiplied by corresponding dimension is ampere meter. So, always remember I in ideal I is the scalar dL is the vector. So, dL is the one which gives the direction to the current element, current whereas here k and J they are the vectors. So, coming next to current element which is like a line current infinite line current as shown here. So, you have again at point P the magnitude of the vector H is given by this formula where alpha 2 and alpha 1 are the angles as shown in this figure. And if it is a infinite line current for finite line current of course, this is applicable then alpha 2 and alpha 1 will come infinite line current alpha 2 will because when this goes to infinity alpha 2 will tend to 0 and alpha 1 will tend to 180 degrees. And then this formula reduces to well known formula H is equal to I upon 2 pi rho. Always remember in case of electrostatic fields and this magnetostatic fields because E and H are equivalent. E in electrostatics or electricity is equivalent to H in magnetic fields or magnetics. So, if E is in we saw in electrostatics E was proportional to 1 over r square is it not? Here H is also 1 over r square variation. This well known formula I upon 2 pi r or 2 pi rho and there 1 over r dependence 1 upon distance dependence is for only the infinite line current case. But for a finite current element it is always 1 over r square variation for H this point to be noted. The question is if it is a infinite what do you mean by infinite charge distribution? No, if it is a infinite line charge it is q upon 2 pi epsilon 0 r l that is you know it is a 1 over again distance dependence for a line charge. So, here another thing to notice which I have also explained earlier del cross H is equal to J this is the point form of Ampere's law and integral H dot dl is equal to I is the integral form of the same equation. And since del cross H in general is non-zero so this will be a non-conservative field. We have earlier seen what is the difference between conservative and non-conservative field. So, this is a this is an example of non-conservative field. Now coming to magnetic flux density we know B is equal to mu naught times H in case of free space. In materials we will see later how this gets modified and how mu r relative permeability comes into picture. Now you know there is often confusion between this what H represents what B represents. So, one can say that H represents a source which is current and its unit of course unit of H is Ampere per meter. So, you can say it is a macroscopic you know representation of magnetic field whereas B you can visualize at a point is it not because H you cannot really visualize at a point. So, you can say that H is representing the current source which is you know producing it. So, B definitely can be visualized at point we say flux density at this point is so much Tesla or Weber per meter square. So, that is why it is you know B can be called as macroscopic you know representation of magnetic field. So, again it is a very you know subtle difference between the two and you know only one of them is good enough to you know represent magnetic fields if it is free space both B and H are not required only one of them is sufficient to you know describe the magnetic fields if one is known other is automatically known. So, then this is a Gauss's law of magnetics closed surface integral B dot ds is equal to 0 if you apply divergence theorem then you will get divergence B dB is equal to 0 that leads to the second Maxwell's equation in point form divergence B equal to 0 which tells that there are no magnetic sources or sinks and flux lines they are always closed. But we should always note that if this surface integral if you make it open surface integral integral B dot ds open surface integral gives you the flux crossing that surface moment you make it closed it becomes 0. This point I already explained free space only one of them is sufficient to describe magnetic fields. Now, divergence B is equal to 0 and we know that divergence of curl of a vector is always 0. So, then we can define B vector as curl of some vector. So, curl of say A vector what is this A we will see in the next slide. So, B we can define as curl of A right. But we know B from you know Bayard-Severt law which we have seen in the previous slide H H was this quantity that quantity into mu 0 gives you B. Actually by using these two formulae del cross B is equal to del cross A and B is equal to this you can then derive expression for A as given here. So, A is integral mu 0 i dL upon 4 pi r and integrated over the you know length L right. Again as I mentioned to you whenever you see in most of the equation whenever i appears involving vectors i will be you know accompanied by dL vector that only gives you the direction of current right. Whereas the same equation if you want to represent in terms of if there is a volume current distribution then this i dL will get replaced by j dV where j is the vector dV is just a scalar. Now we come to a very important quantity in electromagnetic which is called as magnetic vector potential you know because you have potential in electricity which is V and then E we calculate in terms of V as E is equal to minus gradient of V. But in magnetics you know you have only B and H fields and then there is you know no other there was earlier no other quantity in terms of which you can calculate B and H fields. So, then you know using all these basic equations and the properties and the Maxwell's equations and whatever we have seen the researchers coined this magnetic vector potential A and later on I will explain you this potential that word actually is not really a correct word because it is not really a potential because potential means it is like a scalar right. But this A is actually vector. So, Maxwell originally called is magnetic vector potential as electrokinetic momentum vector. So, it basically was actually it should have been called as vector but this magnetic vector potential this you know these words have been very commonly used in now literature and most of us are used to it. But we should remember that it is vector and as Maxwell correctly termed it is electrokinetic momentum vector. How that is momentum vector we will see later when we see time varying fields. So, del cross H is equal to J and then you know B is equal to mu naught H. So, this equation can be recast like this and then B you replace by del cross A and then we know del cross del cross of a vector is given by del of divergence minus del square of that vector. So, this is what these are common vector identity and is equal to mu 0 times J. Remember that we are still in you know we are not describing materials yet. So, we are in free space that is where mu 0 is appearing. Later on we will bring in mu which is mu 0 times mu r. So, now actually we you know divergence A. So, till now what we have done is B is expressed as curl of A. So, curl you know see in vectors by definition of vector a vector is completely defined if it is both curl and divergence are defined. So, till now we have defined only curl of a vector. So, B B is equal to curl of A. So, divergence of A we have not yet defined. So, we will now define divergence of A as this mu 0 epsilon 0 daba B by daba T. Again this is free space that is why here mu 0 and epsilon 0 is appearing and this is called as Lorentz gauge. Now this is not an arbitrary definition. This Lorentz gauge is again it can be proved that it is consistent with continuity equation. This you can see in all textbooks it is you know that derivation is given straight forward derivation. So, this Lorentz gauge is consistent with continuity equation. And now hence this A gets completely defined because we have defined curl of A which is B and divergence of A as given by this expression. So, A as a vector is completely defined. Now at low frequencies this right hand side term will be almost negligible because you know frequency is small. This again daba D by dt or daba V by daba T is given by j omega V in frequency domain. And since these mu 0 and epsilon 0 are generally very small numbers is it not 4.2 10 to the power minus 7 8.85 into 10 to the power minus 12. So, these are the product is small and if the frequency is small this is going to be a very small number is it not. So, that is why this is 0 at low frequencies. So, divergence A becomes 0 at low frequencies and then this is called as coulomb gauge. So, that is why you know in low frequency electromagnetic divergence A is always taken at 0. When you are dealing with high frequency electromagnetics involving wave propagation, antennas and whatnot, divergence A is taken to be by this expression Lorentz gauge. So, this is the difference which clearly separates low frequency and high frequency electromagnetic field computations. So, now if we all agree that this is consistent theory then we will take divergence A equal to 0 and then del square A will be then equal to minus mu 0 times j. Now compare this equation with Poisson's equation in electrostatics which is del square v is equal to minus rho v upon epsilon naught again this is free space in free space and the corresponding expression for v integral rho v dv upon 4 pi epsilon 0 r. So, now this expression for v and the expression that we saw in the previous slide for A you can see the similarity if mu 0 and epsilon 0 appear individually if one is in numerator other will be in denominator we will always find that here mu 0 ideal here you know rho v dv upon 4 pi epsilon 0 r. So, here mu 0 is numerator here epsilon 0 is in denominator. So, you can 4 pi 4 pi is common here it is you know line source that is why ideal is there here it is a volume charge. So, it is rho v dv. So, there is a similarity between the expressions. So, now this is a you can say this is a vector Poisson's equation del square A equal to minus mu naught j is a vector Poisson's equation that can be split into three scalar equations del square A x is equal to minus mu 0 j x the j the x component of j then similarly y component of j and z component of j will be correspondingly be equal to del square A y and del square A z. So, these are basically three scalar Poisson's equations right now little bit you know more understanding of magnetic vector potential we will do. So, of course, we have understood its you know expression and its you know genesis on the point of your definition of divergence and curl, but let us understand in space how it you know works. So, again here i he i is the line current here. So, if the i is directed like this through this del vector see again I have shown here purposely del as a vector here I have not just shown i because this del is the one which will give the current the corresponding direction right. So, we should be consistent whenever we show something we should always remember this small small things that will make our understanding clear. So, for this current along this direction you have the magnetic vector potentials also directed in the same direction because you know in the previous that expression you have on right hand side there is only one vector D L. So, the direction of that vector will be the direction of A is it not. So, D L is in the vertical direction through which the current is flowing. So, A is also vertically directed and the magnitude of A is reducing as you go away from the source because it is inversely proportional to the distance. Now here del cross A is not 0 obviously because it del cross A is B, but how do we you know sort of understand it here suppose we take this as the z direction these are z direction then A z you can see the A z component is changing the direction is fixed, but the magnitude is changing is it not. So, that means A z magnitude is changing with respect to x right and in the curl expressions you have now for example here I have written that we will see that in the next slide also B is given by this expression daba a z by daba y A x hat minus daba a z by daba x A y hat. Now here we are talking of two-dimensional approximation we are not actually that means current is in z direction and we are we are basically you know the B field will be that is why in x y plane is it not right. So, here moment A z is changing with x that means at least one of this term is going to be non-zero that means B exists right now what we have considered here is this. So, this is z and suppose this we are talking of x so what we have shown is here this is you know current and A is we have shown like this and only the we are considering the x component, but if you rotate this by 90 degrees here basically daba A z will change with respect to y are you getting what I am saying right. So, here it is changing in this plane which is z x plane right it is basically the variation with respect to x after 90 degrees rotation in phi you will have A z varying with y and at any intermediate angle A z will vary with both x and y. So, that clearly you know helps us understand this magnetic vector potential and its relations relation with B right and how the A vector varies with you know distance from the source. Now, this slide is particularly very important for 2D FEM calculations that we are going to see throughout this course our course will be dominated by 2D magnetic field computations and that is why this side is quite important. So, psi this denotes flux is open surface integral B dot ds remember again I am repeating close surface integral B dot ds is identically 0 open surface integral B dot ds is equal to the flux causing the that surface right. So, now B you replace by del cross A right and then you apply stroke theorem you will get psi flux is equal to close counter integral C stands for counter close counter integral A dot dl. Now, if the flux is in webers this is meter. So, the unit of A becomes Weber per meter right. So, unit of A magnetic vector potential is Weber per meter. So, line integral of A around any close path is equal to flux passing through the area enclosed by the path right this is what is describing this equation. Now, again let us understand in space what is happening. Now, here you have current again in z direction. Now, here again I have marked the coordinate axis z is vertically up x is in the horizontal direction. So, y is going into the paper because it is like x is horizontal y goes inside z is up right. So, it is basically the fingers of the right hand you turn around from the x to y you will get the z direction. So, x y into the paper and z up right. So, now again you know you have this current source directed along the z direction. So, you have this A 1 and A 2 at two points which again will be in z direction. So, then we are just taking a counter like this okay. If you take the top view of this in the x y plane you will see this current source adjust by dot is it not. So, this will be a dot current is coming out and the field lines are as per the right hand rule again thumb pointing in the direction of current field will be given by the fingers right. So, the field will be as shown there right. So, this is x y now again here x y so z comes out. So, remember here y was going in but here moment you show this as x y z comes out. So, it is basically by the right always go by right hand rule okay. So, now this and if you see here the same thing in 2D I have shown it just by a dot here. So, here of course I have shown the field but if you just have to understand by you know points. So, A 1 and A 2 will be at 2 there will be the such two points in this x y plane right. You should remember the flux is always associated with the corresponding area through which it flows. Now, again you know what is the area that is corresponding to this flux is flowing through such an area where one dimension is along z axis. So, here actually when we say the flux is really flowing flux really does not flow to this paper surface. Flux actually you can always have to visualize that flux is crossing some surface. So, what is that surface either it will be z x z. So, this will be you know either this surface. So, that means suppose these are current source the flux is crossing like this is it not. So, either it will be suppose this is z it will be either z x plane and the corresponding surface or if you turn by 90 degrees in phi it will be z y right. So, always it is crossing some surface which is you know and at interval it will be you know some surface which is either in z x plane or not in z y plane is it not some intermediate angle. But whereas, if this is there it will be z x this will be z y because this is y as shown here y is into the paper right. So, you should be very clear about this. So, flux crossing this surface and it is going in. So, flux goes into this surface right and that is given by integral b dot d s a n hat where a n is the unit normal to the surface. Now, again I mentioned in one of the previous lectures the direction of this a n will be decided by now this counter integral is it not because this is open surface. So, open surface has two possibilities of n either you know it is you know it is either it could be this is open surface it could be this this unit normal or it could be this unit normal. But now we are taking some counter integral. So, that counter integral decides the direction of unit normal for that open surface right. So, I am again and again repeating some of these points. So, that these are very important for you know in general vector calculus point of view and electromagnetic field distribution in space and visualizing those. So, now in this case it will be a y hat for the x z plane surface shown in the figure right it will be a y hat for the x z plane shown. So, now this psi is equal to a close counter integral a dot dl now that actually is you know if you evaluate that it will be simply a 1 l l is the length of this you know segment. So, a 1 l minus 0 why because this length is perpendicular because a is directed like this. So, a dot dl will be 0 on this and on this and on this side a and dl they are oppositely directed because a is vertically up dl because we are taking counter integral. So, dl is down is it not. So, that is why you have got a minus sign. So, finally and then this again is 0 this does not contribute to the line integral. So, then you have got psi is equal to a 1 minus a 2 into l and for 2D approximation again in one of the previous lectures we discussed that whenever we do 2D approximation we take 1 meter depth in z direction and that is why if we do that then l this l will be 1 in z direction and then this will just reduce to a 1 minus a 2. Now, coming to this point again. So, now here if you take two points on one counter these points 1 and 2 are at same distance from the current source. So, then a 1 z minus a 2 z will be equal to 0 is it not because psi is 0 because no flux is crossing the surface. So, that is why they are called as see this concentric counters are called as equi A or equi potential counters or flux counters because on this counter since a a 1 and a 2 magnitude is same. So, no you know flux crosses in fact flux is going along suppose these are the two points suppose these are current and these two points are there. So, the flux is going along the points it is not crossing any surface which is you know subtended by those two points are you getting what I am saying the flux when flux has to cross the surface is it not. So, here those two points the flux is just going along those points. So, it is like a parallel it is going parallel to those points whereas when the flux has to cross it has to go perpendicular for example here the flux crosses that surface is it not. So, it is basically here though that is why the flux is crossing the surface here and that is why there is a difference between a 1 and a 2. Since here flux is going along the you know counter joining points 1 and 2 right a 1 minus a 2 will be 0. So, this is another important point that flux can be set up by using only boundary conditions in terms of A. Now, for example here these one rectangular you know geometry this could be a magnetic material right and you want to set up the flux flowing like this that you can easily do by imposing the boundary conditions like on this side there is a a 1 now remember a 1 means a 1 is z directed. So, this is the x y plane and a all along this will be vertically directed as shown here as I am showing here it is a 2 again you know vertically directed, but the magnitude everywhere is same a 2 right and on these two horizontal you know lines you have homogeneous Neumann condition. I have explained in one of the previous lectures what is homogeneous Neumann condition that is derivative normal derivative of that field variable is 0 that means the value here. So, now for this horizontal you know line then unit normal will be this is it not see why it is normal because normal we always also set with surface is it not. So, then you may be wondering where is the surface here always remember practical things are always 3D. So, here also it is 3D figure approximated as 2D. So, what is the third dimension into the paper and it is 1 meter depth. So, that means this length into 1 meter depth and a corresponding surface form for that surface this will be the unit normal is it clear. So, unit normal is always to a surface right. So, always you know this you have to imagine this is a 2D approximation. So, what in the background there is a 3D actual figure and corresponding surface here is formed by this line and a 1 meter depth you know segment and a corresponding you know rectangle is it clear and the unit normal then this will be the become the unit normal depending upon you know whether it is how the counter integral is taken and all that either it could be this or it could be you know the other way right as I explained earlier. So, so difference in a values matters that means you know you just a 1 minus a 2 is we have seen flux is it not. So, you take the difference accordingly to set a given flux and flux density. That means you know it does not mean that there is no current source anywhere current source is there that current source could be you know distributed current source here somewhere and it could be a distributed current source. That current source I have shown purposely a dot here because then that dot only will produce the flux in this direction. So, there is a distributed current source which will give this condition that a is directed vertically up because this current is coming out. So, similarly a will be coming out here and the magnetic vector potential is same all along. Why I am saying here distributed current source if it is a point source you will not get constant a values on this is it not then it becomes a circular the values will be constant only on the circle. So, that is why I am saying some distributed current source only will give you constant a values on this vertical line right. Now, so vertical boundaries we have diraculate boundary condition right and horizontal boundaries we have homogeneous Neumann boundary condition. Now, this diraculate boundary condition these are not homogeneous because there is some finite value. If suppose this a2 you define it as 0 right because only the difference matters. So, a2 you define as 0 and a1 correspondingly some number which sets up a given flux then this a2z will become homogeneous diraculate boundary condition. Whenever the potential specified is having some finite value it is non-homogeneous if the potential specified is 0 it is called as homogeneous diraculate boundary condition right. So, now this v is equal to del cross a and then you know this is a normal our curl formula and this expression in fact we saw earlier here this expression comes because there is only a z a x and a y are 0 right because a current is directed in z direction. So, a is directed only in z direction. So, a x and a y components are 0. So, that is why you get a simplified expression for v in two dimensional case remember this two dimensional case. So, now this is a very this last point is very very important not described in many you know textbooks of electromagnetic or even in on numerical techniques on electromagnetics. But this actually explains the magnetic field lines and electric field lines very very clearly and corresponding difference between them. So, now on the vertical boundaries here we have defined a z is it not we have defined this a z values two a z values that means daba a z by daba y is 0 because a z is constant on those vertical boundaries right. So, daba a z by daba a y is 0. So, that means this this term becomes 0 right that means only this term exist b y exist is it not only the y component of b exist right. So, in this case the flux line or the equi a line and a b line they flow together because both are in the same direction y direction clear. So, that is why magnetic flux flows along equipotential lines in magnetics right. Now, let us see the corresponding thing in electric field lines case electrostatics. So, in contrast e is equal to minus del v and then you have got this expression for del v and now let us take again a simple parallel plate capacitor case. Now, here I am taking the capacitors plates which are you know vertically oriented. So, this is one plate and is another plate. So, you have v 1 here specified v 2 here okay. Now, the we know this will be the equipotential lines vertical and the e lines electric field lines will be orthogonal. Let us verify that they indeed are orthogonal by through this equation. So, now here on vertical these bondage we are specifying v 1 and v 2. So, here daba v by daba y is 0 is it not daba v by daba y is 0. So, only e x exist right. So, that that is clear that only e s exist and that is what we have got e exist in x direction only. So, now here e and v lines they are orthogonal to each other. So, you can say electric flux which in case of electrostatics it is e or d is the flux remember it is e or d that flows perpendicular to equipotential lines right because see v you are specifying v right on the vertical. So, v is constant on as a function of y. So, as y is varied v is constant. So, daba v by daba y is 0 let us select previous in the previous case of magnetics right. Now, here this shows clearly the same difference between electric flux and magnetic flux. So, here in this slide you have these two plots one for electric field one for magnetic field in case of electric field you have a high voltage conductor in the vicinity of some ground plane or ground conductor which also could be concentric in this case because the field lines are symmetrical about the circle. So, you have here you have equipotential lines circular and the e fields are crossing orthogonally to the equipotential lines and as you see here the there is no electric field inside this conductor because there are no field lines because the conductor is assumed to have very high conductivity tending to infinity and that is the main difference between the electric and magnetic fields. In magnetic field case you have a current carrying conductor here and there is internal as well as external field. So, the field is there inside as well as outside and you can see there are no orthogonal lines crossing each other here. So, that confirms the statement that we made that in case of magnetics equipotential flux flows along the equipotential lines in case of electric field the flux which is no e field flows orthogonally to the equipotential lines. So, that clearly brings out the difference between the electric and magnetic fields. So, with this we end 7th lecture from tomorrow we will further tomorrow's class we will further see the magnetics and later on we will see time varying fields. Thank you.