 lecture in the course engineering electromagnetics. In this lecture, we start our discussions on the important topic of wave propagation. We start discussions on this topic with the help of the following concepts. That is, we start with a review of Maxwell's equations and then go on to consider constitutive relations and then the wave equation. You would recall that during the introduction to this course, we identified a number of important phenomena and we call these action at a distance phenomena mainly related to electricity and magnetism. And we said that it was in connection with these action at a distance phenomena to explain which the concept of fields was evolved namely the electric field and the magnetic field. And we also said that the fields point of view is more general and more accurate than the voltage and current point of view in many situations. Therefore, we start our discussion on the fields today. The first question that would come to mind is what are the laws obeyed by these fields electric and magnetic fields? The answer to this question is very simply that the laws that these fields obey are collectively known as Maxwell's equations. The Maxwell's equations consist of a number of sub equations so to say which are essentially laws proposed by various scientists and explorers. They are also known by the name of the person who proposed these first. Then why is it that this set is known as Maxwell's equations? The set is known as Maxwell's equations because it was Maxwell who established these equations as a consistent set first of all. And he contributed in a unique way to the development of these equations. And that is why the set of these equations is known as Maxwell's equations. These equations were proposed in 1870. Of course, it must have taken time to evolve these. These were proposed somewhere around 1870. And based on these equations Maxwell predicted the phenomenon of wave propagation without the aid of any physical media. This was a very drastic point of view, change from the points of view prevailing at that time. And these concepts, these predictions were verified much later experimentally by the unique experiments of Hertz. In about 1888 these concepts were verified. This was a very unique situation in the development of science where the theory was proposed first and the experimental verification came later. And ever since the Maxwell's equations got established the world has not been the same again. It is been changing at a rather rapid rate. And it will be worthwhile to consider these equations. We write down the Maxwell's equations one by one. We start with the equation which in the differential form reads as follows. del cross h equal to del d by del t plus j. It is the first term del d by del t, the displacement current density term which was introduced by Maxwell, which made all the difference. As far as the interpretation of this equation is concerned, it is easier to consider the corresponding integral form by taking the surface integral on both sides. And then utilizing vector algebra theorems such as Stokes theorem, one gets h dot d l over a closed path equal to surface integral of del d by del t plus j dot d a which is a surface integral. Of course, the special case when the time variation is absent, the so called static case can be obtained very simply and it would be del cross h equal to j. This equation is known as a law. What is that law? Ampere's law including the displacement current density term as proposed by Maxwell. What are the various quantities appearing in this? First is h which is the magnetic field intensity with units of amperes per meter. It is a vector quantity. D is the electric displacement density or simply the displacement density with units of coulombs per meter square charge per unit area. The third quantity that appears in this law is the conduction current density or the current density with units being amperes per meter square. And one can see at a glance that the equation is balanced from the points of view of units of various quantities. What is the interpretation of this equation? On the left hand side what we have is the line integral of the magnetic field over a closed path. This line integral of the magnetic field over a closed path we call magneto motive force or in brief M M F. So, the interpretation runs as follows. The M M F the magneto motive force around a closed path equals the total current. The total current includes the displacement current and the conduction current or any other current that may exist. For example, the convection current term which is not written here, but if it exists in some problem in some situation it will have to be taken into account here. So, we say that the M M F around a closed path equals the total current through the surface bound by that path. Now, one could say a little few words about the nature of the conduction current density J. What could be the agencies causing the presence of a conduction current? It could either be a source for example, an antenna on which some time varying electric charges and therefore, some electric current flows or even in the absence of a source even in the absence of an antenna the medium itself could permit some conduction like a lossy dielectric in which case also there could be some conduction current or we could be considering wave propagation in a good conducting material where again the presence of conduction current is very easily visualized. And therefore, this J the conduction current density could arise because of a number of reasons. This will have a bearing on what we do later on. Next we take up what is known as the second equation of Maxwell. There is no hard and fast rule about the sequence, but here we will be calling this usually the first Maxwell's equation. And then the second one which reads as del cross E equal to minus del B by del t. And let us make some space here. And since it is easier to interpret the corresponding integral form that is what we write. And we obtain E dot d L over a closed path equals minus del by del t of P dot d A where on the right hand side the integral is over a surface. The corresponding static form is del cross E equal to 0 based on which one says that the static electric field is irrotational because its curl is 0. What is this equation known as this is the Faraday's law Faraday's law of electromagnetic induction. The various quantities appearing in this are the electric field intensity E with the units of volts per meter and B which is the magnetic flux density. What would be the units of the magnetic flux density? Webers per meter squared in honor of the great scientist Weber or volt seconds per meter squared. If you work out the units of the magnetic flux based on the Faraday's law of electromagnetic induction straight away or the CGS units having Gauss. Here we would be using Weber's per meter squared as a unit for the magnetic flux density. What is the interpretation for this law this equation of Maxwell? It runs very similar to the interpretation of the first equation that we dealt with. Here we have the line integral of the electric field around a closed path and therefore we say that the EMF electromotive force around a closed path equals the negative time rate of change of the total magnetic flux through the surface bound by the path. Now we do not need this anymore. Now we are familiar with the applications of the Faraday's law of electromagnetic induction and we know that the total flux may change may have a time rate of change either because the flux density itself is changing or because there is some motion involved. So both those possibilities are there. Here in this course usually we will be considering stationary items and therefore it will be the time rate of change of the magnetic flux which will be causing some generation of EMF. The third equation reads as del dot d equal to rho. The corresponding integral form obtained by taking the volume integral on both sides and then applying the divergence theorem will give us d dot dA over a closed surface is equal to rho dV where the right hand side integral is a volume integral. The corresponding static form is not different del dot d remains equal to rho. This is Gauss's law. The new quantity that has appeared here rho it is the charge density with units of coulombs per meter q and the interpretation or the statement of Gauss's law runs as follows. The total electric displacement through the surface enclosing a volume equals the total charge within that volume the statement of Gauss's law and particularly in electrostatics we have seen some applications of this law where one can make out the value of the electric displacement density and therefore the electric field intensity for some symmetric charge distributions in a very simple manner. Next we consider what is the final equation in the set which is del dot B equal to 0 and one can write the corresponding integral equation for this one also. Reading as B dot dA is equal to 0 and the corresponding static equation which reads just the same and one may call this the Gauss's law for the magnetic flux and actually this is the alternative statement for the fact that isolated magnetic poles do not exist and one says that the total magnetic flux through a closed surface equals. So, these are the four equations known as the Maxwell's equations and although in some form or the other they were available before Maxwell or they were available to Maxwell but they were available in a rather isolated and disjoint form and it was Maxwell who unified these and established these as a consistent set and hence so much of credit associated with. These equations include so to say what is called the equation of continuity which reads as del dot J is minus del rho by del t which can be obtained from the first equation and the third equation. One takes the divergence of the first equation the divergence of a curl is identically 0 and then one applies the third equation and that leads us to del dot J equal to minus del rho by del t which is the equation of continuity from which under special circumstances one can derive Kirchhoff's current law. One can write the corresponding integral form and the static form in a simple manner. So, these are the Maxwell's equations which will be our tools for understanding and introducing the phenomenon of wave propagation. The phenomenon which is the basis of all broadcast all wireless communication etcetera but before we can do that we need to simplify these equations a little bit since these contain a large number of variables. And also we are aware that the flux densities and the field intensities are not independent they are related through the properties of the medium in which the flux densities and the field intensities exist. And that brings us to the second sub topic of today's lecture which is the constitutive relations. The constitutive relations help us in incorporating in Maxwell's equations the properties of the medium. These constitutive relations you would be familiar with these read as D equal to epsilon E, B equal to mu H and J equal to sigma E where these epsilon mu and sigma are the permittivity the permeability and the conductivity of the medium. We are dealing with units of Farage's per meter, Henry per meter and Simon's per meter or Mohr's per meter. Now they could be in general very complex situations but to keep things simple one makes a number of simplifying assumptions which also are obtained in many important situations. So they are not completely fictitious or unreal. The simplifications we make are the following. First we consider that the medium we are dealing with is homogeneous which would enable us to consider the values of these constants of the medium epsilon mu sigma as constant everywhere. They will be uniform everywhere. So everywhere these will be constants. It is very easy to visualize situations where this situation does not hold good. For example when a wave is propagating close to the surface of the earth and then the earth and air epsilon and sigma are different but yet this kind of situation one meets with many times. So we assume a homogenous medium permitting us using a constant value everywhere of epsilon mu and sigma. The second simplifying assumption that we make is that we are dealing with isotropic media. What does that mean? Isotropic means something which has the same properties in all directions. By virtue of this assumption we are able to use epsilon mu and sigma as scalars. Of course for a homogenous medium everywhere it will be a scalar quantity. Can there be an isotropic media also? That is for which the permittivity or permeability or conductivity are different in different directions. Certainly they could be dielectric anisotropic media or magnetic anisotropic media. One could consider simple examples. For example the ionosphere through which the signals to and from the satellite pass or from which the short wave signals reflect back towards the surface of the earth. It is an anisotropic dielectric media. Similarly the ferrites are magnetic anisotropic media and they have their own special applications. In such a case when one has anisotropic what will happen to these constitutive parameters? For example if epsilon if the dielectric is anisotropic then one would need to write D equal to epsilon e where epsilon is a tensor. And it will read as equal to epsilon x x epsilon x y epsilon x z epsilon y x epsilon y y epsilon y z and epsilon z x epsilon z y epsilon z z in the most general case. Depending on the choice of coordinate axis one may have only the diagonal terms non-zero or one may have all terms non-zero. Similarly for permeability or conductivity depending on which way the medium is anisotropic. In which case one can readily see that the situation will be much more complex and therefore here will be restricting ourselves to isotropic media. The third simplifying assumption that we make is that we have a source free situation that is the region that we are dealing with is free from sources that is there are no there is no charge density or there is no impressed source current density. The medium itself may have some conduction current density under the influence of an electric field that is a different matter. But the region itself we shall consider a source free. This again is a simplification is not going to hold good when we are specifically dealing with sources for example antennas. In that case we will take the antenna charge density and current density into account. But away from these sources away from antennas we can safely consider a source free medium. In which case these equations simplify and these become del cross H equal to del D by del T, del cross E equal to minus del B by del T and del dot D equal to 0 consistent with our assumption of source free medium and of course del dot B is equal to 0. So, with the help of the constitutive relations we incorporate the medium characteristics and then we further simplify the situation for ourselves by assuming that we are dealing with source free situation. Now, one can incorporate the constitutive parameters constitutive relations in these relatively simplified set of Maxwell's equations. We can make space here and now these read as del cross H equal to epsilon del E by del T and the moment we write it in this manner it should be obvious clear that we are making the assumptions of a homogeneous medium and isotropic medium. Otherwise as we have just discussed they will be more complicated relationships. Similarly del cross E is equal to minus mu del H by del T and then del dot E is equal to 0 or del dot D is equal to 0 it does not make a difference under the source free assumption and del dot H is equal to 0 which is the set of equations we will take as the starting point for introducing wave propagation. Now, the various assumptions that we made hold good under free space conditions. Therefore, the solution that we will get will be the solution under free space conditions or alternatively a perfect dielectric perfect homogeneous dielectric which free space is and of course is the if the dielectric does not happen to be perfect if it does have some loss then we have seen that through the use of the complex permittivity concept we will be able to generalize the solution that we get for the lossless case. So, that is not too much of a restriction we can number these. So, that one can refer to these conveniently and even with the substitution of the constitutive relations we find that it is still quite like a maze and we will like to evolve one equation preferably which governs the behavior of the electric field and say another equation which governs the behavior of the magnetic field. So, that is what we attempt next and this attempt will lead us to the wave equation. We consider the first two equations and depending on how we process these we will either get an equation governing the behavior of just E or an equation governing the behavior of just H. We start with the second equation and take the curl on both sides. So, that we have curl of curl of E equal to minus mu curl of del H by del t. The curl of curl can be written as gradient of divergence del of del dot minus the laplacian E. We can say that this is a vector identity curl of curl can be curl of curl of curl twice curl can be expanded in this manner. On the right hand side we interchange the sequence of the spatial derivative and the time derivative. As long as we are dealing with linear media linear situations the sequence of the derivative is not going to matter. So, that is another assumption we are making. And under the simplifying assumption we obtain minus mu del by del t of del cross H. The first term by utilizing the third equation is 0. So, one is left with only the second term as far as the left hand side is concerned. And then on the right hand side one substitutes from equation 1. And one sees that we get del 2 E equal to mu epsilon del 2 E by del t square no minus will cancel out. Which is an equation containing only one variable the electric field and of course, the parameters of the medium mu and epsilon. By virtue of our various assumptions the medium that we are considering is non conducting. Therefore, sigma is not appearing in this equation. And when we consider the solutions of this it will turn out that these solutions represent propagating waves. And accordingly this equation is called wave equation. Now, since the first and the second equations are completely symmetric depending on which equation we start with we will get the wave equation for the magnetic field. And that will also be very similar del 2 H equal to mu epsilon del 2 H by del t square both of which become the wave equations. Now, these appear very compact because of the use of the vector notation. Let us consider for a moment what does the left hand side imply if one were to expand this for example, in Cartesian coordinates. Now, the left hand side would imply del 2 by del x square plus del 2 by del y square plus del 2 by del z square. And let us say that we consider the x component of the first vector equation. So, that it is E x on this side. So, this will be equal to mu epsilon del 2 E x by del t square. Which is the which is one of the three component equations corresponding to the electric field wave equation. This immediately brings out the power of the vector notation. Similarly, there will be two other component equations concerning the y component of the electric field and the z component of the electric field. And similar three equations related to the magnetic field wave equation. And therefore, to be able to make out something one needs further simplifications. Simplifying assumptions which are not completely meaningless that kind of situations are obtained in practice. The simplifying assumptions that we make are the following. We say that let del by del y and del by del z both be equal to 0. That is we consider a situation where there are no variations with respect to the y or the z coordinate. A drastic simplification on the face of it, but that is the kind of waves which exist far away from sources. So, simple and yet a practical situation. What would remain only the variation with respect to x would remain. And we focus attention on let us say the y component of the electric field. Then the z component of the electric field wave equation will yield del 2 E y by del x square equal to mu epsilon del 2 E x by del t square sorry E y here also. It is the y component that we are considering under the assumptions that there is no there are no y or z variations. And we are considering the y component in this vector equation del 2 E y. Thank you. Now, this is a simple second order partial differential equation. We have handled this kind of equations earlier particularly in the case of the transmission line. Where instead of E y we had the voltage variable or the current variable. And the solutions can be written by inspection. What will be the solutions like? We will have E y equal to some function let us say f 1 of x minus say v naught t. And similarly another function f 2 of x plus v naught t. Where v naught is nothing but 1 by square root of mu epsilon by substituting back either of these in the original equation. One can satisfy oneself that these indeed are the solutions of the starting equation. And these functions of the arguments of this kind are known to be representing propagating waves. One can consider the behavior of one of these functions on the overhead projector. And with the difference that here we are using z instead of x this situation is identical to what we considered for the transmission lines. If we consider the behavior of the function at some time instant t equal to t 1 it may appear like this. At another time instant t equal to t 2 it may look like this. And therefore, in the duration t 2 minus t 1 this disturbance has traveled further away along the direction of propagation. And therefore, this disturbance is traveling in the positive z direction. And this fits in with the definition of a wave which we have seen earlier also which we have seen earlier also. We said that if a physical phenomenon that occurs at one place at a given time is reproduced at other places at later times. The time delay being proportional to the spatial separation from the first location then the group of phenomena constitute a wave. The solution that we have got now fits this description and therefore, these solutions also represent propagating waves. This function represents a wave propagating in the positive x direction. We have deliberately chosen x variable this time to indicate that waves could propagate in any direction. They do not always have to propagate in the z direction. And this function represents waves propagating in the negative x direction. Usually this will be the wave propagating away from the source and this will be the wave propagating towards the source depending on our choice of coordinate system. And depending on the boundary conditions this will have an appropriate magnitude and phase. Or it may not even exist depending on the boundary conditions. And therefore, since the general solution of this equation represents propagating waves we call these equations as wave equations. The solution that we have got now under the simplifying assumptions del by del y and del by del z equal to 0. As I said is of considerable importance and this solution is known as uniform plane waves. And far away from a source most waves can be considered if nothing else at least locally as uniform plane waves. For example, the signals that we receive from satellite or the signals that we send to a satellite or in general signals far away from an antenna would satisfy the description would meet the description of uniform plane waves. So, if you have any questions we like to take those now otherwise we will stop here. Then we stop this lecture here.