 Welcome to the 36th lecture in the course Engineering Electromagnetics. In today's lecture we continue with our discussion on radiation and consider what are called the retarded potentials. You would recall that in the last lecture we saw that if the circuit dimensions become comparable with the wavelength then the circuit is likely to radiate. What we have in mind here when we say circuit dimensions is the separation between the electric charges and currents because these are what are the sources of radiation. Now beyond this qualitative explanation that when the separation between the sources of radiation fields that is the charges and currents becomes comparable to a wavelength then the radiation can take place. We would like to evolve a quantitative relationship between the sources that is the electric charges and currents and the radiation fields. How we go about this is as follows. You would recall that when we had simpler situation for example the static fields we could use Coulomb's law to say that the electric field due to an isolated point charge q is given by E equal to q by 4 pi epsilon r square times r cap where r cap is the unit outward normal from the point charge. If we have a number of such charges then we require to perform a vector addition of the electric field due to the individual charges to find out the total field at a given point. While this was okay if the charge configuration was simple the moment we had a more complicated situation for example charge distribution on some arbitrarily shaped body this approach was no longer feasible. And therefore what we did was we considered the potential V which was given by 1 by 4 pi epsilon and then we considered rho over an infinitesimally small element of volume d V divided this by r and then considered the integration of this kind of terms. And then in terms of this potential V we wrote E equal to the negative gradient of this potential function V. Since the summation here or the integration here involves a scalar quantity this was considerably simpler for an arbitrarily distributed charge. And then the electric field was obtained as the negative gradient of the potential function. In a similar manner coming to the magnetic field we could have used biot and saver's law to obtain the magnetic flux density B which is mu by 4 pi and then we have ideal cross r cap by r square. Where we can consider a current over some filament which is placed in arbitrary manner and then by integrating the contribution to the magnetic flux density at different points due to small small elements we can find out the total flux density at a given point. As a simpler alternative what we had was the magnetic vector potential A which was obtained as mu by 4 pi and then going to let us say a continuous charge distribution, continuous current distribution over a certain volume we could write J d V by r. In terms of which vector potential A we could write B as del cross A or the magnetic field intensity H as 1 by mu del cross A. The integral involved in the evaluation of the vector potential A is simpler than the integral involved in the direct evaluation of the magnetic flux density or the magnetic field intensity. And therefore, for even slightly complicated current distribution we used this alternative. When we want to consider a quantitative relationship between the charges and the currents and the radiation fields there also we have a similar situation. It is quite possible to evolve a direct relationship between the sources and the radiation fields. But since these sources charges and currents could be distributed in an arbitrary manner depending on the kind of antenna we take this direct relationship is going to be difficult to work with. And therefore, drawing an analogy from the simpler situations it will be easier to first obtain a relationship between these charges and currents and some potential functions. And then from these potential functions we will be able to calculate the radiation fields. So, that is the route we are going to take here. So, first of all we need to evolve some potential functions which will satisfy Maxwell's equations and will include the source terms without the sources there is no radiation. And therefore, we start with the Maxwell's equations. We write del cross h equal to epsilon del e by del t plus j. Then del cross e equal to minus mu del h by del t and del dot e equal to rho by epsilon and finally, del dot h equal to 0. Contained in these equations is the equation of continuity which reads as del dot j equal to minus del rho by del t. It is not an independent equation can be derived from the Maxwell's equations themselves. The way we have written the Maxwell's equations it is quite apparent that we have not specified any particular time variation. Secondly, the medium that we are considering is homogeneous and isotropic. So, that the constitutive parameters epsilon and mu are constant scalars again for the sake of simplicity. Finally, what are these J and rho? J and rho are the source terms which are going to cause radiation fields. Then what about the conduction current that may be present in a situation when the medium is conducting? That conduction current term will have to be written separately if the medium is conducting. For example, then we will need to write plus conduction current density which is sigma times e that is e times the conductivity of the medium. And many times to distinguish this J from the conduction current density that may be present one may write a subscript s here as well as here to indicate very clearly that these are the source current density and charge density terms. However, if we keep this at the back of our mind then we can simplify the notation and do not use this and for the moment we consider a perfectly insulating medium. If the medium is not so, then we will add that term and it will be possible to take it into account. So, these are the Maxwell's equations incorporating the source terms that is necessary otherwise we will not be able to relate the radiation fields to these sources. Now, how do we go about it? We can number these equations and let us consider equation 4. What does it say? It says that divergence of the magnetic field intensity is 0. When is that possible? When H can be expressed as the curl of a vector because from the vector algebra we have this identity that the divergence of a curl is identically 0. And therefore, we write H equal to 1 by mu del cross A. It is expressed as the curl of another vector and then equation 4 will be always satisfied. You would notice that the form of H for example, the factor mu has been chosen such that it is consistent with the static's expressions. Now, this becomes our equation 5. Now, if we substitute from 5 this expression for H in equation 2. So, equations 2 and 5 they will give us del cross E equal to minus mu del by del t of H which is 1 by mu del cross A. And by rearranging terms, we get curl of E plus del cross A. So, this is our equation 5. So, we will write H equal to minus mu del by del t of H which is 1 by mu del cross A. And by rearranging terms, we get curl of E plus del cross A. And by rearranging terms, we get curl of del A by del t which should be equal to 0. We have interchanged the order of the time derivative and the space derivative which is permissible as long as these operations are linear. So, therefore, we see that the curl of a certain quantity must be equal to 0. Then we recall another identity which is that it is the curl of a gradient which is identically 0. And therefore, the quantity within the brackets if it is expressed as the gradient of a certain scalar function, then this equation will always be satisfied. And therefore, we write E plus del A by del t equal to minus del V. So, that we have E as minus del V minus del A by del t which becomes our next equation. The choice of the negative sign here has been made so that the expression for or the relation between E and B remains consistent with the static or the low frequencies situation. For example, if in a situation time variation is not significant or this term becomes 0, then we go back to our statics relation that is E is minus del. It is for that reason that we choose this particular form for the potential function V. And therefore, we see that the magnetic field intensity and the electric field intensity can be obtained from these new potentials that we have obtained. These are scalar potential V and the vector potential A. The question that comes now is how are we to obtain the potential function V. And therefore, we see that the magnetic field intensity and the electric field intensity can be obtained from these new potentials that we have obtained. These are scalar potential V and the vector potential A. The question that comes now is how are we to obtain the potential function these potentials themselves. Once we have these potentials we can get these fields fine, but what about these potentials themselves. Then we notice that while we have utilized two of the Maxwell's equations, the other two we have not really utilized and those two will be utilized for obtaining an equation which these potentials are going to satisfy and the solution of those equations will give us these potentials. So, let us carry out this process on the other side. We substitute for these field intensities from 5 and 6 in equation 1 and in equation 3. So, equation 1 gives us del cross H. So, equation 1 gives us del cross H. So, that it becomes 1 by mu del cross del cross A which should be equal to epsilon times del by del t of E which is minus del V minus del A by del t plus of course, J. We can multiply through by mu and then express the curl of curl of a vector as del of del dot A minus del 2 A. Which should be mu epsilon and then gradient of del V by del t minus mu epsilon del 2 A by del t squared plus mu J, where terms can again be rearranged to read as del 2 A minus del 2 mu epsilon del 2 A by del t squared which is equal to mu epsilon gradient of del V by del t plus del of del dot A minus mu J. Similarly, we can write this equation as equation 3 is utilized and we get divergence of E which is minus del V minus del A by del t which should be equal to rho by epsilon. So, that we have divergence of gradient is the Laplacian del squared. So, that it is minus del squared V and minus divergence of del A by del t equal to rho by epsilon and we can change the sign on both sides to get us this. These equations can be numbered as 7 and 8. We still do not seem to be reaching somewhere, the problem is as follows. We have got equations which the potentials A and V should satisfy. However, both equation 7 and 8 contain both the potentials A and V. Therefore, these are coupled differential equations and that is quite troublesome to deal with. We like to have an equation in one single variable say V or A. Apart from this practical problem, the relations that we put down so far are very similar for A and V. Also, leave the values of these potentials A and V somewhat uncertain. For example, we have only specified the curl of A in terms of H. Similarly, it is only the gradient of V which has been specified and one could easily see that this is not a unique value. One could write V plus some scalar function S and this required equation will still be satisfied and therefore, we see that A and V are not uniquely specified. In fact, there is a theorem due to Helmholtz which is stated as follows. The theorem says that any vector field due to a finite source is uniquely specified if both the curl and the divergence of the field specified. This is due to Helmholtz theorem which states that any vector field due to a finite source is uniquely specified if both the curl and the divergence of the field are specified. And from this point of view, we see that there is a certain amount of non uniqueness about the vector potential A and as we pointed out about the scalar potential V. And also we have this problem of the coupled equations that is equations 7 and 8. These problems are removed to a large extent if we write down a relation which right now may seem arbitrary but the relation is as follows. Divergence of A is equal to minus mu epsilon del V by del t. It puts another constraint on the potentials A and V and therefore, the uniqueness will be improved. And the substitution of this condition in equation 7 and 8 shows that they become very well behaved they become decoupled. This condition is known as Lorentz gauge condition. Depending on the time available to us towards the end of the lecture, we will discuss both of these aspects the uniqueness as specified by Helmholtz theorem and the Lorentz gauge condition in a little more detail. Right now let us accept this and see what results it makes possible for us. When we substitute this Lorentz gauge condition in equation 7 and 8 what we get are the equations which read as del 2 A minus mu epsilon del 2 A by del t squared equal to minus mu j. On one hand and on the other hand we get del 2 V or del squared V minus mu epsilon del 2 V by del t squared which should be minus rho by epsilon. Very nice symmetric equations in A alone and in V alone. These equations we can number as 9 and 10. These look pretty much like wave equations with this source term included. So, we reached some distance we have got equations which potentials A and B should satisfy. Next we got to worry about what are the solutions of these equations. And then once again we take the help of the situation that pertains at low frequencies or where there is no time variation. I think we can remove these. When there is no time variation then the equations that we have are del 2 A equal to minus mu j and del 2 V equal to minus rho by epsilon the lower equation is the Poisson's equation. And the upper equation is the corresponding equation for the vector potential A. And we ought to be familiar with the solutions of these. We can consider a situation I mean for which system are we trying to solve this equation that system should be there in front. So, let us put down a system as follows. We take some arbitrary shape on which there is some current density j and charge density rho which are not going to be independent j and rho are never independent they are related to the equation of continuity which need not be uniform over the entire body. So let us fix up some origin and say that these current density and charge density are functions of the position vector r prime. And what we require are the potentials A and V at some field point P which field point is defined in terms of the position vector with respect to this origin r. And conceptually how do we go about this determination we say that we consider a small element of volume here. And if we say that this volume is V prime let this volume element be d V prime which is located at a distance capital R from the elemental source. And then we say that the vector potential A at the point P which is going to be a function of the position vector r of the point P is equal to the volume mu by 4 pi and then j which is a function of r prime d V prime by capital R. And similarly, the scalar potential V is going to be 1 by 4 pi epsilon and then rho by 4 pi epsilon mu which is a function of r prime d V prime upon capital R. Capital R is a variable and so will so is r prime. And therefore, they are appearing under the integration of psi. These are the solutions of these differential equations when there is no time variation. As a first attempt say first order or zeroth order solution let us assume for a moment that same kind of solutions are applicable to the equations that these potential satisfy when time variation is present. That is these sources j and rho are functions of time fine. So, that these become if we just extend these no time variation and now we say with time variation and what we do is we simply write the second variable T and say that the result that is the potentials are also available as a function of time in terms of some processing of the source terms which are functions of time fine. A very simplistic approach, but if it works it is fine. The question is does it work is there any problem in the way we have written these hypothetical solutions for a and b when the sources are time vary. We consider these equations now what are these equations indicating. They are indicating that if there is some change due to time as time instant changes in these source terms j and rho it is instantly at the same instant reflected or observed in the a and v which is holding good at a field point p. Whatever is the change in j and rho instantaneously it is available in a and v at the point p field point p irrespective of the separation or the distance of the point p from this source. Now that is not possible any fields that are established at point p due to radiation from the source has to be established through the mechanism of wave propagation and it is a fundamental concept of propagating waves that a phenomenon occurs at one place and then is repeated at other places at different instance of time and this time delay is proportional to the separation. Time delay and its proportionality with the separation is an integral part of propagating waves that is completely violated here. Whatever is the change here is instantaneously reflected at point p no matter how far away it is. So that certainly is not acceptable and once again being consistent with the way we have been proceeding somewhat intuitively somewhat rigorously we think of a way to modify the same expressions in a suitable way to remove this problem that we have discovered and this problem can be removed if we say that the effect of any time variation here is available at point p after a certain time delay or put in other words whatever is the variation we observe here at some time instant t is because of the variation in the sources at an earlier instant of time. And therefore we rewrite these expressions as follows a r t is equal to mu by 4 pi j r prime and now time variable we write as t minus r by v where the time that a propagating wave would take from each elemental source to travel to the field point is directly incorporated in this capital r is the distance and v is the velocity of wave propagation the velocity of light in the medium concern. So we complete this as d v prime divided by r and in a similar manner we get the expression for the scalar potential v as v of r t is 1 by 4 pi epsilon then rho of r prime t minus r by v d v prime divided by r. From this point of view the way we have incorporated a time delay or we have incorporated a certain retardation in these potentials these are called retarded potentials. Similar solutions to these equations are obtained if we proceed in a more rigorous manner without taking this somewhat intuitive route. Another possibility is to substitute back these expressions in these equations and satisfy oneself that these equations are also satisfied. We will not go into those two alternatives here we will assume that these are the solutions of the equations that a and v satisfy. One thing that I want to point out here is the time delay for a given field point is different for different portions of the elemental for different portions of the source or for different locations of the elemental sources. And therefore, right here one can visualize some kind of an interference pattern appearing in the final result for a and v for an extended source. Because the time delays from different elemental sources are going to be different and therefore, some kind of a pattern called radiation pattern for antennas is going to be there it is going to be present and it is nothing but a kind of an interference pattern. Constructive interference at some points and destructive interference at some other points depending on the relative placement of these sources with respect to the point p for the same position vector r. Next what we would like to take up is the extension of the same framework to sinusoidally time varying situations. Because that is what we use very frequently in practice we can leave the lower part and the upper part can be erased. Whenever we deal with sinusoidally time varying signals we can use the phasor notation conveniently and in phasor notation the equations 7, 9 and 10 become del 2 a plus omega squared mu epsilon a which should be minus mu j. Where now the quantities a and j are in phasor notation and in phasor notation each time derivative is replaced by the factor j omega. And similarly we have del 2 v plus omega squared mu epsilon v equal to minus rho by epsilon which become our equations 11 and 12. Once you substitute a particular form of time variation the wave equation changes to what are called the Helmholtz equations. We have seen this kind of equations earlier for example for wave guides and even transmission lines but the difference now is that the source term is included. And once again we put down a solution for this by the similar kind of intuitive reasoning and we say that in phasor notation the potential the vector potential a at a field point p which is defined in terms of the position vector r and other things remaining identical. So a of r is equal to mu by 4 pi then j of r prime upon r and what will be the equivalent of time delay in phasor notation that is put down as e to the power minus j beta r d v prime. In phasor notation a phase delay is identical or equivalent to time delay that can be seen in a straight forward manner. For example we consider the factor e to the power minus j beta r in phasor notation and if you want to incorporate the time variation explicitly we multiply it by e to the power j omega t and consider the real part which will give us real of e to the power j omega and then the real part. t minus r by omega by beta in this manner which can be written as cosine of omega times t minus r by v since omega by beta is the phase velocity. So the effect of adding a term like this delaying the phase like this is identical to delaying the time in phasor notation and therefore by incorporating this phase delay factor appropriately we can take care of the requirement of time delay proportional to distance as we discussed earlier. Similarly we complete by writing v is equal to 1 by 4 by epsilon then rho as a function of r prime by r e to the power minus j beta r d v prime. Of course it goes without saying that in all these integrals the integral is over v prime the extent of the time delay sources v prime everywhere. If we consider the phasor notation and the phase delay from that point of view these potentials can also be called delayed potentials. So it becomes retarded or delayed potentials both terminologies are used retarded potentials is the more common term utilized. So we have got the expressions for the fields as we saw earlier in terms of the potentials. We have got the equations these potentials should satisfy and we have got the solution for these potentials given a source distribution and therefore the framework of a quantitative relationship between these sources charge density and the current density and the radiation fields is complete. How would one apply it in practice? In practice one would go about it in the following manner we can make space here. The sequence of things will be to obtain the potentials. Now is it necessary for us to obtain both potentials A and V that is not necessary. For one thing the say the scalar potential V is available in terms of the vector potential A. So a separate evaluation of V is really necessary is in practice not done. So the steps are as follows we calculate the vector potential A following this relation which is the sum of our work today. So let us put down these as equation numbers 13 and 14. So using equation 13 we can calculate the vector potential A given the current distribution then we calculate H which is 1 by 8. So mu times the curl of A then as far as the calculation of E is concerned we have several alternatives. The simplest one is to use the Maxwell's equation which reads as del cross H equal to epsilon del E by del t. Now you would ask that what happens to the current density term here. So we say that as far as the region away from the source is concerned where the source is not specifically included E and H are related through this. And therefore once the effect of the source is included in H and if we calculate E from that H we are relating both A and H to that source term. Otherwise we can write these various equations in many different ways. We can convert this to phasor notation it is possible to express E also entirely in terms of the vector potential A all those things are quite possible. But usually this is the way we go about calculating the radiation fields given a certain source distribution. If you have any questions we can take up those at this point of time. Otherwise let me quickly justify the Lorentz gauge condition. It turns out that it is completely consistent with the equation of continuity. For example if we consider equation 9 and calculate divergence of J by taking the divergence of equation 9. And then rearranging terms. And we use equation 10 to calculate del rho by del t by taking the time derivative of the entire equation 10. And if we add these up del dot J plus del rho by del t and use the Lorentz gauge condition. The relationship between the divergence of A and the time derivative of V then the result comes out to be 0 as required by the equation of continuity. And even otherwise the results that we calculate have been found to be quite sound and consistent in practice that is experimentally. And therefore we accept this Lorentz gauge condition that is where we stop today. In the next lecture we shall consider the utilization of these expressions to calculate the potentials A and V and the radiation fields for a very simple source. Thank you.