 Welcome to the 35th lecture in the course engineering electromagnetics. Today we start the last major topic in this course and that is radiation. The topics we consider today are the radiation mechanism, importance of antennas and then we work out a definition of antennas. You would recall that in the beginning of the course we said that there are a number of phenomena which have important applications which can be described only in terms of the associated fields. What we have in mind here are the electric and the magnetic fields, wave propagation and radiation are two such very important phenomena. In the rest of the course so far we have been talking about wave propagation and now we start some discussion on radiation. The word radiation has appeared in our discussions earlier, for example if we said that if we increase the frequency of operation of a transmission line circuit it is likely to radiate and therefore waveguide circuits are better and so on. Therefore it would appear that there is some relationship between the phenomenon of radiation and the frequency of operation or in other words the wavelength of operation. In any case we have seen that the time delay effects are related to the frequency or the wavelength vis-a-vis the dimensions of the circuit and therefore radiation also is not something which is mysterious and magical, one can understand this in simple terms for example the relationship of the circuit dimensions with the operating wavelengths. For this purpose let us consider the following example, we consider the familiar transmission line, two wire transmission line with spacing between the conductors as S, the individual conductors having a diameter D and it is connected with some source of time varying signal. At the far end the transmission line is open circuited and therefore standing waves are going to be formed on this transmission line and the current wave form is shown here over some length of the transmission line, it has a sinusoidal variation. The conductors carry currents which are equal and opposite in the two conductors, for example if the direction here is this way then the direction of the current here is this way and then of course it changes direction every lambda by 2 distance and therefore the current direction is going to be according to the bigger arrows that we have just put. And from the point of view of continuity we are going to have an equal amount of positive and negative charges or positive and negative flowing currents and after all if any radiation is to take place the source of this radiation should be these charges and currents and therefore the situation that we have now is that we have equal amounts of positive and negative charges and equal amounts of currents flowing in the two directions in the forward direction and in the reverse direction. As long as the spacing between the conductors S is much less than a wavelength these possible sources of radiation are spaced very closely together and their overall effect is nullified quite effectively. To understand this better we can consider the example of an electric dipole where of course time variation is not present but otherwise the fact that the fields become very small for very closely spaced sources can be brought out clearly. We consider the electric dipole with equal and opposite charges with a very small spacing between them such an arrangement is an electric dipole that you would recall that we worked out the electric field for such a system by first considering the potential at a point and then considering the negative derivative the gradient of that potential. As a result of this in terms of the coordinates r and theta we obtain the expressions for the radial component of the electric field and the theta component of the electric field. Now the actual expression is not of much interest to us right now it is enough to recall here that these field expressions had a variation with respect to r which was 1 by r cube both of these field components. And therefore the moment r assumes some significant value the value of these field components becomes quite small and this happens because we have equal and opposite sources charges placed very close together we work out these fields in the approximation that l is very small and therefore going back to our example we can say with more confidence that in such a system where the conductor spacing s is much less than a wavelength there is not going to be any significant amount of fields at significant distances from the system fine. Now therefore if we consider the situation change situation when the spacing between the conductors is gradually increased deliberately alright then although the currents and charges became remain equal and opposite but the spacing between them is now more and continuing this process of increasing the spacing further one may obtain a configuration which looks quite like an antenna a dipole antenna is obtained where depending on the length of these arms these spacing between the charges and currents can be quite significant it can be of the order of a wavelength for example for a half wavelength long dipole and then it is quite likely and it is seen in practice that such a system can radiate it can have fields which reinforce each other may be in some direction not in all directions but there is likelihood of fields being quite strong at considerable distances from this system alright. The next question that may come up is that alright the field is strong at sufficient distances from the system because the spacing between the sources has been increased and it has become of the order of a wavelength but how does the field radiate how does it detach itself from the system okay so very qualitative empirical kind of picture can be considered for this purpose and we considered the flared out system that we just saw okay let us say that this is the dipole antenna fed by a transmission line alright and we consider that the voltage that is applied to the dipole antenna is let us say of the form as a function of t so that it looks like this so any certainly time varying voltage is applied so that this is 0 this is t by 4 and this is t by 2 and so on where capital T is the time period of this sinusoidally time varying signal considering the first quarter of the time period time from t equal to 0 to t by 4 let us say that the upper conductor is at a positive potential with respect to the lower potential that is the way the voltage is applied. So that the field lines electric field lines are going to look like this and as time advances these field lines will propagate in the space beyond this system and depending on the velocity of propagation or the wavelength they will have traveled by a distance lambda by 4 away from this system in this direction going from the upper conductor to the lower conductor. These field lines will continue to propagate forward as the time advances however from this time instant onwards the polarity or the voltage sign of voltage between these two conductors is going to change and therefore the new lines that start are going to have an opposite direction the nature is going to be similar but the direction is going to get reversed and keeping these things in mind the picture that one may have at later instant of time say at small t equal to capital T by 2 is going to be like this where the field lines that started in the first quarter of the time period are going to propagate up to a distance lambda by 2 with the polarity as it was earlier and new field lines corresponding to the next quarter of the time period are going to be initiated they will have traveled by a distance lambda by 4 but they will have an opposite polarity to keep a distinction between the field lines corresponding to the first quarter and the second quarter of the time period we have drawn these as continuous and dashed lines. The polarity of these is opposite to the first set of field lines now there is a slight inconsistency what is the charge sign of charge here and here it cannot have field lines coming to this as well as starting from this and therefore actually what is going to happen is that the field lines will detach themselves from the structure which now becomes an antenna and they will form this kind of closed loops. And once we have closed loop kind of electric field lines there is a non zero value of the curl of the electric field and therefore there is going to be some time varying magnetic field also and so on and this way a propagating wave will be emanated from this structure and the process will continue as we consider more and more advanced instance of time. So very qualitatively very simply this is the picture that one can have in mind that how the radiation detaches itself from a structure which is likely to radiate which is going to happen only if the spacing between these possible sources of radiation is considerable in terms of wavelength. And therefore we say that the radiation is not something which is mysterious it is something which can be engineered you could manipulate the transmission structure so that radiation is increased. We adopt that point of view when we want something to radiate when we want to accomplish something through that radiation of course there could be undesired unintentional radiation also that also will take place when the circuit dimensions become comparable to a wavelength. When we have undesired radiation we have problems of electromagnetic interference and electromagnetic incompatibility as the clock rates and the data rates are increasing this problem is now increasing more and more in the circuits that we use in computers and communication systems that is the unwanted undesired radiation here we shall focus on the desired radiation that is the radiation which we arrange on antennas. When we come to antennas it is useful to say a few things about the importance of antennas which will come from the kind of applications antennas have or are likely to have. As far as the applications are concerned we should have no doubt that they have very important applications some of the antennas are a part of our daily life for example the television receiving antenna or the antennas that the police patrol carries on mobiles and jeeps and more recently the parabolic the parabolic reflector antennas put up by the cable TV operators earlier similar antennas but of smaller size were seen for microwave line of sight links. So these antennas shapes are very familiar to us in fact there are situations where one has to use an antenna these situations are not difficult to identify the moment in a communication system one of the two ends becomes mobile or tries to become mobile we cannot have a hard wired communication system and such a situation arises all too often we could consider bigger systems like communication with a ship from shore or communication between ground and aircraft and so on and more commonly even individual customers now like to be in touch while on the move and therefore whenever we have a mobile situation antennas have to appear in the communication system. Another situation where antennas become almost indispensable are when the information from the same source is desired to be disseminated to a large number of people the broadcast situation then also antennas are clearly unavoidable. In fact one could generalize and say that in general antennas serve as the eyes and ears of communication systems now some types of antennas have been mentioned earlier but these are not the only types of antennas we have a large variety of antennas existing some of which we may have seen some of which hopefully we will see later but let me quickly show you some common antenna types so that you know that the subject matter that we are dealing with right now is going to be helpful in describing this kind of structures. First of all we have the dipole antenna that we just saw which can be considered to be derived from a transmission line and many of these antennas could be considered to be extensions of transmission structures where the spacing between the sources has been deliberately increased the dipole antenna is a very clear example of that we could also have the loop antenna circular or square loop antenna which is used as a direction finding antenna very commonly we can have what is called a helix antenna these antennas that we have just seen are constructed out of wire conductors and therefore they are also called wire antennas alright and this category of wire antennas in itself as a separate technique of analysis therefore we say that this is the category of wire antennas another category that one can see and one is somewhat familiar with is this kind of antennas you are quite familiar with the pyramidal horn antenna derived out of a rectangular wave guy alright once again this is an extension of the parent transmission structure where the aperture is flared out right you could have the horn derived from a circular wave guy we could also cut a slot in a metal flange which can be excited with the help of a coaxial line in this kind of antennas the radiation appears to be coming out of an aperture and therefore these are called aperture antennas and there is a separate set of techniques for dealing with aperture antennas then we could have combinations of these individual antennas for example we could form arrays of linear antennas one such array which is very useful is the yagiuda array where we have one single element which is fed therefore this is called the driven element and the other elements are they support currents due to mutual coupling and therefore they become parasitic elements some service directors some other service reflectors you can make out that television receiving antenna is a simpler version of this kind of yagiuda array we could put radiating wave guides close together to have some desirable properties which a single radiating aperture may not have then we could cut slots in wave guides and as long as they disturb the current lines these slots are going to radiate and you would notice that there is no slot at the center of the wave guide that would not radiate alright there are other details of this that we can consider later and then we have the more familiar paraboloidal reflector antenna which is excited by a smaller antenna kept at its focus this may be a small wave guide horn acting as the feed we could also have antennas at microwave frequencies which similar to their optical counterparts for example a lens antenna could be used where this is a low loss dielectric material and it will have a function which is very similar to that of lenses used at optical frequencies having talked about the basic mechanism of radiation and the importance of antennas and some commonly seen antenna types and having mentioned that there are some situations where antennas are unavoidable we now consider situations where although it is not necessary that an antenna is used but it may become profitable or more economical to use the antenna right and therefore we consider the communication systems divided into two broad categories they can be divided into many categories depending on the point of view the point of view here will be clear in a minute the in the first category we put systems which consist of a transmission line network it is an interconnected network of transmission lines the examples are not far to see telephone system is the ubiquitous system cable television more recently and also relatively recently would be the local area networks if they use hardware interconnections many of them do so this is one kind of communication systems this is characterized by very high population density per unit area you will have a large number of consumers or customers whatever you prefer to call these and it is quite economical to interconnect these although through the exchange but in a telephone network each consumer is connected to every other consumer so there is a hardware connection or route available between different consumers so also is the situation for the cable television distribution network let us consider some basic characteristics of this kind of system based on interconnection of transmission lines for the telephone network what is the transmission medium we utilize we use a pair of twisted wires briefly called a twisted pair the bandwidth of the signal that this transmission medium supports is roughly 4 kilohertz usually it is supposed to be 0.4 to 3.4 kilohertz we just to take a figure we say 4 kilohertz and for such a signal for such a frequency the medium offers an attenuation which is typically 2 to 3 dB per kilometer alright similarly for the cable television network since the frequencies are high we use a coaxial cable depending on the number of channels the highest frequency could be tens of megahertz and since we are using a better transmission line so although the frequency is high the attenuation is not too high it is of the order of 4 to 5 decibels per kilometer which attenuation figures will change if we use some other transmission medium for example optical fiber would offer much lower attenuation however the point that we want to make here is that the attenuation behavior is exponential the signal level is going to vary as e to the power minus alpha times d where alpha is the attenuation constant and d is the distance what is the effect of this exponential behavior if we take an example of a system where let us say for a 20 kilometer distance the attenuation is 100 decibels which in a coaxial cable like this is quite typical if we want to increase the operating distance between the transmitter and the receiver by another 20 kilometers then we have to be prepared to encounter and somehow take care of another 100 dB of reduction in the power level and when we recall that 100 decibel means reduction in the power level of 10 to the power minus 10 that is a very large amount of attenuation and it is quite clear that one will require frequent repeater amplifiers so that the signal level keeps above the noise level which is always present in a communication system and therefore this kind of behavior is always going to pose a limitation on the maximum repeater less distance that one can have in such a system and therefore the conclusion as far as this category of communication systems is concerned is that the attenuation behavior is exponential which the communication system designer has to take into account and is going to increase the cost of the system considerably what is the alternative the alternative is that you have a system where you do not use a network of transmission lines use a system which is based on electromagnetic radiation that is at the transmitter end you have an antenna and at the receiver end you have an antenna right now is that going to have a behavior which is different or better than the exponential attenuation behavior we have just noticed in fact at the outset a little bit of thought will show that there is a significant amount of coupling loss between the transmitting antenna and the receiving antenna let me explain this coupling loss as follows let us consider an antenna which say looks like this just to consider a typical antenna which is likely to be familiar and it is going to radiate let us say majority of its power in a narrow angular region which decides the beam width of the antenna to not to load the situation too much against the antennas let us say that the beam width is quite small let us take this beam width to be say 10 degrees or so where the beam width would be the angular separation between the 3 dB points below the maximum radiation all right even with this beam width which depending on the point of view can be considered quite narrow the wave front is going to keep on expanding and for a distance of 6 kilometers since this is roughly one sixth of a radian the wave front dimensions would be about one kilometer by one kilometer now 6 kilometers is not a very large distance and we are considering a fairly narrow beam width antenna and yet the power radiated is distributed over a large area and we cannot by any stretch of imagination hope to have a receiving antenna which will capture this entire power at the most you can build antennas which are a few meters squared in area all right and the antenna cost increases as the size increases and therefore, what we are going to capture is a very small fraction of this radiated power and therefore, we say that as we have projected in the beginning itself when we consider a system based on electromagnetic radiation and we consider some other receiving antenna located at some distance there is a significant amount of coupling loss which will depend upon the beam width of the transmitting antenna then how can such a system be viable it becomes viable from the following point of view the consider the way the power density changes you we can consider let us say some particular power density at say d equal to 6 kilometer what is going to happen to this power density at d equal to 12 kilometers we have double the distance what we expect is that from the same kind of considerations the area over which the radiated power will be distributed now will become 4 kilometer squared and therefore, we say that the power density has become one fourth the power density at half the distance and therefore, we can generalize that the power density varies as 1 by d squared the signal level is not going down exponentially it is going down by 1 by d square for a doubling of the distance the signal has become 1 by 4 times which is a reduction of 6 decibels this corresponds to for this example a reduction of 6 decibels and therefore, to have a comparison let us have a system here antennas and the operating frequency and the separation so chosen that for a distance of 20 kilometers the coupling loss is 100 decibels and now we double the distance we make it 40 kilometers and what is the change in the signal level or what is the total reduction in the signal level that will only be minus 106 decibels instead of incurring another 100 decibels of reduction we incur only a 6 dB reduction in the systems based on electromagnetic radiation or in other words systems which use antennas and therefore, we make another significant point here as far as the second category of communication systems is concerned and that is the attenuation behavior is algebraic for every doubling of distance you keep reducing the signal level by 6 decibels and therefore, it is quite conceivable that beyond a certain distance loss in a communication system using electromagnetic radiation will be increasingly less than with transmission lines alright and it is quite conceivable that a system based on electromagnetic radiation will have a lower cost or less complexity compared to a system entirely based on transmission lines. Now where this breakeven point is going to occur will depend upon the transmission medium that we consider for the transmission line based system and the kind of antennas and frequencies that we consider for the second category as I mentioned if we use a transmission line medium which has very low loss for example, optical fibers which are touching 0.2 decibels per kilometer loss the equation changes drastically but from a conceptual point of view we see that a system based on electromagnetic radiation can be competitive can be considered can be a very good competitive fine. The next thing that we will like to do is to evolve a working definition of antenna which will appear when we consider the functions of an antenna what are the functions of an antenna the examples that we have considered they show that we can have antennas serving the purpose of radiating or receiving electromagnetic energy however we mentioned that they could be undesired an international radiation from a system and that could also satisfy this function but we would not call that an antenna the next thing that one could try out is that antennas act as a directional device they try to direct the radiated energy into a certain narrow angular zone and what is true of transmission is also true of reception they will receive radiation preferentially in that same narrow angular zone however there could be antennas which do not have this kind of directional behavior for example the simple antennas that you use on the cordless telephones they work in any direction and they are not really directional devices alright and therefore this also does not appear to be a functional aspect which can be used as a defining function of the antenna and then we have to look for something else we say that what is the purpose of the antenna the purpose of the antenna is to radiate and to radiate efficiently so supposing we just connect a transmission line or even a wave guide we just leave it open circuited say like this there is a source here and you could look at this as a transmission line or as a wave guide and this is just let left open circuited in free space then you know that there is going to be reflection at this discontinuity which is going to be given by the reflection coefficient representing the reflection coefficient by gamma which will be from the relation ZL-Z0 upon ZL-Z0 where the role of ZL will be played by the intrinsic impedance of the medium in which the structure has been left open usually that of free space but it could be some other medium as well and the role of Z0 will be played by the characteristic impedance of the transmission line or the wave impedance of the wave guide and from this point of view we find that normally the characteristic impedances or the wave impedances that we use from some other practical considerations are different from the intrinsic impedance of free space and such a simple system is not going to radiate efficiently even if we increase the separation in terms of wave length alright and therefore we have to do something to reduce this mismatch and from that point of view we could consider that an antenna is a kind of a matching device between a region where there is a guided wave and a region where you have free space and that becomes a defining aspect of antennas which is applicable to any entity any structure which is acting as an antenna we can say that its antenna is the transition device is the transition region between the guided wave and the free space we can take up here any questions or clarifications that you may have yes this one yes no we are not considering whether it will radiate or not right now we are considering whether it will radiate efficiently or not and straight away we see that there is a mismatch and therefore there will be a significant power which is going to be reflected back it cannot come out of the system and therefore we say that an antenna is a region of transition or a transition device and many of the antenna types that I showed earlier for example the dipole antenna or the wave guide horn antenna are they take this shape of a transition region to reduce the mismatch between the guided wave region and the free space region any other questions yes please when we attach a matched load to the end of the transmission line the power from the transmission line will be transferred to that matched load efficiently that part is fine how do we say that the matched load will radiate efficiently that can be disappeared within the load itself how do we extrapolate that the matched load will radiate efficiently we have no reason to say that okay in the subsequent lectures we will be talking about the techniques with the help of which we can work out the radiation properties of some of the very rudimentary simple types of antennas we will not have time to discuss the more elaborate antennas for example the aperture antennas. So this is where we stop today in the lecture today we have considered in very simple terms why radiation may take place and we said that radiation can take place if the distance between the sources charges and currents is made comparable to wave lengths then we considered what are the various important functions that an antenna serves what are the different types of antennas we identified situations where antennas are indispensable and we also saw where communication system based on electromagnetic radiation that is using antennas could become competitive with another class of systems where we use an interconnected network of transmission lines and finally we evolved a definition of antennas which is applicable to all antenna structures thank you welcome to the 36th lecture in the course engineering electromagnetic in today's lecture we continue with our discussion on radiation and consider what are called the retarded potentiates you would recall that in the last lecture we saw that if the circuit dimensions become comparable with the wave length 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 wave length 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 feeds 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 squared 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 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 biotin severed'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 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 these 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 root 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. So, 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 alright. 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. 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 statics expressions alright. 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 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 alright. So, therefore, we see that the curl of a certain quantity must be equal to 0 then we recall another identity which says 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 frequency situation. For example, if in a situation time variation is not significant or this term becomes 0 then we go back to our static 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 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, 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 mu epsilon del 2 a by del t squared which is equal to mu epsilon grade gradient of del v by del t plus del of del dot a minus mu j. Similarly, 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 location 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 as 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 b should satisfy. However, both equations 7 and 8 contain both the potentials a and b. 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 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 equations 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 equations 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 have 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 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 which is a function of r prime d v prime upon capital R. Capital R is a variable and so is r prime. And therefore they are appearing under the integration sign. 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. 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. Find 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 varying? 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. R is the distance and v is the velocity of wave propagation the velocity of light in the medium concerned. 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 rho of r prime t minus r by v d v prime divided by capital 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 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 waveguides 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 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. And similarly, we complete by writing v is equal to 1 by 4 pi 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 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 frame work 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 mu times the curl of 8. 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 edge 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.