 Welcome to the 37th lecture in the course engineering electromagnetic. In this lecture we continue our discussion on the fundamentals of radiation and we take up the topic of herdsian dipole today. The herdsian dipole is also known as the alternating current element or the oscillating electric dipole. The reason for these various names will become obvious as we proceed further. You would recall that in the last lecture we considered the procedure that one could follow for relating the sources of radiation that is the time varying currents or charges to the radiation feeds. And we said that it is easier to first calculate potentials the vector magnetic potential and the scalar electric potential first from these sources and then calculate the radiation fields from the potentials themselves. So today the herdsian dipole that we consider will be an illustration of the application of this method. First let us show what we mean by the herdsian dipole. The herdsian dipole is a very short length in some textbooks you would find the length being represented by d l to emphasize the fact that it is a very short length. And when we say short or long in the context of our course what we have in mind is that the length the overall length of this entity l is much much less than a wavelength. When it satisfies this requirement or this condition then it is very short. The advantage of taking up a very short let us say conducting wire is that we can without too much of an error say that the current is uniform whatever current is flowing is uniform over the entire length. Otherwise as you have seen on the transmission lines the current has to vary in a sinusoidal fashion depending on the terminations. So if we say that this length is very small compared to a wavelength then we can safely say that the current is uniform over the entire length which simplifies our procedure considerably as we will see. We also assume that the wire on which we have assumed a uniform current which is of length l much less than a wavelength is very thin that is the diameter is much less than the length which also will help us in keeping things simple. We place this hergene dipole of length l and carrying let us say a current i at the origin of the coordinate system. The coordinate system that we use is the spherical polar coordinate system and since we will be dealing with it for the next few lectures let us spend a little more time on the coordinate system. The coordinate system is shown here in some more detail and of course with reference to the more familiar Cartesian coordinate system. The coordinates of a point p are r theta phi in this system where r is the radial distance from the origin theta is the angle the radius vector makes with the z axis and phi is the angle the projection of this point p on the x y plane makes with the x axis right. Different kind of surfaces are generated if we assume the value of each of these coordinates to be constant. For example r equal to constant generates a spherical surface only a small part of which is shown in the diagram phi equal to constant generates a plane containing the z axis and theta equal to constant generates a conical surface with the axis coincident with the z axis. The unit vectors are r cap theta cap and phi cap r cap is the direction in the radially outward direction away from the origin theta cap is a direction which is normal to r cap tangential to the sphere with this radius r and has a direction in which theta is increasing and phi cap again is normal to the plane constituted by r cap theta cap and has a direction in which phi is increasing. So these are the three mutually orthogonal unit vectors for this spherical polar coordinate system. One thing that should be kept in mind or should be emphasized is that these directions r cap theta cap phi cap are not fixed unlike the x y z directions in the Cartesian coordinate system. These direction change what is the actual orientation of these directions depends upon the values of the coordinates theta and phi. For example for theta equal to 0 degrees the theta cap direction is normal to the z axis and is parallel to the x y plane. On the other hand for theta equal to 90 degrees the theta cap direction is perpendicular to the x y plane and is the same direction as the minus z direction. And similarly the plane that is constituted by phi equal to constant will be a different plane if the value of this constant is taken differently. So it is this kind of coordinate system in which we are placing the Herzian dipole. And what we wish to do is to calculate the radiation fields at some point P which has coordinates r theta and phi. What are we required to do for this purpose? The first step that was listed last time was to calculate the vector potential A. And for the vector potential A the expression that we wrote was the following. We wrote A as a function of r is equal to mu by 4 pi integral of j r prime e to the power minus j beta r upon r dv prime. Where it will help to consider the general picture so that we are clear about the meaning of these symbols r r prime and capital R. We considered this kind of a extended source on which we have the current density and the charge density which are varying with time. And with respect to some origin the position vector of different points on this extended source is designated as r prime. Depending on the element that we consider here r prime will have different values. The field point was considered to have a position vector r and the distance of the field point from different elements on the source was taken to be capital R. And we integrated the contribution due to this kind of small volume elements at point P to get the overall vector potential. In the present case where the source is located symmetrically with respect to the origin and going back to our general expression for the vector potential A for the we have already seen that r can go to small r. Secondly since the source is so small the length is much less than a wavelength. This small r is going to be virtually the same for different portions of the source. If we consider the distance of the point P from these different portions it is going to be virtually the same. And therefore capital R can be taken to be small r which can be considered to be constant for this such a small source that we are considering. This facility will not be available for general antennas. Next let us consider what is the integration of the current density over this source. We can consider the integration to be of the following type j and then r prime is the same thing as r then dv prime could be considered to be a result of an area integration and a length integration. In general it could be a triple integral of this kind. The source that we are considering is very small in length and its diameter cross section is even smaller. That approximation we have already stated and therefore as far as the area integration of this current density is concerned that will give us the current that is flowing in the source. We were saying that the integral of the current density as far as the cross section is concerned is going to give us the current flowing on the source. And then idl is what we are required to calculate and once again dl is so short that this also goes to simply i times l where i is the current flowing on the source and l is the length of the source. So in this simple case this seemingly difficult integral becomes very simple and straight forward and therefore what we get is also the current that we have we are considering is z directed since the source we are considering is oriented along the z axis. So keeping these things in view we get an expression for the vector potential which has only a z component in this case since the source the current has only a z component which becomes az equal to mu i times l by 4 pi r and then e to the power minus j beta r alright. This factor drops out of the integral sign because of the various simplifications in this case and other things as we have just discussed run out to be very simple. This is the first step the common first step in dealing with this kind of problems that is from the current distribution you calculate the magnetic vector potential not always. But normally capital R which is the distance of the field point p from various portions on the source is a variable okay and that complication we are avoiding in this case by assuming a very small source right that is true the next step if you recall is the calculation of the magnetic field intensity and what is the relation we follow for calculating that we write h equal to 1 by mu del cross a curl of a multiplied by 1 by mu that is how we can calculate the magnetic field intensity h right. What is the expression for the curl of a that is what is going to be required next in spherical polar coordinates the curl of a vector let us write it in terms of a general vector say f is given by r cap by r sin theta and then we have del by del theta of a phi times sin theta that is the first term in the radial component of this curve minus del a theta by del phi. Similarly we have the other two components which read as theta cap by r and then 1 by sin theta del a r by del phi minus del by del r of r times a phi and then the last component phi cap by r times del by del r of r times a theta minus del a r by del theta okay. I started to write a general expression but then I have written the components of a and therefore this will need to be corrected and I think this should do the job good. Therefore what we see is that we require the components of the vector a in this case for which the curl is to be calculated. So far what have we got we have got the z component of the vector potential a and therefore next what we need to do is to identify its components in the spherical polar coordinates and then going back to our picture of the spherical polar coordinates you can see that if a z directed quantity is available its radial component will be that component times cosine theta and similarly the theta component can be calculated. So looking at this we can write the expression or expressions for the other two components which will be well the spherical polar components that is a r will be a z cos theta and a theta will be minus a z times sine theta. It can be seen in a relatively straight forward manner that a quantity which is entirely z directed will have no phi component and therefore a phi is 0. So we have got the three components in the spherical coordinates for the quantity of interest. There is no phi component there is a radial component and a theta component but those also are functions of only r and theta if you look at the expression for e z. So the components that are non-zero are not a function of phi. Now that is no coincidence the source that we are dealing with is completely symmetric with respect to phi so there is no variation with respect to phi. Also the current that is causing the radiation has no component along the phi direction therefore there is no phi component in the vector potential also. So this kind of qualitative checking up can be done in most of the problems. Keeping these components in mind if we look at the curl expression we find that to calculate the magnetic field the phi component is 0 therefore these terms are not going to contribute anything. We have only the r and the theta components which are non-zero further these components also do not have any variation with respect to phi. Therefore these terms also are not going to contribute anything more than a 0. Therefore for the magnetic field the only component that will be there will be the phi component alright and we have the theta and the r component of a available those can be plugged in here and we can get the phi component of the magnetic field and let us note down this result somewhere and we say that h phi is equal to i l sin theta divided by 4 pi and then we have j beta by r plus 1 by r squared into e to the power minus j beta r. Now let me ask you a simple question the current that we have taken up is it time varying or is it constant with time? It is time varying we are using phasor notation and therefore the current has sinusoidal time variation which is there implicit in all these field expressions and can be recovered using our standard method that is as far as the magnetic field is concerned. This is the second step in this kind of problems the first step was the calculation of the vector potential a the second step is the calculation of the magnetic field. So the second step also we have crossed what will be the third step the third step will be the calculation of the electric field for which we can go back to the Maxwell's equations and see that we have del cross h equal to epsilon del e by del t in general in phasor notation it is j omega epsilon times e and therefore e can be calculated as 1 by j omega epsilon times the curl of h where to calculate curl of h again we will need to go back to this kind of expression and now as far as the magnetic field intensity is concerned we have got its component there is only one component five component and that also is independent of five and on that basis one can see that we are going to have only the r component and the theta component as far as the electric field is concerned. Therefore by plugging in this magnetic field intensity component five component in the curl of h and then dividing the curl of h by j omega epsilon we get the following field expressions we have e r equal to minus j eta i l by twice pi beta and then cosine theta times j beta by r squared plus 1 by r cube and then of course e to the power minus j beta r which has to be there in these fields to take into account the phase delay the hallmark of delayed or the retarded potentials and finally we have e theta equal to minus j eta times i l by 4 pi beta sin theta times minus beta squared by r plus j beta by r squared plus 1 by r cube e to the power minus j beta r and of course as we have seen the other three field components that is h r h theta and e five r 0 somebody would say that these three are complicated enough so we do not need the other field components anyway the field that we now get for the simple source is quite complicated at least so it looks therefore we spend a little time here on the importance of this source the importance of the source stems from two three different considerations first of all it is perhaps the simplest source that one can think of to illustrate the method of the field calculation the method of radiation field calculation for the this kind of sources secondly any extended source can be considered to be build up of this kind of building blocks so if you know the field due to the building block then we can hope to find out the field of the overall system right thirdly even when we do not do that that is considered an actual antenna composed of a number of this kind of building blocks there are many antennas which because of various constraints are quite small much smaller than a wavelength and for those antennas these results would be directly or with small simple extensions be applicable therefore the source that we are talking about is quite important but what about these various terms in the field components for that purpose let us consider the magnetic field first and then we will consider the electric field components in the magnetic field there is only one component the five component we have two terms one is varying as one by r the other is varying as one by r square it is obvious that the component which varies as one by r squared is going to become less and less as this distance are from the source increases and it will only be the one by r term which will remain significant after a certain distance so from that point of view the one by r term is known as the distant field term or the far field term because this is the term which is going to be significant at significant distances from the antenna as we will see it is this term which contributes to average power flow away from the source and therefore it is many times also called the radiation field term or the radiation term but that we have yet to see what about the one by r squared term this is called the induction term or the induction field the reason for this kind of nomenclature particularly for calling one by r squared term the induction term is the following supposing we were to calculate the magnetic field because of such a small system such a source the current element directly extending the results for the no time variation case then we would just apply biotin severed slope and what we would write is that H is equal to one by four pi and then integral let us say we write the general expression IDL cross r cap by r square okay and now visualize the system we are dealing with then you will find that in our case IDL cross r cap will have a phi orientation and then the expression is going to be identical to the one by r squared term we are going to have this as IDL sin theta by four pi r squared the additional factor e to the power minus j beta r can be explained or understood as the effect of a finite time of propagation of this field from the source to the field point alright and therefore this is called simply the induction term there will be a certain distance a certain value of r where the two terms become equal and beyond which the distant field term will dominate over the induction field term since other factors are the same this break even distance can be calculated by equating j beta r j beta by r to one by r square okay if the distance is satisfying this condition then the magnitudes of the two terms are going to be equal and therefore this implies that r is equal to one by beta maybe we can take the magnitude alright since the velocity of wave propagation is omega by beta beta can be substituted in terms of v and omega and that gives us beta equal to omega by v so that it is omega here and v here which can be simplified to become lambda by 2 pi or approximately lambda by okay so at a distance which is of this order lambda by 6 the two terms become equal in magnitude at shorter distances the induction term will be larger in magnitude and at greater distances it is the distant field term which will be more significant but the importance of this term will become clearer when we consider the power flow that is as far as the magnetic field and the two terms that appear in the expression for the magnetic field are concerned going now to the expressions for the electric field we find that in addition to the one by r and the one by r squared terms we have one by r cube terms as well in these field expressions what is the origin why these terms are arising that is what we will like to understand now you would recall that we had considered in a previous lecture the field due to an electric dipole which had a one by r cube variation of the field components so let us go back to that and see if that can throw some light on this issue we consider the electric dipole okay which consists of equal and opposite charges which are separated by a very small distance let this distance be l the length of the electric dipole alright and then at a point P which is such that it has a distance small r from the center of the electric dipole and this angle is theta the field components read as er equal to 2ql cosine theta divided by 4 pi epsilon r cube and the theta component of the electric field has an expression which is ql sin theta divided by 4 pi epsilon r cube and if you remember e phi is 0 okay there is no electric field in the direction normal to the plane containing the electric dipole which can be argued out from simple considerations these expressions appear quite similar to the one by r cube terms that have appeared in the electric field components for the herzen dipole alright now this kind of fields appear when we have equal and opposite charges situated at some distance where are we having charges in our source we assumed a certain uniform current flowing over a very short length right now if we consider the equation of continuity which says that the divergence of j is equal to minus del rho by del t which says that if there is a net outward current flow at a certain point there must be and a corresponding depletion of charge density at that point okay so current flow cannot be separated from the build up or depletion of charge if from a certain point the current is flowing out the charge there must be getting depleted and if to a certain point the current is flowing in the charge there must build up we can look at it in an integral form for example if we integrate this over a closed surface okay then we are going to have j dot da integrated over a closed surface which becomes minus del by del t rho dv where we have used the divergence theorem alright and therefore if this becomes equal to i in a certain problem then this is equal to minus del q by del t alright talking of the magnitude therefore we see that the in phasor notation therefore i should be equal to minus j omega q and therefore if we have assumed at a certain point a certain value of i then there must be a q there which is equal to minus i by j omega okay going back to our source you would recall that we have considered a uniform current and if there is a system like this where we have assumed uniform current and let us say the current is flowing in this direction at a certain point of time there must be a corresponding depletion of charge here given by such an expression and a build up of charge here again in magnitude given by the same expression okay and therefore since we have assumed nonzero current at these ends there must be corresponding charges building up or negative charge building up at these two ends conceptually therefore the small source that we have considered can be looked at as let us say a system consisting of this kind of two spheres or two small disks which are connected by a conducting wire of length l and where we have a charge plus q and a charge minus q which depends upon i in this manner and of course the polarity and the magnitude of sign keeps on changing with time since i is a sinusoidally time varying quantity to be on the safe side so that we do not have to make too many considerations we consider that our source is like this where it is connected where these two spheres or disks are connected by a very thin wire so that the distributed capacitance of the wire is very insignificant compared to the capacitance between these two spheres further we assume that the radius of these spheres is much less than the length which itself is much less than a wavelength in which case we can assume a uniform current over this connecting wire. Now it was precisely such a system which was used by herds for experimental verification of the waves predicted by Maxwell from his system of equations known as Maxwell's equations and therefore this is just the herdsian dipole all right he actually built such a such an antenna in this manner and then he was able to produce propagating waves and then detect them he could observe standing waves being formed by these waves and so on that was an outstanding series of experimentations so therefore we call it a herdsian dipole and from other points of view therefore you can see that it can equally well be called an alternating current element since it is a wire of a very short length supporting sinusoidal time varying current or an oscillating electric dipole because it is just like a dipole and then the charges involved in this dipole are varying with time. So whatever point of view we take we can call this system in a number of different ways and therefore if we substitute this kind of expression for q in the fields due to an electric dipole then let us see what we are going to get we will get n e r which will be 2 i l cos theta by 4 pi omega epsilon r q with a negative sign and a factor j here the other field component e theta turns out to be minus i l sin theta by j omega and then 4 pi epsilon r q which except for a difference of sin which will depend upon the time instant we have taken and the science we are using here these expressions are just the same as we obtained for the electric field components in the herdsian dipole or the alternating current element that we are considering alright. From this point of view these field components are sometimes also called as the electrostatic field terms 1 by r q terms are the electrostatic field terms because of their similarity with the fields due to an electric dipole alright. In any case these will become almost negligible at significant distances from the source. Now this kind of field terms are arising here because we have by virtue of our assumption assumed a discontinuity in the current at the ends of the source right. So this kind of charges will arise only when there is a discontinuity in the current. So if there is a circuit with a uniform current with no discontinuity then these charges will not build up and these electrostatic field terms will not appear. However if there is any antenna which is significant in terms of wavelengths it cannot have a uniform current distribution and therefore there will be a small or large amount of discontinuity between various portions of the circuit and then this kind of electrostatic field terms can be expected to be present. This is where we like to stop today. We have considered the herdsian dipole and the various simplifying approximations that go with it and we have calculated the vector potential and the fields corresponding to such a simple source and we have tried to offer an explanation for the various terms in the field that arises from such a source. Thank you.