 Welcome to the 15th lecture in the course Engineering Electromagnetics. In this lecture also we continue with our discussion on wave propagation. The aspects that we consider in today's lecture are the following. We first consider the power flow in terms of the pointing vector and then go on to consider complex pointing vector along the lines of complex power flow and then we go on to discussing the wave propagation in conducting media and start with deriving the wave equation for conducting media. Taking up the first topic the power flow, you would recall that in the previous lecture we introduced the concept of pointing vector and we said that the pointing vector represented by P and given as the cross product of the electric field vector E and the magnetic field vector H is a measure of the power flow per unit area. Now this concept of expressing the power flow in terms of the field vectors is new to us. We are more familiar with power being described in terms of the product of voltage in current and therefore it will be interesting to see how this new concept of expressing power in terms of the product of the field vectors correspond to or is consistent with the other descriptions that we are familiar with. So for this purpose let's first take the example of a uniform plane wave and we try to determine the power flow associated with the uniform plane wave per unit area through some alternative means and then see how that corresponds to the statement of the pointing theorem that the pointing vector P equal to E cross H is the measure of the power flow per unit area. So let's say that there is a uniform plane wave with associated fields E and H propagating in some direction along one of the axis of the Cartesian coordinate system. Now how can we arrive at a description of the power flow? So for that purpose we write the power flow per unit area is equal to let's consider a unit volume and consider the energy stored in that unit volume which will be the energy stored per unit volume or the energy density. Now if we consider the speed at which this energy stored in a unit volume crosses the cross section of that unit volume that will give us the power flow per unit area like we have been doing the sums in mathematics finding out the volume of water flowing out of a pipe given the cross section of the pipe and the velocity of water etc. So similarly we can write here power flow per unit area is the energy density multiplied by the velocity which should give us in magnitude and direction since we are using velocity the power flow per unit area. Now as far as the first term on the right hand side is concerned the energy density that we know what is the energy density associated with the electric field and what is the energy density associated with the magnetic field. So using that we write half epsilon e square plus mu h square which is a scalar quantity this we multiply by the vector v not representing the velocity of the wave that should be the power flow per unit area. Then we manipulated in simple manner to see how it corresponds to the pointing vector. We are aware that for a uniform plane wave E and H are related through the intrinsic impedance of the medium. Overall electric field and the overall magnetic field are related in this manner E by H is equal to the intrinsic impedance eta of the medium. So we use that relation and replace E in the first term by eta H and H in the second term by E by eta. So that we get half and then epsilon or epsilon eta E H on one hand and mu by eta E H on the other hand multiplied by the velocity vector. We further realize that since eta is equal to square root of mu by epsilon for a perfect dielectric having no conductivity. These terms both of these the factors multiplying E H they are equal and they are equal to half square root of mu epsilon E H plus square root of mu epsilon E H times v not or it is E H by v not times v not recognizing that for a uniform plane wave v not is 1 by square root of mu epsilon mu and epsilon being the permeability and the permittivity of the medium alright. But therefore this is equal to E H times a unit vector in the direction of the movement of the wave the direction in which the wave is propagating. And we have already derived a relation for uniform plane waves that it is the cross product of E and H which is in the direction of the propagation of the wave and also for a plane wave uniform plane wave E and H are mutually perpendicular. Using these two results this is simply E cross H in magnitude and direction. So what we have done is that for a uniform plane wave starting with some description of the power flow per unit area we have reached a result that it is equal to the pointing vector E cross H and therefore this new notion that we have introduced is quite consistent with the alternative descriptions of power flow. The example that we have taken involved fields waves in infinite media or in free space are described in terms of fields and therefore that the power flow is coming out in terms of fields is perhaps quite alright. But we have stated earlier that the fields description is a more accurate description is a more general description and voltage and current descriptions are special cases of the fields description. So is it possible to see some in some situation where the power flow is given in terms of the voltage and current product how that corresponds to the pointing vector E cross H that should be even more interesting and for that purpose we consider the example of let us say a coaxial cable. We consider the usual coaxial cable with the with an inner and outer conductor in this manner and we consider that there is a voltage that is applied between the inner and the outer conductor. The inner and the outer conductors are connected through some resistance R and a current I flows in this direction in the inner conductor and an equivalent opposite current I flows in the outer conductor. We know that the power flow in this assuming that this is a DC situation there is no time variation is going to be the product of the potential difference between the two conductors and the current that is flowing in one of the conductors that is VI. We want to see how it corresponds to the pointing vector description for the power flow. So we need to evaluate the electric and the magnetic field associated with the system and if we redraw the cross section let us say this is the x and the y axis and this is the radial direction and this is the angle theta so that theta cap is in a direction normal to the radial direction. Let the radii of the inner and outer conductors be A and B and let this be the z direction coming out of the plane of the blackboard. Now what is the magnetic field between the two conductors in the space between the two conductors? It will be I by 2 pi R in the direction theta cap as I methodally oriented symmetric with respect to theta same value at all values of theta since it is a symmetric structure but varying with R. Strongest close to the inner conductor and weakest close to the outer conductor. What is the electric field? This we have derived earlier we can use the expression straight away it is V upon R natural log of B by A and if the inner conductor is at a greater potential then it is the radial direction in which the electric field lines would be there. Now we find out the pointing vector P which will be E cross H which will be V i upon 2 pi R squared natural log of B by A in what direction? R cap cross theta cap therefore that will be the z cap direction. What is the interpretation of the pointing vector that we have been giving? We have said that it is a measure of the power flow per unit area. So for total power flow associated with the system we need to integrate it we need to perform an area integral over the space where the fields are non zero. So that is the space between the two conductors and if we consider some element of area which is concentric with the system and has a thickness which is dr and has a radius R then the power flow through this area will be the pointing vector times the area of this element which will be V i by natural log of B by A then into 2 pi R squared times the area of this element which will be 2 pi R dr. What we are doing is we are writing P dot da which is equal to this and if we integrate this over the surface over which the fields are non zero we will get the total power flow so integrating from A to B. So the total power flow is going to be V i by log B by A and integral of dr by R from A to B now the integral is exactly equal to the term in the denominator and therefore this is equal to V i the power flow. So it shows very clearly that the concept that we have stated now that the pointing vector equal to E cross H is a measure of the power flow per unit area is completely consistent with any other alternative description of the power flow as we saw for the uniform plane wave and as we have seen for the coaxial cable. Now one thing is very noteworthy here what was the area of integration for determining the total power flow the area of integration was the entire region where the fields were non zero that was the space between the two conductors in the coaxial cable and it excluded the conducting regions. So from this point of view where is the power flowing it is flowing in the non conducting region in the region between the two conductors and therefore this is at some variance with our somewhat mistaken notion that the power flows in conductors and the greater the power flow the thicker the conductors. Actually conductors only provide a means for conducting current and to guide the wave associated with the power flow or the for the transfer of power flow. And their thickness is decided from the consideration of keeping the losses as low as possible and from the point of view of providing sufficient mechanical strength. So that is the advantage of the fields point of view we get a clearer picture of the phenomenon of power flow and the transfer of power. From this point onwards we now shift to the second topic that we will state for today and that is the complex pointing vector. You would recall that we use the concept of complex power at low frequencies. So we draw upon the same notion here and introduce the notion of complex pointing vector through the notion of complex power and we consider a situation where we have voltage in current phasors which are let us say like this with respect to some reference the voltage phasor let us say is like this. It has a magnitude V and it makes an angle theta V with respect to the reference and as time advances it rotates that is the concept of the phasor representation. Let us say the associated current phasor is like this with an associated magnitude I and an angle theta I. So that the angle between these two phasors theta V minus theta I is theta. Now we try to go back to the time varying voltage and current from these phasors so that the time varying voltage is going to be real of the voltage phasor V times e to the power j omega t which becomes magnitude of V and then cosine of omega t minus theta V that one can see here straight forward manner. Similarly the time varying current is going to be magnitude of it is going to be real of I times e to the power j omega t and through similar considerations it is going to be magnitude of I times cosine of omega t minus theta I. That is all right you are right omega t plus theta V and omega t plus theta I there is no reason for it to be negate. Now let us consider the instantaneous power flow instantaneous power flow we represent this as W is going to be the product of time varying V and I and therefore it is going to be magnitude of V times magnitude of I and then cosine of omega t plus theta V and cosine of omega t minus theta I which can be expressed as a sum and difference of these two angles thank you so that it is magnitude V times magnitude I by 2 and then we have cosine of theta V minus theta I plus cosine of 2 omega t plus theta V plus theta I. Now if we consider the average power averaged over a period of sinusoidal time variation then the second term is going to average to 0. This has an average value which is 0 and as far as the average power is concerned it is only the first term that will remain and we obtain W average equal to half magnitude of V magnitude of I times cosine of theta V minus theta I. Continuing on the other side we rewrite this as the average power equal to half magnitude V magnitude I into cosine of theta where theta is the difference between the phasor angles of the voltage phasor and the current phasor. Corresponding to this one also sometimes talks of the reactive power which is written as half magnitude V magnitude I times sin theta and cos theta is known as the power factor in power system terminology. Now let us introduce the notion of complex power and we say that let complex power W be half V I star where star stands for the complex conjugate of the current phasor which one can see is going to be half magnitude of V magnitude of I and then e to the power j theta V minus theta I or half magnitude of V magnitude of I e to the power j theta which is W average plus j W reactive and therefore we are in a position to say that the average power flow is equal to half real of V I star where V and I are the voltage and the current phasors. So, here we have introduced the instantaneous power, the average power and the complex power and how the average power can be calculated when we have phasor quantities available. W average is half real of V I star. So, following this same notation we are now in a position to write that the instantaneous power flow or the instantaneous pointing vector and designating it like this is going to be e cross h and the complex pointing vector is going to be what? It will be P equal to half e cross h star writing exactly corresponding relations to what we wrote for the voltage and current description. So, that the average pointing vector P average becomes half real of e cross h star. For sinusoidal time varying signals e and h phasors would be available to us and this is how we can calculate the average power flow per unit area or the average pointing vector. Continuing further if we want to specialize this to the case of a uniform plane wave propagating in the z direction then what would be the power flow in the z direction per unit area that we can work out for a z directed uniform plane wave. We will have x and y electric and magnetic field components and the cross product in such a case can be written easily and we will have P z in such a case power flow in the z direction per unit area equal to half real of e x h y star minus e y h x star which of course could be converted to uniform plane waves propagating in other directions through a similar procedure. Next we like to apply this to some specific cases and see what is the effect of polarization on the power flow description and some important differences in the case of the linearly polarized waves and those of the circularly polarized waves would emerge. We apply this to uniform plane waves and we first consider the case of linear polarization. Sir, this expression for P z is itself real e x h y star. Not necessarily. Sir, e x can be written in terms of h y h y star is mod f i 4. We need to apply it to different cases. For uniform plane waves. Yeah, for uniform plane waves for example for the elliptically polarized waves it may not come out real. There can be situations where this may not be real. That is why that is put. For the linearly polarized uniform plane wave what kind of electric field expression can be right? We consider that the wave is propagating in the z direction therefore what can be right for the electric field vector? For the wave to be propagating in the z direction we can have x and y components of the electric field vector. That is one thing. Second thing is for it to be linearly polarized they must be in phase. Both x and y field components must be in phase. That is the most general description of a linearly polarized wave. One or the other field component x or y may be absent. That is a special case. And therefore we write e equal to let us say a x cap plus b y cap. And of course in phasor notation factor e to the power minus j beta z for a wave propagating in the z direction. Now how do we obtain the magnetic field intensity vector? We recall the relations between the electric and the magnetic field components in terms of the intrinsic impedance. And for this case they will become e x by h y equal to eta and e y by h x equal to minus eta. So that h x is equal to minus e y by eta and h y is equal to e x by eta. And therefore here we will have 1 by eta on one hand. And the x component is minus b by eta x cap and the y component is a by eta. Eta is already taken out common. So we do not need to write it here minus b plus a y cap and of course e to the power minus j beta z. And now we apply the formula that we have derived for a wave propagating in the z direction in terms of the products of the various components and taking the complex conjugate. And in that case we get real of e x times h y star that gives us a square. These two factors cancel out because of the product of the quantity with the complex conjugate. And similarly the other term gives us plus b square applying minus e y times h x star. Assuming that a and b are real this is simply going to be e square by 2 eta or alternatively it will be half eta h squared h squared standing for the square of the magnitude of the magnetic field intensity. Now the appearance of the factor half can be related to the sinusoidally time varying field quantities. We are considering a linearly polarized wave and in the x y plane depending on the ratio of a and b the field vector electric field vector is oscillating along this direction is having different magnitudes. But the orientation remains unchanged and it is varying sinusoidally with time and therefore on an average the power flow becomes half e squared by eta or similarly half eta times h squared quite reasonable. If we consider a circularly polarized wave let us see what happens. How will we construct a circularly polarized wave? Here we stick to z directed plane wave that is all right. So, we will have x and y components. But for the wave to be circularly polarized the magnitudes of these two components must be identical and there must be a phase difference of 90 degrees. So, let us say we write it as x cap plus j y cap times e naught and e to the power minus j beta say which is a circularly polarized wave. If we look at it carefully it will be a left circularly polarized wave. This describes the phase shift in the wave as the wave propagates in the positive z direction. Now, the magnetic field description can be obtained in a similar manner using these relations and we are going to get h equal to 1 by eta and then h x is minus e y by eta. Therefore, it will be minus j times x cap and h y is e x by eta. So, that it is simply plus y cap and e naught e to the power minus j beta z. And now we apply this relation that we have written for a uniform plane wave propagating in the z direction and the real power flow per unit area given by these terms. This is e y and this is h x star. So, what do we get? We get half and of course, e naught squared and then e x h y star will give us 1 and e y h x star will give us minus 1 with an additional minus sign. So, this is plus 1 and here one can see that if the phase difference were not 90 degrees one would not get a quantity which is already real. Then the real part will have to be considered which is simply eta also comes here which is simply e naught squared by eta the factor 2 in the denominator goes away or e squared by eta or alternatively eta times h squared. Now one can understand the difference between this final result for the circular polarization and this result for linear polarization in two different ways. One could either consider that the circularly polarized wave is a superposition of two linearly polarized waves each one having the same amplitude. So, each one will have a result like this and therefore, the total power associated with the circularly polarized wave is twice that. So, it is e squared by eta or else one could consider the behavior of the circularly polarized wave electric field vector. And what does it look like? It is an electric field vector with magnitude e naught rotating in the x y plane in this case in this direction. So, as a function of time is the electric field vector magnitude changing it is not changing. And therefore, from the point of view of mathematics a factor half is not required when we consider the average power whichever root appeals to you this is the result for circularly polarized waves and this is the result for linearly polarized waves. This small difference one should keep in mind. Also many times the RMS values are specified corresponding to phasor quantities and if they are given then one can see that this is going to be 1 by eta times e RMS squared or alternatively eta times h RMS squared which extension is quite straight forward. While we are discussing the polarization and the power flow associated with waves of different polarizations one important character of polarization must be mentioned. Supposing an antenna is transmitting a wave which is x polarized linearly polarized wave and the electric field vector is oriented in the x direction. Now if this signal is to be received the receiving antenna must be able to respond to this x directed polarization. If it is so oriented if the receiving antenna is so oriented that it responds to y directed polarization not to x directed polarization the signal received at the receiver will be nil. In such a case we would say that the receiving antenna is oriented for a polarization which is orthogonal to the polarization transmitted by the transmitting antenna. This is the reason why our radio and TV receptions improve when we maneuver the antennas and when they correspond when their orientation corresponds to the polarization of the incident wave then we get a strong signal and not otherwise. It is very easy to see that if it is a linearly polarized wave then what would be the orthogonal polarization? For x directed linear polarization y orientation will be the corresponding orthogonal polarization. So this kind of orthogonal polarization can be utilized to increase the information handling capability capacity of communication systems. They could be operated transmitting two different signals in two different in two orthogonal polarizations which would be non interfering with each other. Now just as for linear polarization one can identify an appropriate orthogonal polarization is the same thing possible for circularly polarized signals. The answer is yes and one can show this using the following example. We have just seen that this is one kind of circularly polarized wave left circularly polarized wave. So let one wave be represented as e 1 equal to x cap plus j y cap let us say e 1 and e to the power minus j beta z. And let us consider another wave e 2 which is right circularly polarized all we need to do is to change the 90 degree phase difference. If this is ahead in phase by 90 degrees if we retard this in phase by 90 degrees it becomes a right circularly polarized wave. So we write e 2 equal to x cap minus j y cap say e 2 e to the power minus j beta z both travelling in the positive z direction. The corresponding components time varying components can be obtained by multiplying by e to the power j omega t and taking the real part. So here we will get e x 1 the time varying part equal to e 1 cosine omega t cosine of omega t minus beta z actually but then we can consider the behavior at z equal to 0. Similarly what will be the time varying e y 1 realizing that there is a factor j here this will be minus e 1 sin omega t again at z equal to 0. Similarly we can put down the time varying field components here e x 2 at z equal to 0 will be e 2 cos omega t and e y 2 will be e 2 sin omega t. Now let us take a dot product of the time varying electric field vectors that is e 1 dot e 2. So that the corresponding components get multiplied and get added that is how the dot product is going to be taken. So we get this as e 1 e 2 which comes out as a common factor and we have cos squared omega t minus sin squared omega t or it is e 1 e 2 times cosine of 2 omega t and if we consider the average value of this quantity that is going to be 0 over a period of sinusoidal time variation. So from this point of view these two polarizations are orthogonal to each other and antenna oriented or designed to receive right circularly polarized wave will receive on an average a 0 signal if a left circularly polarized wave is incident on it. So for circularly polarized waves also the concept of orthogonal polarization is possible and many times when we need to increase the capacity of communication systems which are using circular polarization we can use this concept. We could transmit signals in right circularly polarized manner for one signal and using the left circularly polarized wave for the other signal and these would not interfere. If there are any questions up to this point we can try those out now. Otherwise we go on to considering the third topic for the day and that is the consideration of wave propagation in conducting medium. Now do we expect any difference in the behavior of a propagating wave in a conducting medium compared to that in free space or in a perfect electric. It is for the latter situations that we have got all the results. Now we want to consider if there are any differences that arise when we allow the medium to be conducting that is the medium is not a perfect electric. Where would a difference occur? We go back to the Maxwell's equations and we see that the first equation del cross h equal to epsilon del e by del t. This is what we wrote when we considered a perfect electric medium or free space. Of course we made the simplifying assumptions of the medium being homogeneous and isotropic and further source free. When the medium is conducting there will be a conduction current associated with the electric field. So a conduction current term must appear here when we want to consider wave propagation in conducting medium. This is where a difference would occur. Of course where j is equal to sigma times e sigma being the conductivity of the medium. Of course if sigma is 0 then we go back to the perfect electric situation. We write the other Maxwell's equations that is del cross e equal to minus mu del h by del t del dot e equal to 0 and del dot h also equal to 0. So we are still considering a source free region in which we will consider the behavior of either the electric field alone or the magnetic field alone. And this conduction current term which is appearing is because of the finite conductivity of the medium not because of any source being present in the region of consideration. The procedure is going to be similar to what we did earlier for perfect dielectric. We number these equations say 1, 2, 3 and 4 then from equation 2 if we take the curl on both sides we get del cross del cross e equal to minus mu curl of del h by del t. Then the curl of curl of e can be written as del of del dot e gradient of divergence minus del square e the Naplesian operation on the electric field vector. And on the right hand side we interchange the time derivative and the space derivative these being linear operations so minus mu del by del t of del cross h. And now from equations 3 and 1 equation 3 will allow us to see that the first term on the left hand side is 0 and equation 1 will allow us to replace del cross h in terms of the right hand side. And therefore, we get minus del square e equal to minus mu del by del t of epsilon del e by del t plus j or which is sigma e. And therefore, we get del 2 e minus mu epsilon del 2 e by del t square minus mu sigma del e by del t equal to 0 which is the wave equation for the electric field in an infinite conducting medium. And of course, if sigma goes to 0 we get back the wave equation for perfect dielectric meeting. This is where we stop today. Today in this lecture we have given examples of power flow in different situations and shown how the concept of power flow in terms of pointing vector is consistent with these different descriptions. We introduce the notion of complex pointing vector and then we have gone on to introduce the wave equation in conducting media. Thank you.