 Welcome to the 16th lecture in the course Engineering Electromagnetics. We have been discussing the wave propagation for the last few lectures. Continuing these discussions, the topics for today's lecture are the following. We consider the wave propagation in a conducting medium, then see the difference between conductors and dielectrics, then go on to introduce the concept of the depth of penetration. And if time permits, we consider the concept of surface impedance. Taking up the first topic listed for the day that is the wave propagation in a conducting medium, you would recall that we wrote down the wave equation for a conducting medium as follows, del 2e minus mu epsilon del 2e by del t squared minus mu sigma del e by del t equal to 0. This is the wave equation in a conducting medium involving the electric field, a similar equation involving the magnetic field which also would be called wave equation can also be written. We consider the characteristics of the wave propagating in a conducting medium by considering sinusoidally time varying signals. So for sinusoidal time variations and that is using phasor notation, the wave equation that we have written gets simplified to del 2e plus omega squared mu epsilon e minus j omega mu sigma e equal to 0. In phasor notation, any time derivative can be replaced by the factor j omega which can be rewritten as del 2e minus j omega mu into sigma plus j omega epsilon e equal to 0. And then introducing a new symbol that is del 2e minus gamma squared e equal to 0, we can write this where gamma is j omega mu into sigma plus j omega epsilon whole square root, where gamma is a complex quantity involving the frequency and the characteristics of the medium. In general, it will have a real part alpha and an imaginary part beta. The nature of the propagation in a conducting medium can be understood by making the simplification as we did for considering wave propagation in a perfect dielectric. That is we consider that there is no variation of the fields with respect to two of the coordinates. So assuming that there is no variation with y or the z coordinate, we can write the Laplacian operator del squared as del 2e by del x squared. For the simpler situation when only x variations are non-zero minus gamma squared e equal to 0, where one can write the solution almost by inspection as e equal to say c 1 e to the power minus gamma x plus c 2 e to the power plus gamma x. This is in phasor notation and one can recover the actual time varying electric field vector by multiplying by e to the power j omega t and taking the real part. Carrying out that procedure and considering the first term, we get the first term in the solution and the time varying electric field which will now be a function of both x and t turns out to be c 1 e to the power minus alpha x and then cosine of omega t minus beta x. Now, as far as the second term is concerned cosine of omega t minus beta x, we have established already that this kind of terms or terms with this kind of an argument represent waves propagating in the positive x direction. As far as the additional term that appears here is concerned, now we do not expect power generation of power amplification in a passive medium. And therefore, this must represent the regular attenuation of the wave as it propagates in the positive x direction. Similar interpretation can be arrived at for the second term in the general solution which would represent waves propagating in the negative x direction. Here one should make a remark that since we expect the wave to attenuate as it propagates in its direction of propagation, one should choose the appropriate square root of the complex quantity gamma. So, that we get attenuation of the wave as it propagates that is a matter of mathematical choice. So, from this point of view alpha is called the attenuation constant, beta is called the phase shift constant and gamma the new symbol we have introduced is called the propagation constant. In terms of the phase shift constant, now one can write the phase velocity v p or v phase which will be omega by beta. And also what would be the expression for the characteristic impedance or the intrinsic impedance of the conducting medium that we are considering. If you recall for a perfect dielectric we obtained eta as square root of mu by epsilon which we could rewrite as j omega mu upon j omega epsilon. And now considering Maxwell's first equation that is del cross h equal to j omega epsilon e plus j which we write as sigma e that is it is j omega epsilon plus sigma times e. And when we compare it with the expression the corresponding expression that we get for the perfect dielectric case we see that j omega epsilon is now replaced by j omega epsilon plus sigma. So, instead of obtaining the relationship between the electric and the magnetic field vectors by going through the Maxwell's equations one can simply make use of this substitution. And therefore, here if you replace j omega epsilon by j omega epsilon plus sigma we should get the correct value of the characteristic impedance or the intrinsic impedance. And therefore, eta for conducting media turns out to be j omega mu upon sigma plus j omega epsilon whole square root. Which expressions are of course, the general expressions if sigma is allowed to approach 0 they will take the form that we derived for the special case when the medium was a perfect dielectric or free space. From the expression for the propagation constant gamma that is gamma equal to this much and gamma equal to alpha plus j beta one can write down the general expressions for the attenuation constant alpha and the phase shift constant beta. That is a simple mathematical exercise and on that basis one gets alpha equal to omega then mu epsilon by 2 within the square root sign. And then we have 1 plus sigma squared by omega squared epsilon squared minus 1 as far as the attenuation constant is concerned. This can be handled as a simple exercise. And similarly, the phase shift constant beta turns out to be a very similar expression omega times mu epsilon by 2 and 1 plus sigma squared by omega squared epsilon squared plus 1. All terms are the same, but there is a difference in sign here which depending on the values of the parameters of the medium can make a lot of difference. So, these are the general expressions for the attenuation constant which would have the units of nephers per meter and the phase shift constant which will have the units of radians per meter. Now, fortunately or as the practice goes it is possible to divide or categorize the various types of media that one uses in practice in two broad categories. In one category we put conductors and in the other category we put dielectrics. Normally, we tend to use good conductors and good dielectrics or good insulators because these materials in these categories are low loss. Materials that do not fall in these categories would be involving a greater amount of loss. Therefore, one tends to avoid their use and for these two categories it is possible to obtain simpler expressions for the attenuation constant and the phase shift constant. But first we have to specify the basis of this categorization. How would one put some materials as conductors and some other materials as dielectrics. So, that would be the basis of this categorization. That brings us to the second topic for the day that is conductors and dielectrics. For this categorization once again we draw upon the Maxwell's equation derived from Ampere's law that is del cross h equal to j omega epsilon e plus sigma e. That is this equation written for the general case of conducting media. On the right hand side now one can identify the two terms as follows. The first term of course, we are using the phasor notation. The first term is the displacement current density. And similarly the second term is the conduction current density. And the broad rule that one can think of for classifying materials as conductors or dielectrics would be that in a material that we want to classify as a conductor the conduction current or the conduction current density should dominate. And in a material that we would like to classify as a dielectric current insulator the displacement current or the displacement current density term should dominate. So, depending on this relative value of these two current densities it will be possible to categorize or classify the various types of material that come to our hands. And therefore, one writes the ratio of the conduction current density and that of the displacement current density. What will be this ratio equal to? For a given value of the electric field it is going to be sigma by omega epsilon. And the value of this ratio can be used as an index for categorization. And we say that for conductors we have the value of sigma by omega epsilon as much much greater than 1. How much greater than 1? Perhaps 10 is the lower limit it must be greater than 10 giving a typical numerical figure. And similarly for dielectrics we should have sigma by omega epsilon much much less than 1. And the materials that fall in between may be called quasi conductors or quasi insulators or semiconductors whatever one feels like. But as I said that radio frequencies usually one tends to use materials which are which fall either in the category of conductors or in the category of dielectrics because these would be low loss materials. One can consider the nature of these terms for conductors and dielectrics. For example here for conductors sigma and epsilon the conductivity and the permittivity are relatively independent of frequency. Which means that it is quite possible that a material which is a conductor at let us say some low frequency does not behave as a conductor at least according to this criterion at a much higher frequency alright. For example we can consider copper it has a conductivity which is 5.8 into 10 to the power 7 most per meter. And let us consider the value of this ratio at a very high frequency let us say 10 gigahertz. So, in which case it becomes 2 pi into 10 to the power 10 and then the permittivity of copper. Permittivity of good conductors is almost as much as of free space. And therefore, we may write here 1 by 36 pi into 10 to the power minus 9. Which is going to be a very large figure something which is of the order of say 10 to the power 8 or so. Even at a very high frequency that we have considered that is 10 gigahertz. However, if we increase the frequency to a much greater value it is quite likely that this does not behave as a conductor. For example, at x ray frequencies this could behave as a dielectric material. The point we are trying to make is that the behavior of a material as a conductor is dependent on the frequency. For the kind of frequency we will be considering martens like copper will behave as conductors. Similarly, for dielectrics the ratio sigma by omega epsilon the entire ratio turns out to be relatively independent of frequency. And this quantity is given a different name it is called tan delta or the dissipation factor. It is equal to sigma by omega epsilon which of course, is much less than one for dielectrics. For example, for mica the tan delta is say point 3 0's and then 2 and which is relatively independent of frequency. At audio frequencies as well as at radio frequencies the dissipation factor or tan delta for mica is going to be so low. And therefore, it will be a dielectric it will be an insulator at the frequencies of interest to us. There exist media which come into play in wave propagation which fall into play in between say the two categories conductors and dielectrics. And one important medium like that is earth. What is the conductivity of earth? It is different for different locations and in different seasons also it could be different taking a typical figure. The conductivity is let us say 5 into 10 to the power minus 3 most per meter. And the permittivity is let us say 10 times that of free space. And then let us say if you want to find out the highest frequency up to which the earth could be called a conductor what would be that frequency. So we say that let sigma by omega epsilon be equal to 10. Any frequency at which this happens this value is reached that is the highest frequency at which the earth can be called a conductor. At lower frequency it will definitely be a conductor and at higher frequencies it will be difficult to call it a conductor based on this criteria. And therefore, we say that the corresponding omega is going to be 10 epsilon by sigma. One can plug in these values and the corresponding frequency would come out to be 900 kilohertz a little less than a megahertz. This is the kind of frequencies that one uses for medium wave transmission and for such frequency range earth behaves as more or less a conductor. The utility of putting various materials that are used at the frequency of frequency range of our interest into these two different categories is that it is not necessary for us to use the complete general expressions for the attenuation constant and the phase shift constant. We can use simplified expressions of alpha and beta which hold good for conductors or for dielectrics which would be simpler to deal with. And therefore, next what we do is to put down the corresponding expressions for dielectrics and for conductors. Simplified expressions for dielectrics. For a dielectric we have stated that sigma by omega epsilon should be much much less than 1. And therefore, in the part of the expression 1 plus sigma squared by omega squared epsilon squared we can use the binomial approximation giving us this equal to 1 plus sigma squared by 2 omega squared epsilon squared. And therefore, we find that alpha the general expression for which is written here simplifies to the following value it is omega times square root of mu epsilon by 2 and we have sigma squared by 2 omega squared epsilon squared which further simplifies to sigma by 2 square root of mu by epsilon. We shall compare this expression for the attenuation constant with the expression for the attenuation constant that we obtain for transmission lines under low loss approximation and we will find a very interesting comparison. Right now we say that the simplified expression for the attenuation constant for the dielectrics is like this. On the other hand the expression for beta for this approximation is going to be omega times mu epsilon by 2 and then we have sigma squared by 2 omega squared epsilon squared plus 2 which also can be simplified further giving us omega times square root of mu epsilon into 1 plus sigma squared by 8 omega squared epsilon squared which goes to show that the presence of a small amount of conductivity causes a correction term to be added to the expression for the phase shift constant. Similarly, the phase velocity which is going to be omega by beta can be seen to be equal to omega by omega square root of mu epsilon into 1 plus sigma squared by 8 omega squared epsilon squared so that it is v naught into 1 minus sigma squared by 8 omega squared epsilon squared and once again we find that the presence of a small amount of conductivity causes the appearance of a small correction term in the expression for the phase velocity of the wave. Here v naught is equal to 1 upon square root of mu epsilon. The velocity of the wave if the medium had been lossless. In a similar manner one writes the expression for the intrinsic impedance eta which is j omega mu upon j omega epsilon plus sigma whole square root which we simplify as j omega mu upon j omega epsilon into 1 plus sigma by j omega epsilon whole square root which can be seen to be equal to in the approximation that we have been using as square root of mu by epsilon into 1 plus j sigma by 2 omega epsilon that is the intrinsic impedance for a dielectric medium is not entirely real it is a complex quantity it has a small reactive component. The j has appeared in the numerator instead of the denominator so once you take that into account it should be alright. So these are the simplified expressions that one can utilize when one has ensured that the medium that one is dealing with is a dielectric medium. In a similar manner one can write down the simplified expressions for materials which come under the category of conductors. For conductors we have already stated the criterion that is sigma by omega epsilon is much much greater than 1 and therefore the expression for gamma the complex propagation constant which was j omega mu into sigma plus j omega epsilon this itself can be simplified to read as j omega mu sigma under this approximation and therefore we can write this as omega mu sigma by 2 whole square root into 1 plus j. Signifying that for this case the real part of gamma and the imaginary part of gamma are both equal and now we do not need these expressions anymore and therefore we say that for this case both alpha and beta are equal to omega mu sigma by 2. Now since sigma is going to be large for conductors this is going to be a large quantity typically it will be a large quantity and therefore both the attenuation constant and the phase shift constant will be large for conductors. So based on this going to the overhead projector one can note down some conclusions of general significance for the conducting media. For conductors the first conclusion that we can write is the attenuation constant is very large and therefore the wave attenuates greatly as it propagates within the conductor. The second conclusion the phase shift is very large because both alpha and beta are equal what would be the consequence of the phase shift being very large the phase shift constant being very large the phase velocity will be very small. In fact the velocity of an electromagnetic wave in a conductor will have the same order of magnitude as the velocity of sound in A it becomes that low and the last conclusion is that the characteristic impedance is also characteristic impedance is also very small that can be seen from an expression for the characteristic impedance for good conductors. For good conductors the characteristic impedance is going to be j omega mu by sigma alright because the second term in the denominator is negligible which is equal to omega mu by 2 sigma whole square root into 1 plus j that is once again it will have a real part let us say R and an imaginary part j where both R and x are equal. So, these are conclusions of some general significance for conductors. These observations that for a conductor the wave attenuates very greatly as it propagates in the conductor brings us to the next topic of discussion today that is the depth of penetration. We have seen that the wave attenuates greatly in a conductor and therefore it would be able to penetrate into a conductor only to a small extent and this penetration is known as the depth of penetration. One defines the depth of penetration as the depth at which the wave attenuates to a value 1 by e which is roughly 37 percent of the value at the surface. We imagine a wave impinging on a conductor it will have some value of the field vectors at the surface and since it is a conductor the wave will attenuate rapidly and it will decrease in amplitude and at a certain depth it will go down by a factor of e and that distance will be called the depth of penetration. How we can obtain an expression for the depth of penetration can be seen here what is required from the definition is that e to the power e to the power minus alpha x and at x equal to delta this value should be equal to e to the power minus 1 and therefore quite simply the depth of penetration delta comes out to be 1 by alpha. While the concept of depth of penetration would apply to all materials but it is particularly relevant to conductors and therefore we can use the simplified expression for alpha which we have got for conductors and therefore this is going to be equal to 2 by omega mu sigma when we have conducting materials at hand and one can make out that for a conducting material sigma will be large and delta will be inversely proportional to the square root of frequency and the greater the frequency for a given material the lower will be the distance up to which a wave impinging on the conductor will be able to penetrate inside the conductor which to some extent could explain why it is simpler for us to receive the medium wave broadcast signals inside houses but it is relatively more difficult to receive the FM or the television signals the later two signals requires special antennas it is somewhat related to the depth of penetration phenomenon. One can obtain some idea of the typical values for materials that are used for example for copper if we choose a frequency of 1 mega hertz and calculate delta sigma was already specified for copper as 5.8 into 10 to the power 7 most per meter we know all quantities and delta turns out to be 0.0667 millimetres as we progressively increase the frequency the depth of penetration delta will go down by this relation at 100 mega hertz delta would be only 0.00667 millimetres and at 10000 mega hertz it will be 0.306 667 millimetres or in other words 0.667 micrometres or microns which is a very short distance very small distance within the surface. One can consider some other media and see what kind of depth of penetration one obtains for these media one of these medium media that we consider is say seawater sorry to be able to calculate the depth of penetration we require the constants conductivity and the permittivity for seawater we have the conductivity as 4. And the permittivity of course can be taken as 80 times that of free space and therefore the corresponding value of delta which one obtains at 1 mega hertz comes out to be 25 centimetres which is a result of some significance because one could use radio waves of appropriate frequency for underwater communication as well and there the an idea about the depth of penetration would help us in choosing an appropriate frequency for communication. Similarly for fresh water for which one has sigma equal to 10 to the power minus 3 so many moles per meter and epsilon remains 80 times epsilon naught the depth of penetration delta at 1 mega hertz is considerably large it becomes 7.1 meter so that given a certain frequency it will be far easier to carry out underwater communication in say fresh water lakes than in seawater. So in this manner these various special cases and the concept of the depth of penetration can help us in analyzing what happens in different situations or for designing systems appropriate. In today's lecture we have considered wave propagation in conducting media and we have seen that the various media can be put in two broad categories conductors and dielectrics and finally we introduce the concept of the depth of penetration. Thank you.