 Now, next is the circular waveguide. So, a circular waveguide has a circular cross-section. So, it has a circular cross-section. So, we have boundaries for all theta and at all r equal to a. So, this is the radius and this is the radius is a, this is the radial direction, the radius is a and it is free to propagate in the z direction. So, now again all components of the electric field and magnetic field will satisfy this wave equation. So, del perpendicular is the transverse del and again here kc square is equal to omega square by c square minus kg square. So, since this is a circular waveguide, we will solve this in cylindrical coordinates. So, in cylindrical coordinates del perpendicular square can be written as 1 by r del by del r of r del by del r plus 1 by r square del 2 by del theta square. So, this two-dimensional wave equation in r and theta, if you substitute here, this becomes like this. Now, here psi can be either E z or B z. So, you know that you need to solve only for E z or B z. So, you know from Maxwell's equation that curl of E, curl of E is equal to minus del B by del t and curl of B is equal to 1 by c square del E by del t. So, as we did in the previous lecture, you can simply write each component of, component from this equation and each component from this equation. So, you will get three equations here, three equations here. And you know that the variation, the exponential variation is e to the power of i, k, g, z minus omega t. So, if you take the derivative with respect to z, you will get i, k, g and if you take the derivative with respect to time, you will get minus i omega. So, in these three equations and in these three equations, if you substitute del by del z with i, k, g and del by del t with minus i omega, you can see that all the transverse components that means E r, E theta, B r, B theta, they can be obtained from E z and B z alone. So, you just need to find out the, you just need to solve for E z and B z and then using that, you can get the values of E r, E theta, B r and B theta. So, we have to solve the two dimensional wave equation only for E z and B z. And the boundary conditions are that E z is equal to 0 at r is equal to a, that means the tangential component of electric field is 0 and del psi by del m that means where psi is equal to B z is 0 for r is equal to a, this is the normal component of magnetic field is equal to 0. So, you just have to solve the wave equation for E z and B z with these boundary conditions as we did for the case of rectangular wave ride. So, let us say the TM case. So, here for the TM case, B z is equal to 0 and E z will definitely exist. So, now we have to solve this for E z and from E z we can derive E r, E theta, B r, B theta. So, we have to solve just one equation for E z. Now, we know the form of electric field is like this. So, the amplitude here now depends on r and theta, r and theta are the directions in which the boundary is applied to the system. So, the wave is free to propagate in the z direction. So, it is a propagating wave in the z direction, but it will form a standing wave in r and theta direction. So, this is the wave equation for written in terms of E z. We can expand del perpendicular square and we get this expression. So, this is the total wave equation in the z direction. Now, as before as we did for the rectangular wave guide, we can write E z r theta as r which is a function of r only and capital theta which is a function of theta only and solve this equation by using the method of separation of variables. So, substituting instead of E z in this equation, if we substitute instead of E z capital R and capital theta. So, we get like this. So, so this is the expression. Now, we divide throughout by r and capital theta. So, here this is derivative with respect to r. So, theta comes out and if you divide with r and theta, theta is cancelled here and similarly from here this expression, this is derivative with respect to theta. So, r comes out and when you divide by capital R capital theta, r will get cancelled. So, we are left with this equation. Now, we, so this is the expression. Now, again we can rearrange this and write it as this expression here. So, now if you notice the left hand side, it is a function of r only and the right hand side is a function of theta only. So, again as before this is possible when each side is equal to the same constant. So, you can make this as equal to, you can equate both the left hand side and right hand side as equal to some constant m square. So, now we get two, so here we have kept it equal to m square. So, now you get two equations, one is the radial equation and one is the equation in the theta coordinate. So, let us write the equation in the radial coordinate. So, we get here r by r del r by del r r del r by del r plus kc square r square is equal to m square and in the theta coordinate we have del 2 capital theta by del theta square plus m square capital theta is 0. Now, this equation in theta, this is again the equation of a simple harmonic oscillator and it has solutions of the form of cosine and sine theta. So, you can write the solution capital theta as a m cos m theta plus b m sin m theta. Now, the relative amplitude of a m and b m, they determine the orientation of the field in the right. So, that means, which of the two components dominate. For a circular waveguide and for any particular value of m, the theta axis can always be oriented. So, you can always orient the theta axis such that one of them goes to 0. So, let us do that for simplicity. So, we orient the axis such that b m is equal to 0. So, we are left with just one component, theta is equal to a m cos theta. So, the variation in theta, so we are solving here for ez. So, the variation in theta in capital theta in theta is cosine or is cosine form. Now, let us solve the second equation. So, here this is the equation in the radial direction. So, this is the equation. So, we can take m on the left hand side and then we multiply this equation throughout by r and rearrange. So, we get this expression r del by del r multiplied by r del r by del r plus kc square r square minus n square times r is equal to 0. So, now, here both in the numerator and denominator we can multiply by kc, kc is independent of r. So, we can just divide and multiply by kc here as well as here. So, now, we have this net resultant equation. This is a Bessel's equation in terms of kcr and it has two solutions. One is the Bessel function jm kcr and the other one is the Neumann function. Now, let us denote kcr with capital X. So, we can write this expression in terms of x. So, the solution of, so this as I said is the Bessel's equation in terms of x or kcr and it has two solutions, the Bessel function and the Neumann function. So, Bessel functions are, so I plotted it here. So, Bessel functions are basically functions of they are of various orders. So, we have j0, this is the Bessel function of 0th order. Then we have j1 which is the Bessel function of first order. Then we have the Bessel function of second order and so on. And similarly, we have Neumann function, the Neumann functions again of the 0th order, the first order and the second order and so on. Now, if you notice this from these equations, we see that at x is equal to 0 or at r is equal to 0, the Bessel function, it has a finite solution. So, here it is, so it is if you see jm x is proportional to x to the power of m at x is equal to 0. However, if you see the Neumann function, it is infinite at x is equal to 0. This is infinite. Now, since the field must be finite in a waveguide at r is equal to 0, you cannot have infinite field, it is an unphysical solution. So, we reject the solution of solution that is the Neumann function and we accept only the Bessel function as the solution. So, now, so the solution for this equation is now r is jm kcr. So, now we can write the total solution of the ez field. So, ez was r, capital R into capital theta. So, capital R now is jm kcr and capital theta is am cos theta which we had just found out. So, the complete solution for TM mode inside the waveguide is given as ez r theta zt is am which is some constant jm kcr. So, it is the variation is of the form of Bessel function in the r direction and cosine variation in the theta direction and then it is a propagating wave in the z direction propagating with a propagation constant kg. So, now let us see the boundary conditions for the TM mode Bz is equal to 0 and ez exists. So, the boundary conditions are that at r is equal to a. So, let us see this, this is the radius. So, r is equal to a, now we have this ez component. So, ez component is here. So, at r is equal to a this is ez component and this ez component is the tangential component. So, by definition or by boundary condition ez has to go to 0 at r is equal to a for all theta. So, in this equation if we substitute ez is equal to 0 at r is equal to a we get 0 is equal to am jm kca. Now, what does this mean? This means that jm is 0 for all the roots of the Bessel function. So, where is jm 0? So, jm is 0 at this for j0, j0 is 0 at this point, it is 0 at this point, at this point and so on. Similarly, j1 is 0 at this point, this point, this point and so on. And similarly, j2 is 0 here, here and so on. So, these are the roots of the Bessel function. So, the above equation has infinite number of roots. So, we can write this now as xmn is equal to kca. Now, here xmn is the nth 0 of the Bessel function so nth 0 means it is telling us about the root of the Bessel function. So, wherever this is 0, this is the root of the Bessel function. So, xmn is the nth 0 of the Bessel function jm. So, physically n represents the number of cycles of variation of ez. So, how many times ez goes to 0? So, let us say for this case if ez has to be 0 here. So, if we take j0, it can be 0 here or it can be 0 here. So, the variation can be so for different modes, one variation could be like this, another variation could be like this, in the radial direction of ez. So, kc is equal to xmn where xmn denotes the nth 0 of the Bessel function jm xmn by a. So, cutoff frequency from here you can calculate omega cmn is given by cxmn a. Okay. So, the various zeros of the Bessel function can be read from the table. So, the table is shown here. So, for example, the first 0 of the Bessel function j0 for j m is equal to 0 and the first 0 is at so the first 0 is at almost approximately 2.5. So, we see that it is 2.405. Similarly, the second 0 is at 5.5 to the third 0 is at 8.654. So, these values can be read from the tables. Now, for te case for the te case ez is equal to 0 and Bz will exist. So, now we have to solve for Bz. So, again as before, since the boundaries are applied in the r and theta direction. Okay. So, it is the circular waveguide is bounded in r and theta direction and open in the z direction. So, the equation can be written as Bz r theta. So, the amplitude is a function of r and theta only and propagating in the z direction. Okay. So, we have to solve the wave equation for Bz and again we know that e r e theta b r b theta they can be obtained from Bz. So, this is the wave equation and we write it for Bz. We can expand this in cylindrical coordinates and then again as before solving it with separation of variables. Bz is a function of r and theta where r is a function of r only and capital theta is a function of theta only. So, again we get this an expression like this. So, we can solve it and now we have to apply the boundary condition that the normal component of magnetic field is 0 at the boundary at r is equal to a for all values of theta. So, we apply the boundary condition del Bz by del r is 0 at r is equal to a for all values of theta. So, or in other words now, so we know the solution is of the form of Bessel function. So, now it will be we have to differentiate the Bessel function with respect to r and that is equal to 0 at r is equal to a. So, in other words jm prime kca is equal to 0. So, jm prime is what it is the derivative of the Bessel function with respect to r. So, again you can get this from tables these values can be got from tables. So, here jm prime xmn is equal to 0 and so xmn prime which is the derivative. So, this is the nth root of the derivative of the Bessel function jm. So, this is equal to kc into a from here kc you can get it as x prime mn divided by a and again you can calculate the frequency of the mn mode. So, here x prime mn is the nth 0 of the derivative of the Bessel function that means j prime m. So, now Bz can be written as again am jm kcr cos m theta and propagating in the z direction with a propagation constant kg. So, these are the Bessel function and the derivative of the Bessel function. So, this is and both the values the zeros of the Bessel function as well as the derivative of the Bessel function can be found out from these graphs or by reading tables. Now, let us come to the rectangular cavity. So, so far we were seeing that we had applied boundary conditions in two directions and the wave was free to propagate in the third direction. Now, we are applying boundary in the third direction as well. So, now it should form a standing wave in all the three directions. So, there will be no propagation just the standing wave form in all the three directions. So, first let us consider the rectangular cavity. So, rectangular cavity for the Te mode Bz is equal to 0. So, remember there is no propagation in the cavity because now you have applied boundary in all the directions. So, when you say Te transverse electric, so you would wonder it is transverse with respect to what because now there is no propagation of the wave. So, by convention we take it as transverse to the z direction. So, transverse electric means that means the electric field is transverse to the z direction. So, Bz is equal to 0. So, Bz will exist and now if you solve it again, if you solve the wave equation in x, y, z and you apply the boundary conditions that the normal component of magnetic field is 0 and tangential component of electric field is 0, you will get a standing wave in all the three directions. So, you see here a cosine variation in x cosine variation in y and a sinusoidal variation in z and it is no longer of the propagating wave. So, there is no kg involved here. So, earlier you had this expression e to the power of i kg z minus omega t. So, now this kg term is not there, we are just left with this time variation. So, there is only time variation, there is no variation, there is no now propagation in the z direction, it is a standing wave in all the three directions. Again as before you can find out the cutoff frequency. So, now you have boundaries in three directions. So, you will have m, n and p where m, n and p are integers. So, omega c m and p is given by is equal to pi c under root m square by a square plus m square by b square plus b square by d square. So, you see that the cutoff frequency of all the modes it depends only on the dimension of the system, others are all constant. Similarly, now for the tm mode, tm mode can also exist in the cavity, we have vz is equal to 0 and if you solve it, you get ez is equal to e0. So, sinusoidal variation in x, sinusoidal variation in y and the cosine variation in z and no propagation just a time variation, the fields are varying in time. So, just remember the case of the stretch string. So, here it is in the stretch string also there is no variation with space, the ends have zero displacement and the center point has maximum displacement, but there is always variation with time. So, just like that here, there is variation with time, but no variation with space. So, the cutoff frequency is again given by the same formula for both te and tm modes. And here m, n and p are integers and they represent half way variation in the fields in the x, y and z direction respectively. Next is the cylindrical cavity, now cylindrical cavity also again similar way we can solve. Now, here we put n plates at z is equal to 0 and at z is equal to l. So, here it is now closed in all the three directions. A cylindrical waveguide was open in the z direction, but now we have put in plates here at both the ends. So, now it is closed. So, a cylindrical cavity is generally used for acceleration and you can solve for the fields in the cavity by for e z and b z for either tm mode or te mode. And I have written the results here directly. So, here for the tm mode b z is equal to 0. And when you so you can put e z in the wave equation and then put the boundary conditions that the tangential component of electric field is 0 and normal component of magnetic field is 0 and solve for it and then you will get e z like this. So, you see that the e z has a Bessel function dependence in R and cosine function in theta dependence in theta and z. So, this is similar to the case of the waveguide. Now, since it is bounded in the z direction also. So, you have a cosine variation in the z direction and again it is a propagating it is not a propagating wave it is a standing wave in all the three directions r theta and z. From e z you can find out the values of e r, e theta, b r and b theta and they are shown here. So, everywhere notice that the variation in r depends upon the Bessel function or the derivative of the Bessel function whereas the dependence in the theta and z direction are sinusoidal. You can calculate the frequency. So, here the frequency the cutoff frequency or the frequency of the Mnp modes where Mnp are integers is given by this formula it is c by 2 pi under root xmn square by rc square rc is the radius of the cavity plus p pi by l square l is the length of the cavity and as before as in the case of waveguide xmn is the nth 0 of the Bessel function jm and xmn is simply kmn into rc. Similarly, you can solve for similarly you can solve for Te mode in this case e z is equal to 0 and you can find out the value of bz. So, again bz if you see in the radial direction it has dependence it has a Bessel function dependence and in theta and z direction it has cosine sign variations this is true for all the components. So, you can find out the cutoff frequency here the cutoff frequency is different from the the formula for cutoff frequency is different from that of the Tm mode. So, here it is c by 2 pi x prime square mn by rc plus p pi by l square. So, here this is the nth root or the nth 0 of the derivative of the Bessel function. So, there it was for the Tm mode it was nth 0 of the Bessel function jm. So, let us see how the modes are in a cavity. Now, let us say we have so, both Tm and Te modes can exist depending on the values of m, n and p. So, let us say you have mode 1 at frequency fc 1, mode 2 at frequency fc 2, mode 3 and 4 at frequency fc 3 and so on. So, there will be there will be many such modes. Now, since boundaries are applied in all the three directions the electromagnetic wave is a standing wave in all the three directions. So, there is no propagating wave. So, only discrete frequencies are allowed. So, only when this electromagnetic wave that you try to put inside the cavity when it matches with these values that means it matches with the frequency of these modes. So, it is equal to fc 1 or it is equal to fc 2 or fc 3 only then the electromagnetic wave will enter inside the cavity. At any frequency in between these frequencies the electromagnetic wave will be reflected there because there is no propagation in the case of waveguide. So, if you had put in the wave in between the two frequencies it would have adjusted the wavelength of propagation and would propagate inside the waveguide. But here since it is forming a standing wave in all the three directions. So, here what is happening is that only when you feed in power at these frequencies the electromagnetic wave enters inside the cavity at all other frequencies it is reflected back. So, only for these frequencies the wave will enter the cavity and it will form a standing wave pattern corresponding to that mode. So, just summarizing what we did today we saw a rectangular waveguide. So, here this is the cutoff frequency for Mn mode for both Te and Tn the formula is the same. For circular waveguide we have different formulas for the cutoff frequency for the Mn mode. So, here it is in the case of Te mode it is we take derivative. So, X prime Mn this is the Nx0 of the derivative of the Bessel function. And then from waveguides we went to cavity. So, in the cavity again in the rectangular cavity the formula for cutoff frequency it depends upon the dimensions of all the three systems and we saw that the formula is the same for both Te Mn P modes and Tm Mn P modes. In a circular cavity the formula is different for both Te and Te and Te Mn P modes. So, there will be P here. So, to summarize in a waveguide the wave forms standing wave in the direction in which there are conducting boundaries. In order to satisfy the boundary conditions at these boundaries. In the direction in which there are no boundaries it is still going to be a travelling wave. So, in a waveguide any frequency above the cutoff frequency is propagated and the wave propagates in the modes that are allowed. In a cavity since there are conducting boundaries in all the three directions the electromagnetic wave is a standing wave in all the three directions there is no propagating wave. So, only discrete frequencies are allowed and only at these frequencies of the modes the wave will enter the cavity and form standing wave pattern corresponding to that mode. Any other frequency is reflected back. So, in the next lecture we will see the field pattern for different modes in a cavity we will try to understand how the field patterns look for look like for different modes in the cavity and we will try to understand that how acceleration is done using these modes of a cavity. So, because our actual aim was so remember Alvarez what did he propose he said that to use the electric fields associated with the electromagnetic waves in a high Q cavity. So, now we know how the electromagnetic waves in a cavity look like and then we will see that how using these fields we can accelerate. So, that is in the next lecture.