 This is Dr. Rupali Shalke from Valchin Institute of Technology, Sholapur. In this video lecture, we are going to discuss with the wave equation in a lossless media. The learning outcomes of this lectures is that at the end of this lecture, students are able to derive the wave equation in a perfect dielectric media and free space. Also, they are able to ribs in the wave equation in phasor form. They can solve related problems. Before starting with the derivation, let us see what is exactly an electromagnetic wave. Electromagnetic wave consists of the electric field and the magnetic field. They are, they do not require any media, so they can travel through the vacuum. They are also called as a transverse wave because the electric field and the magnetic field are perpendicular to each other as well as they are perpendicular to the direction of propagation. If you see in this diagram, we are plotted the three axis, x axis, y axis and z axis. We are representing the wave is propagating in a x direction and the magnetic field electric field is given in a y direction and magnetic field in a z direction. From this, it is clear that electric field and magnetic field are perpendicular to each other and they are perpendicular to the direction of propagation. Same representations can be seen by the sinusoidal waves, time varying field. If it is here, the electric field is representing in a y direction. Electromagnetic field is propagating in a x direction and the magnetic field in a z direction. Magnetic field is represented by the red colour and electric field is represented by the pubelines. Before deriving the wave equation, let us Maxwell's equations which we have derived in a previous lectures. The Maxwell's equations gives a relation between the electric field and the magnetic field. The Maxwell's equations are derived from the Faraday's law and the Ampere's law. The Maxwell's equation derived from the Faraday's law is given by del cross e bar is equal to minus dow v by dow t and Maxwell's equation derived from the Ampere's law is del cross h bar is equal to sigma e bar plus dow d by dow t. Before deriving the wave equations, we have to consider the assumptions or values for the particular terms that is for perfect dielectric media, the sigma value is equal to zero and the epsilon and mu value are as follows. The same thing is for the free space also. For the free space, we are considering sigma is equal to zero while the values for the mu and epsilon are changing here. Let us start with the derivations of the wave equations in a lossless media. For that, we will consider a Maxwell's equations which is derived from the Faraday's law. As we know that the relationship between the b and h, v bar is equal to mu h bar. We will substitute the value of b as mu h bar. Let us expand del cross e bar, the del function in a x, y and z coordinates. As we are assumed, our electric field in a x direction, y direction and magnetic field in a z direction and wave is propagating in a x direction. So, the variations will be dou by dou x value will not be equal to zero while dou by dou y and dou by dou z will be equal to zero and the electric field we are assumed in a y direction. That is why this term will also not be equal to zero, but e x and e z term will be equal to zero. Similarly, as we are assumed our magnetic field in a z direction, this term will not be equal to zero while h x and h y term will also will be equal to zero. Calculating and equating the a z terms, we get the equation 2. Now let us consider a Maxwell's equations derived from the Ampere's law. That is del cross h bar is equal to sigma e bar plus dou d by dou t. Here as we know the relation between the d and e, d bar is equal to epsilon e bar. We will substitute the value of d bar as a epsilon e bar and as we for a free space, we are considering sigma is equal to zero. Therefore, these equations will reduce and after expansion of del cross h bar, this will be the terms in a x y and z coordinates. Now as per the assumptions, again we will substitute the value for in this case h z will not be equal to zero while other h x and h y will be equal to zero and as signal is varying along the x axis, therefore this term will not be equal to zero while dou by dou y and dou by dou z will be equal to zero and on the right hand side, e x and e z will be equal to zero while e y will not be equal to zero because we are assumed electric field in a y direction. After substitution of this value and we will equate the ay terms and the equation 2 will be reduced. Now, using this equation, equation 2, we will differentiate the equation 2 with respect to the x. After differentiating equation 2 with respect to x, it will be double differentiation and rearranging the differentiation term. Now, substitute the value of the dou e y by dou x from the equation 1. This value will substitute in the equation above equation 3. If you observe this equation, the dou del square h z dou x square will be equal to mu epsilon dou square h z by dou t. This is the equation on the both the side, right hand side and the left hand side, we see that the equation is in a terms of h z. That is why we say that this is the wave equation for the lossless media in magnetic term or in a h z term. Similarly, let us consider now we will consider the equation 1. We will differentiate the equation 1 with respect to x so that we will reduce in the equation 4 and we will substitute the value of del h z by del x from equation 2, which is equal to minus epsilon dou e y by dou t. And then equation 4 will be in this form that is del square e y del x square is equal to epsilon e del square e y by del t square. The wave equation, if you observe again, this equation is in the terms of e y and the both the side. Therefore, this equation is for a lossless media in e y term. Now, this wave equations can also be represented in a phasor form for this. Let us consider the time varying field. For the time varying field, the wave equation for the lossless media can be written by replacing dou by dou t as a j omega that is a harmonically varying. From the equations, we will substitute dou by dou t is equal to j omega as it is a square that is why we will take the square terms and dou by dou x is replaced by del operator which is with respect to the space that is why it is we are taking it as a dou t. Now, here in the space of j mu epsilon j omega j square omega square, we substitute it as a gamma square. The gamma square is called as a propagation constant. Similarly, for the wave equation for lossless media in e y term, the equation get reduced to del square e y bar is equal to gamma square e y bar. The gamma square is nothing but in this the gamma square e y value will be equal to minus omega square mu epsilon minus sign because it is a j square. j square is equal to minus 1, but what is propagation constant is given by propagation is constant is nothing but a alpha plus j beta where alpha is called as a attenuation constant and beta is called as a phase constant. Now, pause the video and think on this. The magnetic field of uniform plane wave in a free space if it is given by this particular equation z 0 in bracket cos 2 pi into 10 raise to minus 90 plus 30 pi x a y bar then what could be the direction of the wave propagation, what could be the direction of electric field magnetic field at x is equal to 0 and t is equal to 0. Now, if you see that the direction of wave will be in a negative x direction and from this equation if you substitute t is equal to 0 and x is equal to 0 then this will be cos 0 cos 0 h bar will be equal to h 0 a y bar h 0 will be its magnitude amplitude then h bar will be given by h 0 a y therefore, the direction of the magnetic field will be in a y direction as it wave is travelling in a x direction magnetic field is in a y direction the only remain state is in a z direction therefore, the electric field at t is equal to 0 and x is equal to 0 it will be in a z direction. These are the few references through which we are collected the data.