 These are Maxwell's equations which describe the behavior of electric and magnetic fields and also how they relate to each other. We also know that light is a form of electromagnetic wave and in this video, in this video we will use all four of these Maxwell's equations and develop some basic ideas about light or electromagnetic waves. Now to apply Maxwell's equations we need electric and magnetic fields, we need both of these fields. So we will assume a way in which the fields are arranged, we will assume a way in which the fields are configured and to keep the maths simple, we will take a very simplified electromagnetic wave where the electric field has only a y component and the magnetic field has only a z component and we will assume that the fields are moving in the positive in the positive x direction with some speed that we don't know of. We will then test whether our assumed fields, whether they follow Maxwell's equations. And if yes, what are some of the conditions? What is the speed with which they should move? Should they be transverse or should they be longitudinal? These are the basic ideas that we will develop throughout the video. And in doing so, we will fully see the power and glory of all the Maxwell's equations as they come together to explain the nature of light. Before we consider each of the four equations, let's imagine that all space is divided into two regions, the first region and this is the second region, by a plane that is perpendicular to the x axis and we can call this a planar wave front. Now at every point to the left of this plane, there is uniform electric field in the y direction and uniform magnetic field in the z direction. But both these fields, both these fields are zero, they are zero to the right of this plane. And we can also say that this planar wave front, this wave front is moving to the right, that is in the positive x direction with some speed c and we do not know what this speed is right now. Again, this is a very simplified electromagnetic wave that we are considering over here and we won't worry ourselves with the problem of actually producing such a field configuration. Instead, we will just ask whether it is consistent with the laws of electromagnetism, that is with Maxwell's equation. All right, now let us first verify if this wave satisfies the first and second Maxwell's equation. Here we have the waves and I'm writing this on the top just so that we know which one is electric field and which one is magnetic field. Now to verify these waves satisfy the Gauss's law of electricity and Gauss's law of magnetism, that is the first two Maxwell's equations which you can see over here. To do this, we take a Gaussian surface, a rectangular box with sides parallel to x, y, xz and yz planes. Here we see that the electric field is the same on the top and the bottom side of the Gaussian surface. The number of electric field lines that are coming outside the box and the number that are going in, that's the same. So that means that the total electric flux is zero. Similarly, the magnetic field lines, they are coming outside the left side of this Gaussian surface and an equal number of magnetic field lines are entering the Gaussian surface from the right hand side. So the total magnetic flux through the box is also zero. Even if part of the box is in the region where E and B is zero, so electric and magnetic field is zero to the right of the plane so there is none over here. Even in this case, we see that the electric and magnetic flux is zero. So integral of E.da, this is equal to zero as it should be because there is no queue enclosed, there is no charge enclosed in the Gaussian surface. So the right hand side is also zero. And also the second equation is also satisfied because we see that the total magnetic flux is zero so integral of B.da is zero for our assumed wave. So even the second Maxwell's equation is satisfied. Now when we started, we assumed that electric and magnetic fields, they only have one component. That is, electric field has only a component in the y direction and magnetic field only has a component in the z direction. We never assumed any x component for either of these two fields. We never assumed any component along the direction of motion. But if electric or magnetic fields, if either of these two, if they had an x component parallel to the direction of propagation and if the wavefront, if the wavefront were inside the box, then there would be flux through the left hand side, like there would be flux coming in through the left hand side of the box but not the right hand side because this part, this region had no electric or magnetic field. So there is no flux coming out from the right hand side or from the front side of the box, from this side of the box. There is flux entering the box though. But the right hand side of these two equations, they must be zero because there is no queue enclosed and B.da is always zero. So if electric and magnetic field, if they had an x component, then they would violate Maxwell's first and second equation. Then the two equations will not be followed or obeyed. So to satisfy the first and second equation, the electric and magnetic fields, they must be perpendicular to the direction of propagation. None of their component can be parallel to the direction of propagation. And therefore the wave, the wave must be transverse in nature. Both these fields, they must be perpendicular to a direction of propagation. They can't be parallel, otherwise the Maxwell's equations will not be followed. So that is one, that is first insight into the nature of an electromagnetic wave. Now let's test whether this wave satisfies Faraday's law. So here all these wave friends are moving to the right and let's say in a time delta t, in a small time interval delta t, this wave friend moved to the right and in doing so, it covered a distance of c dt because the wave friends are moving with a speed of c. Now for the third Maxwell's equation, that is Faraday's law, we need to carry out a closed loop integral of a dot dl. And for that, we will introduce a rectangle and apply the closed loop integral of a dot dl across this entire rectangle. This rectangle has a side length of a. And let's say that our rectangle is big enough so that the right side of this rectangle is still in the space where the electric and magnetic field is zero. Now if we look at this wave friend which has moved forward a distance of c dt, we see that it has swept out an area, it sweeps out an area, sweeps out this much area through the rectangle. And we are interested in this area because this can give us some idea of how much the magnetic flux has changed. Let us look at this entire situation from the side. So now we are looking from this entire setup standing here. So we have the y axis going forward, the z axis will be coming towards us and x is to the right. And you have the electric field lines going up in the positive y direction and the magnetic field lines which you see as blue dots over here, they are in the positive z direction. So now here is our rectangle, let's name it efgh with side length of a. And the wave front has traveled a distance of c dt, sweeping out this much area in a small time interval dt. We can take an area vector pointing in the positive z direction. We can even take it pointing in the negative z direction, the area vector, that doesn't really matter. But once we pick an area vector, we are then constrained to choose one path along this loop either clockwise or anticlockwise. Since we have picked an area vector which is pointing in the positive z direction, then according to the right hand curl rule, our thumb pointing upwards, we see that the curl of the fingers would be anticlockwise. So the path of moving around the loop will be in the anticlockwise direction, that is a dl vector. Now we can integrate e.dl counterclockwise around the rectangle. Let's do that. So we have integral, closed loop integral of e.dl and this is equal to the integral of e.dl along the sides he, ef, fg and gh. So let's start with integral of e.dl along the side he. Now here we see that the electric field and the dl vector, they are making an angle of 90 degrees with each other. And when we open up the dot product, we get a factor of cos theta. So here cos 90, that would be zero. So integral along the sides he and even along the sides fg, that would be zero because cos 90 is zero and that makes the entire dot product as zero. So I'm just writing zero for he and also for gf, I'm writing zero. And now we have integral along the sides fe. Here we see that there is no electric field in this part of the space. So if there is no electric field, the integral would again be zero. So we have one more zero. Finally, we have integral along the side gh. Now here dl and electric field, they are making an angle of 180 degrees and cos 180 is minus one. So e.dl, that would be the magnitude of electric field that is e into dl and the total length of this side is a and that is multiplied by cos 180 degrees, which is minus one. So integral of e.dl around this rectangle, this comes out to be equal to minus ea. So the left-hand side of this equation is non-zero. It has some value, it is minus ea. And to completely satisfy Faraday's law, there must also be some change in the magnetic flux through this loop. And for that, there has to be, there has to be a component of magnetic field in the z direction, either pointing towards us or pointing away from us. Only then there will be some change in magnetic flux, right? If there is a magnetic field in the x or y direction, there will be no magnetic field lines passing through this area and no change in magnetic flux. So there has to be some magnetic field pointing in either positive z or negative z direction. We assumed it to be in the positive z direction. So let's see if that assumption holds. Let's see if that assumption is correct. Now, during a time interval of dt, the wave front moves a distance of cdt to the right, sweeping out an area and the magnitude of this area, that will be a into cdt. So area, area is ac into dt. That means that during this time interval, the magnetic flux, that is d phi, through the rectangle efgh, increases by an amount that is equal to magnetic field strength into the area, that is ac dt. And d phi by dt, d phi by dt, if we divide both sides by dt, d phi by dt would just be equal to bac because dt would just get cancelled. So rate of change of magnetic flux, this is equal to ba into c. Now, if we substitute these two equations in the Faraday's law, let's see what do we get. All right, upon substituting, we get minus EA as equal to minus bac. Removing the minus sign and canceling A, this comes out to be equal to E, that is equal to CB. And this is like in a very important insight because this tells us that our wave will only follow or satisfy Faraday's law if the wave speed, that is C, and the magnitudes of electric and magnetic fields are related in this manner. But let's say, let's say if we assume that the magnetic field is pointing in the negative z direction, let's say if the configuration assumed force, if it was like this, with electric field still in the y direction and magnetic field in the negative z direction, and the field line still moving to the right in the positive x direction with some speed C. So is this field configuration possible? There won't be any change in the closed loop integral of E dot dL because electric field is still in the positive y direction, but the magnetic field is in the negative z direction. So the rate of change of magnetic flux in this expression, we will get a negative b and this will be minus bac. Then after substituting both of these into the Faraday's law, we will get an extra negative sign in this final expression. This would be E equals to minus Cb. But since electric fields, wave speed, and the magnetic field all have positive magnitudes, no solution would have been possible then. So turns out if the directional propagation is in positive x, fields cannot be arranged with electric field in the positive y and magnetic field in the negative z direction. However, this field configuration will hold if we assume that the wave is moving in the negative x direction. Then it will satisfy the Maxwell equation. And I strongly encourage you to maybe try that later, try that on your own. One thing that you would have to do is assume every electric and magnetic field on this side of the space, and then this side of the rectangle will be in a space where there is no electric and magnetic field, basically just flipping this entire scenario. And finally, you should be arriving at this condition which needs to be followed if the waves are to obey the third Maxwell equation. This also tells us that the direction of propagation of the wave is determined by how the electric and magnetic fields are oriented. So if electric field is in the positive y and magnetic field is in the positive z, then using the right hand curl rule, if we curl our fingers from E to B, and if we do that, if we curl our fingers from E to B like this, we notice that the thumb points in the right direction, and that is the direction of propagation. Similarly, in this case, if we curl our fingers from E to B, we will notice that the thumb will point in the negative x direction. So the direction of propagation is always in the direction of E cross B, and this is again one major insight into the nature of electromagnetic waves. Finally, we will carry out a similar calculation using Ampere's law, the last Maxwell's equation. There is no conduction current, so the factor of IC conduction current in the last Maxwell's equation is zero. So Ampere's law becomes this. Now to check if our wave is consistent with Ampere's law, we have moved a rectangle, we have moved a rectangle so that it lies in the x z plane. And we again look at the situation at a time when the wave front has moved forward or traveled forward through the rectangle for a time interval delta t, sweeping out some area, sweeping out some area through the rectangle and moving forward a distance of c dt. Now when we look at the situation from the top, when we look at it from the top, it looks somewhat like this with the presence of a rectangle. Now we are looking at it from the top, so z goes down, x is to the right, and now the magnetic fields, these blue lines, you see them moving in the positive z direction as assumed, and the electric field lines, the pink dots, they're in the positive y direction again as assumed. Some of the part of the rectangle is in the region of electric and magnetic fields, and the other part, and this side EF is in a region where there is no electric or magnetic field. Again the side length is A, and the rectangle's name is still the same, that is EFGH. Now following the last equation steps, we again take an area vector and we can take it pointing outside in the positive y direction, which will constrain us to choose one path around the rectangle, and again using the right hand rule that will be in anti-clockwise direction, even for this case. At this point I want you to pause the video and apply this closed loop integral of B.dl around this rectangle, see the value that you get, and also see what is the rate of change of electric flux through this rectangle in a time interval dt. The process is very similar to what we just carried out for the third equation, but here instead of E.dl, we have B.dl, and instead of magnetic flux, we have electric flux. Okay pause the video and give this a shot. All right, hopefully you have given this a shot. So before we look at the closed loop integral of B.dl, first let's draw the dl vector that will be counterclockwise because the area vector is pointing outside the plane of the screen, so the curl of the fingers come out to be in an anti-clockwise direction. So this is your dl vector. Now B.dl across the sides, HE and GF will be zero because the angle between the magnetic field and the dl vector is 90 degrees, cos 90 is zero, so the dot product comes out to be zero. Also the integral across a length EF is also zero because there is no magnitude of magnetic field in this region of the space. But across a length GH, there will be some magnitude and turns out the angle between the magnetic field and dl vector is zero degrees in this case, cos zero is one, and the length of this side is A. So on solving this integral, this comes out to be equal to B into A. And if you look at the change in electric flux, d phi, d phi, this is equal to the magnitude of electric field that is E into the area that was changed. And in this case, the area that was swept out was A into CDT. So this is EAC into DT. And the rate of change of electric flux, this would just be E into AC. Now when we place these two equations in the Ampere's law, we get BA equal to mu naught into epsilon naught into EAC. A over here just gets cancelled. And finally, this equation comes out to be equal to B that is equal to mu naught epsilon naught into C into E. Now if we compare this expression with the expression that we arrived for Faraday's law, which looked like this, for comparing these two, we can keep electric field on one side and take everything else on the other side. So when we do that, this is E that is equal to one upon mu naught into epsilon naught into C. And that is multiplied with the magnetic field strength. And E is already equal to CB. So for our wave to follow Ampere's and Faraday's law, both of these two equations should be true. And the only way that these two equations can be true is if this factor, if this factor is equal to this factor. When we make them equal, we see that the wave speed, this is equal to one upon mu naught epsilon naught into C. We only take this C to the left hand side, we get C squared. And on removing the square, we get C, the wave speed that is equal to one upon mu naught into epsilon naught. This is a wave speed. This is a speed with which the electromagnetic wave should move if it has to follow all the four Maxwell's equations. And when we plug in the value of these constants, we get the speed of electromagnetic wave as three into 10 to the power eight meters per second. So only if the electromagnetic waves of all wavelength, all frequency, only if they are traveling with this speed and vacuum, only then they will be following the Maxwell's equations and only then they will be physically possible. So we arrived at quite a few insights into the nature of electromagnetic wave in this analysis. Let's summarize them over here. The waves have to be transverse, the direction of propagation is always in the direction of E cross P. And the wave is consistent with Ampere's and Faraday's law provided these two equations hold, provided the electric and magnetic field and the wave speed is related in this manner where the wave speed has to be equal to three into 10 to the power eight meters per second. And it is completely mind-blowing that just with a pen and a paper, we are able to derive the speed of light using nothing but the Maxwell's equations.