 A laser emits a sinusoidal electromagnetic wave that travels in vacuum in the negative x direction, and the electric field is parallel to the y axis. The wavelength is 10 micrometers with E max as 1.2 megavolts per meters. The question is to figure out or write the wave equations for electric and magnetic field as functions of time and position. All right there's a lot going on here so let's look at each piece of information closely. Okay so we have a laser which is emitting an electromagnetic wave in the negative x direction. Let's draw some axis and a laser positioned on the x axis pointing in the negative x direction. Now this laser emits an electromagnetic wave and we know that the electric field, the electric field is parallel to the y axis which means that the field will oscillate along the y axis and it is moving in the negative x direction. So the electric field could could look like this. We can see the oscillations are parallel to the y axis or along the y axis and the wave is moving to the left in the negative x direction. Now this wave is traveling in vacuum so the speed would be equal to 3 into 10 to the power 8 meters per second. The wavelength is 10 micrometers and wavelength is the distance between adjacent maximas or minimas in a sinusoidal wave. So this distance right here the distance between two adjacent maximas this right here is the wavelength and the wavelength is 10 micrometers so that means 10 into 10 to the power minus 6 meters. One micrometer is 10 to the power minus 6 meters. The maximum value of electric field is 1.2 mega volts per meters that would be the amplitude which is right here. This right here is the amplitude and let's call the amplitude as E0 instead of Emax let's call it as E0. We need to figure out the wave equations for electric and magnetic fields as functions of time and position. Now before we do that let's quickly recap what a traveling wave equation looks like. Let's say let's say we have a wave which is moving to the right in the positive x direction. Now since this is a transverse wave the displacement of the particles will be perpendicular to the wave travel so that means the displacement would be along the y axis or parallel to the y axis. Now this wave is moving to the right so this point right here it will go down the other point would come up so for any point on the wave its displacement will vary or change with time and at any given instant in time if you take a snapshot of the wave different positions on the wave can have different displacements. So the wave equation the wave equation should depend upon x that is position and t that is the time. Now this is a sine wave right here so the equation the equation can look like this because the displacement of the particles on this wave is in the y direction we can add a j cap over here to show that. This shows that the particles are only moving vertically that is along the y axis. Now the negative sign over here it means that the wave is moving in the positive x direction and therefore you see a x over here as well because at different positions of x you have different displacements of the particles on this wave. If this wave let's say was moving in the positive z direction but still oscillating in the y direction and in that case you would have kz instead of kx. Now for this wave for now we can only see the electric field oscillating sinusoidally and we know that the wave is moving in the negative x direction so there must be a plus kx plus omega t in the wave equation for this electric field and because the wave is oscillating along the y axis there should be there should be a j cap as well. So let's see how the wave equation can look like. This right here is a wave equation. E0 is the amplitude of the electric field and because it is oscillating in the y direction we have a j cap over here. We don't know where the magnetic field will oscillate but one thing that we do know is that it will be perpendicular to the electric field oscillations and that it will move along with the electric field in the negative x direction. So let's see the wave equation for the b field, the magnetic field and this is how it can look like. It is incomplete because we don't know where the magnetic field is oscillating. So maybe let's try and figure that out. Let's try and figure out the direction of oscillation of the magnetic field. So let's make some space over here. So far we know the direction of the oscillation of the electric field and the direction where the wave is moving. The direction where the wave is moving or propagating is the same direction as the wave velocity. So let's pick a position. Let's say we pick this position right here. So the electric field at this point is pointing vertically up and the direction of propagation is to the left and let's call that vector which is pointing in the direction of propagation. Let's call it s and the direction of propagation that is along the vector s, it's given by the cross product of e and b. Now over here s cap would be negative i cap because it is because the direction of propagation is in the negative x direction and e cap would be because the electric field at this point is in the positive y direction. So if we write that this is how it can look like. Now we do not know where the magnetic field is oscillating and we can actually use two ways to figure out the direction of b cap over here. We can use the right hand rule for cross products. b cap should point in a direction such that on curling our fingers from j cap to b, the magnetic field, our thumb should point in the minus i cap or the negative x direction and when we do this, when we curl our fingers in this way, we will be able to see that b cap is in the negative z direction that is the minus k minus k direction away from us or the plane on the screen. So the direction of magnetic field at this point could look like this. It is pointing away from us that is in the negative z direction or away from the plane on the screen. We can also use this diagram to remember the results that we get from applying the right hand rule. If we move in a clockwise direction from i to j to k and if we do i cross j, we get k cap and similarly if we do j cross k, we will get i cap and so on and so forth. If we do k cross i, we should get j cap. Now over here, we are supposed to get minus i cap. So the only way that we can get that if our magnetic field is pointing in the negative k or the negative z direction. So again, that means that the magnetic field at this point, it must be pointing away from us and in the negative z or negative k direction. Now let's try and figure out the direction of magnetic field again at a different position. So let's say we pick this point over here. Now at this point, the electric field is vertically down and the direction of propagation is still in the negative. It's in the negative x direction and we can use the right hand curl rule again to figure out the direction of the magnetic field oscillation. So our thumb should be pointing to the left and if we curl our fingers from A to B and I strongly encourage you to do that, you should be able to see that the magnetic field should be pointing towards you or outside the plane of the screen. And in this case, when the electric field oscillation is in the negative y direction that is minus j cap, you get a magnetic field oscillation in the positive z direction or plus k cap. That is the magnetic field should be pointing outside from here and it should be pointing inside or away from the plane of the screen at this point. So magnetic field oscillations, they can look, they can look like this. Now if you try and complete the equation over here, we can add a negative k cap which shows that whenever the electric field is oscillating in the plus z direction, you get corresponding magnetic field oscillation in the negative z direction. Now let's try and figure out the value of B naught. That is the maximum value of magnetic field in its oscillation. So let's make some space over here. Now we already know the maximum value of that is the amplitude of the electric field and we know the speed with which the wave is moving. It's moving with the speed of flight because it is moving in a vacuum and there is a relation which connects the speed of the wave with the magnitude of the electric field at any given point and the magnitude of magnetic field at that same point and that expression looks like this. So using this, we can figure out the, so using this, we can figure out B naught because we already know the maximum value of electric field oscillation that is 1.2 megavolts per meters. Now when we solve this, because we should get B naught as this is equal to E naught divided by C and E naught that is E maximum that is equal to 1.2 megavolts per meter. So this is 1.2 into 10 to the power 6 volts per meters and that is divided by 3 into 10 to the power 8 meters per second. Now when we work this out, we get the magnitude of the magnetic field oscillation to be as 4 into 10 to the power minus 3 Tesla because that is the unit for the strength of the magnetic field. So in case of B naught, we can write 4 into 10 to the power minus 5 Tesla. Now all that remains is to calculate the value of the wave number k and the angular frequency omega. So let's do that k is given by 2 pi divided by lambda which is the wavelength and wavelength over here is 10 to the power minus 5 meters. So when we work this out, when we place, when in place of lambda, we write 10 to the power minus 5 and when we work this out, we get the value of k to be as 6.3 into 10 to the power 5 radians per meters. Radians per meters because those are the units. So you have 2 pi radians divided by the wavelength which is measured in meters. Then we have omega. Omega is 2 pi into F and we can work on this. We can manipulate this relation. We can write in place of F, we can write 2 pi into C divided by lambda because frequency, the wave speed and the wavelength are related by the wave relation which is that the wave speed equals wavelength multiplied by the frequency and we can further work this out because we already know 2 pi by lambda. We just calculated it over here. So we can write this as, we can write this as k into C and when we multiply 6.3 into 10 to the power 5 with the speed of light, we get, we get omega as 1.9 into 10 to the power 14 radians per second. Now we can place the values of k and omega into our equations and when we do that and also the value of B naught and when we do that, finally we get our wave equations. This is how the wave equation for electric field can look like. You have the maximum value of electric field and the values of k and omega placed and the corresponding wave equation for magnetic field can look like this. Notice the direction is in the negative k because in the electric field oscillates in the plus j direction, you get a corresponding magnetic field vector in the negative k direction. If there would have been minus j over here, if we would have picked this vector, then we would have written plus k because in that case the magnetic field oscillates in the plus z direction.