Electromagnetic Waves as Sinusoidal Waves

Hey, guys. In this video, we're gonna talk about electromagnetic waves mathematically described as sign of soil waves. Okay, let's get to it. Now, as we know, a common electromagnetic wave could be represented as a Sinus soil wave or sorry as an electric and magnetic field, both of which are Sinus soil. The image above me was the image that we saw before, and this is the most common image for an electromagnetic wave. We have, in this case, an electric field pointing in the X direction right. We have an electric field in the extraction. We have a magnetic field in the Y direction right, and we have propagation in the Z direction. This makes our wave as we know a trans verse wave, because indeed, the direction of propagation is perpendicular to both directions of oscillation, the direction of oscillation for the electric field and the direction of oscillation for the magnetic field. So if we want to describe this mathematically, we would write them as signed functions. Okay, for example, the electric field there would be some maximum electric field, which is that amplitude of oscillation times sign off K X minus omega T omega. We already know because we've seen this in oscillating functions before is the angular frequency of oscillation. But K is something new. It's called the wave number, and I'll explain what it is in one second now. To make this a vector, we have to say in what direction the electric fields oscillating. In our case, it's oscillating in the X direction, so I'll give it and I had. The magnetic field is gonna be described the exact same way because if you look at the waves, the oscillations in the above picture, they're identical except that the magnetic field is 90 degrees from the electric field. So this is gonna be some maximum magnetic field. Be Max times Sign of once again K X minus Omega teen. Okay, and I need to give it a direction. And I'll say it since it's moving in the Y direction. Sorry, since it's oscillating in the Y direction thes air Jay's, by the way. I made a little mistake here. Those excess air, not exes. Those excess our zeez That's the position along the propagation direction. Okay, those are Z's since it's propagating in the Z direction. All right, now what? The wave number is is it related to the wavelength? It's just two pi divided by the wavelength. Okay. And we know how to relate the angular frequency to things like the frequency and things like the period. Because we've done it many times before. Okay, so let's do a quick example and get out of here. The following are the Electric and Magnetic Fields Committee of the Equations. The following are the electric and magnetic fields that describe a particular electromagnetic wave. What is the wavelength of this wave? What is the period? Okay, I'm gonna address the period first just because of what we're both most comfortable with. But it really doesn't matter how you do them, because one answer does not depend upon the other. This is the angular frequency in both cases. That's the quantity that we know is related to the wavelength. Sorry to the period. We know that the angular frequency is two pi over the period, so we know that this is going to be two pi over 4.19 times 10 to the 15. And so sorry. I forgot to step there. And so if I wanna isolate the period, I gotta multiply the period up and divide the wavelength over. So the period is two pi over the wavelength, which is going to be two pi over 4.19 times 10 to the 15. That's the angular frequency. And this whole thing is going to equal 1.5 times 10 to the negative 15 seconds, which is an incredibly small number because the frequency is an incredibly large number. Okay, we would expect the period to be very, very, very small if the frequency is very, very, very high. Okay, now we want to address the wave number. Okay? We have our equation in red above in the green box for what the wave number is. And we know that these numbers are both the wave numbers and the wave number is related to our wavelength, which is what we want to find so I can multiply the wavelength up and I could divide that over and I could get the wavelength is two pi over K, which is two pi over 1.4 times 10 to the seven, which is going to be 450 or so times 10 to the negative nine meters I've chosen to write it like this for a particular reason, because this can then be written as nanometers, which is blue light. Okay, so those are two answers. Now, these two numbers have to actually be related toe one another for a multitude of reasons. Okay, First of all, we know that Lambda divided by the period has to be the speed of the wave, which is, in this case, light. And you'll see that if you do take Lambda and you do take the period you divide them, you will get. See, that's one way to confirm that. This is in fact, a correct wave. Another way is to just take Omega and divided by K, both of which are already represented in this equation in the functions that will also equal the speed of light. And finally, we know that the ratio of imax toe be Max has to be the speed of light. Okay? And if I wrote these functions correctly, I checked the multiple times, so I'm pretty sure I did. All three of these relationships should be true. Alright, guys, that wraps up our discussion on electromagnetic waves as Sinus sort of waves. Thanks for watching