Everyone. So let's take a look at this example problem. There's actually a lot going on in this problem. So I'm just gonna jump right in. We're told that sunlight is initially un polarized light with an average intensity of about 1350 near the earth's surface. Now, the first part here says that if sunlight passes through two polarizer that are angled at 90 degrees with respect to each other, we want to find the intensity of the light after passing through the second polarizer. So I'm just gonna jump straight into the steps here. First thing we want to do is just draw a diagram and label what our um initial light is. So this is gonna be my diagram. I've got my initial light, which is un polarized, right? It's gonna kind of look like this. Uh or you can draw it sort of like that, I guess uh like that. All right. So this is my initial lights. This is initially un polarized and the intensity is 1350. All right. So this passes through one polarization actually one polarization filter and these things are angled 90 degrees with respect to each other. Now, it doesn't doesn't tell us which specific angles, but it turns out it actually doesn't matter because remember the only thing that matters between the two polarization filters is the angle that is between them. So what I can do here just really simply is I can say that the first one is sort of vertically polarized and the second one is going to be horizontally polarized, right? So this is just gonna be on a polarization of 90 degrees. And then this one is just going to be zero degrees such that the angle between them is just 90 degrees. That's the right. OK. So um basically what happens is when this light passes through the first polarization filter, it's gonna be vertically polarized. This is gonna be my I one and then when it passes through the second polarizer, it's gonna be polarized horizontally. And this is gonna be my I two. OK. We wanna calculate what this I two is. So let's just go ahead and set up our um steps for the NTH polarizer. There's two of them here. There's, this is the first one and this is the second one here. So let's just go through the first one really quick. So the first one has initially un polarized light that's passing through. So we're gonna use either one half rule or the cosine square rule. Hopefully realize that it, we're gonna use the one half rule here. So what happens here is that I one is just gonna be one half of I zero. In other words, it's just gonna be one half of 1350 and that's gonna equal 675. All right. So this I one here is 675 when it passes through. Now, let's take a look at the second polarizer. Now, the second polarizer has initially polarized lights, but then it gets polarized to a different angle. So we're gonna have to use the cosine squared rule. So this I two says this is gonna be I one at times the cosine squared of the angle. Now remember this angle here, it doesn't have to do with the angles of the individual axes, but the angle between them this at 90 this one's at zero. So the angle that's between them is 90 degrees. So that's what we plug inside of our formula here. So this just says that this is gonna be 675 times the cosine squared of 90. Now remember the rules for cosines whenever you have a cosine of 90 degrees that always cancels out. So cosine squared of 90 is also just equal to zero. So basically when you get, when you calculate this is that the intensity of the second light here is gonna be zero watts per square meter. And this actually has nothing to do with the fact that the intensity drops by a half or anything like that. This intensity could be whatever number. Basically what this means here is that whenever you have two polarizer ankled 90 degrees with respect to each other, this is sometimes called cross polarization. And that means that it actually cancels out all of the lights that's passing through it. One way you can think about this is that the first filter polarizes it vertically. And then with the second filter, there is no components of this vertical uh vertically polarized light that can survive once it passes through the second filter. So you kind of just cancel each other out. All right. So that's really important here. That's the answer to the first part. Now let's jump into the second part here because basically what the second part says is that we're gonna take a third filter and we're actually gonna sandwich it in between the two filters, right? It says that a third filter with a transmission axis of 30 degrees to the horizontal is inserted between the first two. So we're gonna stick another polarization filter in here and see what happens. We're gonna see what the, what the intensity of the sunlight is after it passes through all three polarizer. OK. So let's just go through the steps again, right? So here we have a transmission axis like this and I've got initially un polarized lights like this. This is gonna be my un polarized light, which is 1350 but now we're actually gonna have three polarizer. There's the first one, then there's the second one and then there is the third one. All right. And just as before what's gonna happen here is we're gonna have this first one is gonna be at 90 degrees. The third one is gonna be at the horizontal like this. And the second one is actually gonna be 30 degrees with respect to the horizontal. So basically, this is your horizontal and it's gonna be oriented kind of like this such that this angle here is 30 degrees. So this is 90 this is gonna be zero degrees. OK. So what happens here is that after it passes through the first filter, it's gonna be vertically polarized. That's gonna be I one. Then when it passes through the second one, it's gonna be polarized at this new angle here, which is 30 degrees, that's I two. And then when it goes through the third one, it's gonna be horizontally polarized and this is gonna be I three. All right. So now let's go through each one of our polarizes. This is the first and equals one. This is the second one and equals two and equals three. All right. So here was, here's what happens. So for the first polarizer, we're gonna use the one half rule again, nothing's changed from before. It's just gonna be I one is equal to uh 675. So it's just gonna be half of 1350. It's gonna be exactly what it was before what it was from part A nothing has changed. This I one here is 675. OK. So it's the second filter. Now what happens? The second filter says that I two is gonna be I one times the cosine squared of theta. Now, what's really tricky about these types of problems is when you have multiple angles is that also you can get confused with your angles. So it's a really good idea to label them the angle that is between these two first filters. I'm gonna call theta 12 because it's the angle between the 1st and 2nd polarizer. All right, then this angle over here, the angle between the 2nd and 3rd, I'm gonna label this as theta 23, the angle between the 2nd and 3rd. OK. So what this says here is that I two is gonna be I one times the cosine squared of theta 12. All right. So this is just gonna be 675 times the cosine squared of theta 12. All right. Now, what's that angle? Now, remember just be very careful. You always want to actually calculate what the angle is between the polarizer really, really important here. So don't just stick in 90 don't just stick in 30. You have to figure out what's the angle between them. So what happens is if you sort of like project this out, this is the 90 degrees. This is actually the angle that you need, not the 30 degrees. So theta 12 is actually just 90 minus 30 which is 60 degrees. That's what you plug into this formula, right? So just be very careful. 675 times the cosine squared of uh this is just gonna be 30 or sorry, 60 degrees. Now, what you should get uh is you should get, let's see, um you should get uh 168.75 and that's watts per meter squared. So that's actually what comes out the other side, 1 68.75. So here, what you can see here is that once you've inserted this third polarizer, because it's not at 90 degrees, there's actually some amount of light that survives. So, whereas initially before you had these two polarizer and they completely cancel each other out to zero. When you stick one in the middle, you actually now have some of the light that passes through, which is kind of weird. It's kind of counterintuitive, but that's actually what happens. All right. Now, let's keep going. We've got one last step here, the third polarizer. This says that the I three is gonna be I two. And again, we use the cosine squared rule. So this is gonna be I two times cosine squared. But now we're just gonna use the, from 2 to 3, we're gonna use this angle over here. All right. So here what uh what this says is that I two is going to be 168.75 times the cosine squared of this angle theta over here, this angle is gonna be the angle between this 30 degrees and zero. So in other words, this is gonna be 30 minus zero. This just equals 30 degrees. That's what we pop into this equation here. So this is gonna be the cosine squared of 30. And what you end up getting here is 100 26.6 and this is watts per meter square. So this is actually the final answer, by the way, that's what survives uh after it passes through the third polarizer, and it's gonna be ho horizontally polarized 126.6 watts per meter squared. So anyway, there's a lot of moving parts in that problem, but it's kind of just tedious setting up the diagram and everything. But hopefully it made sense. Let me know if you have any questions and let's move on to the next video.