Unpolarized light falls on two polarizer sheets whose axes are...

Question

Unpolarized light falls on two polarizer sheets whose axes are at right angles. (a) What fraction of the incident light intensity is transmitted? (b) What fraction is transmitted if a third polarizer is placed between the first two so that its axis makes a 58° angle with the axis of the first polarizer? (c) What if the third polarizer is in front of the other two?

Accepted Answer

Hi, everyone. Let's take a look at this practice problem dealing with polarizer. This problem says, imagine you're wearing sunglasses with polarized lenses and you're looking out through a car's windshield, which is also a polarized layer. There are three parts to this question. Part A what happens to the amount of light that reaches your eyes? Yet these sunglasses and the windshield are polarized at an angle of 65 degrees relative to each other. For part B. Now imagine your sunglasses and the windshield are perfectly aligned with their axes parallel. If you add a third polarized glass in front of the windshield, add an angle of 90 degrees to the windshield, how much light reaches your eyes? And for part C, if the sunglasses and windshield are at 90 degree angles to each other without any additional polarizer, how much light reaches your eyes? In this case, we give him four possible choices. As our answer for choice. A for part A, we have 2.5% for part B, we have 0% for part C, we have 50% for choice B for part A, we have 8.9% for part B we have 0%. And for part C, we have 0%. For choice C for part A, we have 2.5%. For part B, we have 9%. And for part C, we have 0%. And for choice D for part A, we have 8.9% for part B, we have 50%. And for part C, we have 0%. Now, for part A, we need to figure out how much light has reached your eyes after passing both through the windshield and the sunglasses and the windshield and sunglasses act as polarizer and the incoming light, which is the sunlight is actually un polarized light. So we're going to do is take that un polarized light and pass it through each polarizer individually and figure out the intensity after it passes through each one until it reaches your eye. So the first intensity that we need to calculate is the intensity after the un polarized light has passed through the windshield. And so we'll call that intensity I one and that's gonna be equal to one half multiplied by our un polarized intensity, which is I not. And this is because anytime, uh un polarized light passes through a polarizer, it loses half of its intensity. So the polarizer basically B blocks out half of the light. So now that we have the intensity after the windshield, we're gonna use it to calculate the intensity after passing through the sunglasses, which is actually what goes to your eye and to do this, we're gonna need to use Malice's law. So, recall for Malice's law that we have the intensity after the sunglasses, which will be I two is equal to the intensity coming into the sunglasses, which is gonna be I one multiplied by cosine squared of theta where the is the relative angle between the polarization of the sunglasses and the windshield. So we know what that angle is. So we can plug that in and we can also substitute in for um I one in terms of I not. So we'll have I two is equal to or I one will have one half multiplied by I knot. That's gonna be multiplied by the cosine squared of our angle, which in this case is 65 degrees. So we can calculate numerical values for the cosine squared of 65 degrees multiplied by one half. And they'll give me I too is equal to 0.089 multiplied by I not. So here I just kept two significant figures for the coefficient in front of I not. Now, we ultimately want to write um our intensity as a percentage relative to I not. So we're gonna divide both sides by I not. So we'll have I two divided by I kno is equal to 0.089. And I wanna want to write that fra that ratio as a percentage because that's what we're looking for. So we'll multiply this by 100%. And so I'll get I two divided by I not, that's gonna be equal to 8.9%. So this could be my answer for part A. Now, for part B, we're gonna do something similar. We have a slightly different set up here. The sunglasses and windshield are now perfectly aligned. So there's no angle, the angle between those two polar polarization is gonna be zero degrees. But we have a third polarizer in front of the windshield. So that polarizer that's in front of the windshield will again block out half of the intensity of the un polarized light. Again, we'll have I, one is equal to one half I not where I went here is the light that is after this additional polarizer that we um added to the system. And now we can calculate the intensity of the light after the windshield. So we'll call that I two, that's gonna be equal to I one multiplied by cosine squared of the angle in this case, which is gonna be 90 degrees. Now, the cosine of 90 degrees is zero. So that means that I too, it's gonna be equal to zero. So that means that no light is being transmitted um through the windshield. So that means that you have no light going to the sunglasses and therefore no light reaching your eyes. So that means we actually have 0% of the light reaching your eyes since we have blocked out all of it, that's gonna be our answer for part B Report C, we again have the same setup that we had in part A except our angle now is 90 degrees. So if we look at the light that is reaching your eyes, that's gonna be I two. And since we have the same set up, we'll have the exact same equation that's gonna be equal to one half I not cosine squared. But in this, in, in this case, instead of 65 degrees, we're gonna have 90 degrees. And again, cosine of 90 degrees is going to be zero. So we'll have I two equal to zero. So your sunglasses have blocked out all the light and therefore we have 0% reaching your eyes. So that's gonna be our answer or part C. So now I have answers for all three parts, I can look at my answer choices and the correct answer choice corresponds to answer choice B. So just a quick little recap of what we've done here for each of the parts, we broke the problem down into calculating the intensity after each one of the polarizer. And since we are starting with un polarized light, the intensity after the initial polarizer, the first polarizer was always gonna be equal to one half of our initial intensity. And that's because the polarizer will always block out half of the intensity or half of the light of un polarized light for subsequent polarizer, we use malice's law to calculate the intensity after each one of those polarizer. And once we reached our eye and we have that intensity, we would write that intensity as a percentage of our initial intensity. So I hope that this has been useful and I'll see you in the next video.

Key Concepts

Malus's Law

Unpolarized Light

Multiple Polarizers

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