Welcome back, everyone. Let's check out this example problem. So we have a light ray that's going to come in and hit this glass block at some angle. This is going to be my angle of incidence, and it's going to refract. And as it refracts, it's going to go sort of deeper into the glass block until it eventually leaves. And that's actually what we want to calculate. We want to calculate the distance in centimeters below the top of the block that the light ray travels before it exits. Alright? So let's go ahead and set up this problem here. We're now dealing with some kind of refraction problem. This is the angle of incidence. This is my 70 degrees. That's theta 1. And then what happens is after it enters the glass, remember, it's going to be higher end, so it's going to go more normal. So this is air over here. So this is just going to be n equals 1. That's my n one. And this is going to be glass. So this is n2 = 1.46. I just get this from this table over here. So clearly, it's going to bend more towards the normal. And so therefore, you're going to get a lower angle when you go through to the glass block. So in other words, this theta 2 here is probably going to be a smaller number. Alright? So how does this help us?
Well, if you look here, if we have this angle, can we use that to actually solve for y? Well, if you look at the sort of triangle that we've actually made with the light ray, you can see that the refracted ray sort of makes a triangle with the vertical of the glass block and also the horizontal, the boundary itself. So you look at this little triangle here. Now if we could figure out this angle over here, which I'm going to just call theta x, which gives respect to the x axis. Well, I still can't solve the problem because I have 2 unknowns. I have theta x and I need y. Right? I have y. But there's actually one other thing we know about this triangle. We do know that the ray hits exactly halfway across the glass block, in which the length is 7.5 centimeters. So what that means is if it's exactly halfway, this is just going to be the this distance here that's going to be exactly half of 7.5, which is 3.75 centimeters. And it's okay I leave the measurements in centimeters because that's ultimately what I'm trying to find. Alright? So this is 3.75 centimeters. Remember, to solve a triangle, you need 2 pieces of information, or you either need either need 2 sides, or you need a side and an angle, something like that. So if I could figure out what this theta angle this theta x angle is, then I could definitely use this angle and this distance to figure out this y. Alright? But can I actually figure out what this theta x is? Well, if you take a look at this sort of triangle that we've made, we actually come up with another relationship between theta x and theta 2. Remember, this looks just like, this just is a complimentary angle. This theta x and this theta 2 actually just add up to 90 degrees. So we can actually just set up an equation for this. We can say that theta x is really just equal to 90 minus theta 2. Right? Those things have to add up to 90, so whatever this thing ends up being, this is just going to be 90 minus that. Alright? So this theta x here is 90 minus theta 2. So now can I go and solve for that theta 2? Well, of course, we can, because that's just Snell's law. Right? This Snell's law means that this theta 2 relative to the normal, is going to be related to the angle of incidence and the n one. So we're just going to actually use Snell's law to figure out what this angle is, and then we can go back and figure out what this theta x is. And, actually, we can go ahead and solve our triangle from that. Alright? So that's sort of the game plan here. So I want to set up my Snell's law, my n one sin theta one, n2 sin theta 2. Alright. So I have what my n one is. Right? That's just the index of refraction for air. Got my angle of incidence. And I also have what the index of refraction for glass is. So I can just go ahead and figure out this theta 2. Alright? So So plugging in some numbers, this is going to be 1 times the sine of 70 equals, and this is going to be 1.46 times the sine of theta 2. Alright? Now when you work this out, what you're going to get here is you're going to get, you're actually just going to get 0.94 equals 0.94 equals 1.46 times the sine of theta 2. You can actually divide this over to the other side and then take the inverse sine. What you should get for theta 2 is the inverse sine of and you can actually do the division or you could just plug it in as a fraction, 0.94 divided by 1.46. And what you should get here is 40 degrees pretty much exactly, like, just to, you know, pretty much round it to 40 degrees. Alright. So this angle here is 40 degrees, and as expected, we got a lower number because we're dealing with a higher index of refraction. Right? Higher and more normal more and more normal. So if this is 40 degrees then, that means we actually can go back and we can solve for our theta x. Remember, Remember, we said that theta x is just 90 minus theta 2. So what this means here is I'm just going to pull this down. This theta x is equal to 90 minus and this is just going to be the 40 that sticks in here. This is going to be 40, and you just get 50 degrees. Alright? So now that we've solved for this theta x, which we now know is 50 degrees here. Now we can go ahead and we can use the angle and this side over here to figure out what my y distance is. Now remember, we're always just going to use pretty much tangent because we'll never gonna know the hypotenuse of the triangles. So let's set up our tangent equation. We say that tangent theta, tangent of theta x is opposite over adjacent. So opposite over adjacent. This is your opposite relative to this angle, and this is your adjacent. Right? So if you set this up, this is just going to be your y value divided by 3.75. And again, we can keep it in centimeters. So now all I have to do is just plug in some numbers here. Right? So I've got the tangent of 50 degrees equals y over 3.75, and I could basically just go ahead and finish this off here by moving this up to the other side. So 3.75 times the tangent of 50 degrees equals the y distance. And if you work this out, what you should get is you should get 4.47 centimeters. And that is your final answer. Alright? So this is going to be 4.47 centimeters. Alright, folks. So that's it for this one. And let me know if you have any questions. Thanks for watching.