Welcome back, everyone. Now that we've talked about refraction, in this video we're going to cover a related concept called total internal reflection. I'll show you what it is and the equation that you need to solve problems, and we'll do an example together. Let's check it out. Now remember from our discussion on refraction, we said that when light enters a material with a lower index of refraction, \( n \), a lower \( n \), then it bends away from the normal.

Alright. So, imagine I have a light source that's inside of some material here, and imagine we've got a material with a \( n_1 \), and then we say that this index of refraction \( n_2 \) is less than \( n_1 \). Basically, what happens is if you start drawing out some rays, then they're going to bend away from the normal as they refract. So, for example, for 0 degrees, they'll just go straight through, but for this line over here, it'll start to bend away from the normal. And then for this line, which is a little bit steeper, it'll get a little bit farther away from the normal like this.

Now, if you keep sort of drawing some of these rays out, eventually what happens is that these refracted rays are starting to get closer and closer towards the boundary between the two materials. Eventually, what happens is there is a special angle called theta critical. And at this critical angle of incidence what happens is the refracted ray ends up being perfectly parallel to the surface. So this is parallel to the surface over here, and I'll highlight this in yellow. So there's a special angle basically for which you actually have a ray that goes perfectly parallel to the surface. And what we can say here is that this angle theta is equal to 90 degrees. Remember these angles over here are always measured relative to the normal, so it's not 0, it's 90 degrees. Alright?

Now let's keep going with that. Now, let's say we have another ray that's at an angle that's even greater than that critical angle. What do you think is going to happen there? Well, if you look at the pattern here, you have refraction and these refracted rays start to get more and more horizontal. Then afterwards, when they're purely horizontal, if you have anything that's larger than that beta critical, you actually have a ray that just comes in and it bounces off as if it were kind of like a mirror. So at angles that are larger than this critical angle, the light actually does not get refracted, but it instead gets totally reflected inwards or internally. This is actually called total internal reflection. Alright? It's basically this situation over here.

Now what I like to do is kind of think about this as a number line. Right? So you can kind of think about this as, for all of these angles here that are less than theta critical, you just get refraction. So, yeah, let me just write this over here. So you get refraction. Then what happens is you have this theta critical over here, and then for any angles that are greater than theta critical, you actually just get reflection.

Now what you'll need to know about these kinds of problems is how to solve for this special angle, this critical angle here, and that actually just comes straight from Snell's law. Remember that, for this critical angle, what happens is that the theta 2 is going to equal 90 degrees. So if you look at your equation, what happens is that this theta 2 ends up being 90 degrees. And remember that the sin of theta or sin of 90 is just 1. So one of the terms in Snell's Law just drops out. And if you go ahead and solve for this theta critical by moving some of these terms around, eventually we'll end up with this expression here, which says that theta critical is equal to the sine inverse of \( n_2 / n_1 \). This is the equation to calculate that critical angle, and the most important thing you need to know about this is that this situation of total internal reflection happens only when \( n_2 \) is less than \( n_1 \). As we said in the earlier part of the beginning of the video, it only happens when you have light that enters a material with a lower index of refraction. Otherwise, the equation won't work. Alright? So that's really all you need to know about this critical angle.

Let's go ahead and take a look at this example problem here. This example problem actually is just going to use the image above that we've been working with. Basically, it just tells us that material 1 is glass, so this is going to be glass, and then material 2 is going to be air. Alright? And what's the angle for which light will be totally reflected inwards? Basically, what they're just asking us to do is to solve for theta critical. Alright? So with theta critical, and we have the \( n_1 \) is equal to glass, and we know that that is equal to 1.46. And then we have \( n_2 \) is air, so that's just going to equal 1.

So if you look at your theta critical equation, \( theta_{crit} \), this just equals the sine inverse of and this is going to be, remember, \( n_2 / n_1 \). So, basically, you should always have a number that is less than 1 when you work this out. Now you could just plug this in as a fraction, but when you solve for this, what you're going to get here is you're going to end up with an angle of 43.2 degrees. So that is the angle, that special angle for which you'll have total internal reflection here. So basically, what that means here is if you have glass and air, this critical angle, for what you get a 90-degree angle of refracted ray, is equal to 43.2 degrees. Alright, folks. So that's it for this one. Let me know if you have any questions.