But first, let's go over this problem.

Light travels from air to water.

At what incident angle is the reflected light completely polarized?

Well, let's find out.

So let's say we have air on top and water beneath.

And so here we have the normal line.

And this is going to be the incident ray.

And this is the reflected ray.

And that's going to be the refracted ray, which may not be so narrow.

So this is the incident angle.

This is the reflected angle.

And this is the refracted angle.

Now according to the law of reflection, the incident angle is equal to the reflected angle.

And according to Snell's law of refraction, n1 sine theta 1, or the incident angle, is equal to n2 sine theta 2, or the refracted angle.

Let's say this is n1 and this is n2.

Now for most incident angles, the reflected angle is partially polarized.

However, there is a specific incident angle where the reflected angle is completely polarized.

So our goal is to calculate that angle.

That angle is known as the polarizing angle.

And so that's theta p.

Now because the angle of incidence is equal to the angle of reflection, we can call both of these angle p.

Now this is going to stay the way it is.

So how can we calculate this angle?

There's something that you need to know about the reflected ray and the refracted ray in order to calculate that angle.

And that is that at the polarizing angle, these two rays are at right angles.

They're 90 degrees with respect to each other.

And a full line is 180 degrees.

So 180 minus 90 means that these two are complementary.

They add up to 90.

So we can say that theta p plus theta r is equal to 90.

Now even though the reflected ray is completely polarized, it's good to know that the refracted ray is only partially polarized.

So let's get back to this formula.

So theta p is 90 minus theta r.

So the polarizing angle is 90 degrees minus the refracted angle.

Now let's go back to Snell's Law.

So n1 sine theta p, the incident angle is the same as the polarizing angle.

And that's equal to n2 times sine theta r.

Well, you know what, I should have solved for theta r instead.

So let's do that.

So theta r is going to be 90 minus theta p.

So this is what I needed.

So now going back to this equation, it's going to be n1 sine theta p is equal to n2.

And instead of sine theta r, we can replace that with 90 minus theta p.

Now what do we need to do with that?

There's something that we need to do, but what is it exactly that we can do with this?

Perhaps you have taken trigonometry, and if you have, there's something called co-function identities.

So cosine theta is equal to sine 90 minus theta.

So cosine is the co-function of sine.

So whenever the two angles add up to 90, sine and cosine are equal.

So for example, cosine of 10 degrees is equal to sine of 80, because 10 plus 80 is 90.

Cosine 20 is equal to sine of 70.

And you can confirm this with your calculator.

And cosine 30 degrees is equal to sine 90 minus 30, or sine 60.

And so we can replace sine 90 minus theta p with cosine p.

So now we have this equation.

So at this point, what I'm going to do is I'm going to divide both sides by cosine theta p.

So on the right side, these two will cancel.

On the left side, sine divided by cosine is tangent.

And so if we divide both sides by n1, now we can get the polarizing angle.

So tangent theta p is equal to n2 over n1.

Now you need to know the direction in which light travels.

So light is going to travel from n1 to a material with an index of refraction of n2.

So as we saw, it went from air to water.

So n1 is going to be air, because that's where the light is coming from.

And it's going to water as it refracts to it.

So n2 is for water.

Now n for water is actually 1.33.

When I wrote 2, I was meaning like n2.

So let's fix that.

Here it is for water.

So now let's calculate the incident angle at which the reflected light is completely polarized.

So we need to calculate theta p.

So therefore, we need to use the arctangent function.

So the polarizing angle is going to be arctan n2 divided by n1.

By the way, this relation is known as Brewster's law.

And sometimes this is referred to as Brewster's angle, particularly when air is involved.

So if you need to calculate Brewster's angle, you could simply use that formula.

It's the equivalent of the polarizing angle.

So in this case, we're going from air to water.

So n2 is associated with water, so that's going to be 1.33.

And n1, that's for air, so that's 1.

So it's arctangent, 1.33.

So theta p is 53.06 degrees.

Now what is the angle of refraction for which light is transmitted to the water?

So we can use Snell's law to get the answer if we want to.

So this is the polarizing angle.

So n1 is 1 times sine of 53.06, and n2 is 1.33 times sine theta r.

I'm going to show you another way to get the answer, but I want you to be familiar with both ways.

The second way is easier, by the way, so just keep that in mind.

So let's divide both sides by 1.33.

So it's going to be sine 53.06 divided by 1.33, and so that's 0.601, and that's equal to sine theta r.

Now theta r, the refracted angle, that's going to be arcsine of 0.601.

And so that angle is 36.94 degrees.

Now an easier way to get that answer is to use this formula.

Now recall that we said that the refracted angle and the polarizing angle are complementary.

They add up to 90.

So the refracted angle is just 90 minus the polarizing angle.

So 90 minus 53.06, that will give us the same answer of 36.94 degrees.

So as you can see, that's a more simpler approach to get this angle.

Now let's move on to the next problem.

Number two, what is Brewster's angle for light that travels from air to glass?

So feel free to pause the video if you want to work on this problem.

So we're going from air to glass.

And the index of refraction of air, that's going to be 1.

Glass is going to be N2 because light is traveling from air to glass.

And so the index of refraction for glass is 1.5.

So here we have the incident angle, the reflected angle, and the refracted angle.

So both of these will be considered to be theta p, and this is theta r.

So theta p is the same as the Brewster's angle, so you can replace theta p with theta b if you want to.

So the Brewster's angle is equal to N2 over N1.

Well that's tangent of the Brewster angle.

The Brewster angle itself is arc tangent, N2 over N1.

So N2 in this example is 1.5, and 1 is 1.

So it's just the arc tangent of 1.5.

And so that's going to be 56.3 degrees.

Now if you need to calculate the angle of refraction, it's going to be 90 degrees minus the Brewster angle, or the polarizing angle.

And so that's going to be 33.7 degrees.

But this is the answer that we're looking for in this problem.

So that's how you can calculate the Brewster's angle for a simple problem like this.

Number three, what is the polarizing angle for light that travels from diamond to glass?

So we're going to say this is diamond, and here we have glass.

So this is the incident ray, here we have the reflected ray, and a refracted ray.

Actually this should be wider, because we're going from a high index of refraction to a material with a low index of refraction.

So the ray is going to bend away from the normal line.

So this is going to be theta p and theta r.

So N1 is for diamond, we're starting from that side first, so that's going to be 2.42.

And N2 is glass, which is 1.5.

So the polarizing angle, tangent of the polarizing angle, is N2 over N1.

And the polarizing angle is going to be arc tangent of N2 divided by N1.

So in this case, N2 is 1.5, N1 is 2.42.

And so the polarizing angle is 31.8 degrees.

The refracted angle is going to be 90 degrees minus the polarizing angle.

So that's 90 minus 31.8, and so that's going to be 58.2 degrees.

So now you know how to calculate the polarizing angle and the refracted angle.

And so that's it for this video.

Thanks for watching, and have a good day.

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