Let v1 be the velocity of light in air and v2 the velocity...

Let $v_{1}$ be the velocity of light in air and $v_{2}$ the velocity of light in water. According to Fermat's Principle, a ray of light will travel from a point $A$ in the air to a point $B$ in the water by a path $A C B$ that minimizes the time taken. Show that $$\frac{\sin \theta_{1}}{\sin \theta_{2}}=\frac{v_{1}}{v_{2}}$$ where $\theta_{1}$ (the angle of incidence) and $\theta_{2}$ (the angle of refraction) are as shown. This equation is known as Snell's Law.

Let $v_{1}$ be the velocity of light in air and $v_{2}$ the velocity of light in water. According to Fermat's Principle, a ray of light will travel from a point $A$ in the air to a point $B$ in the water by a path $A C B$ that minimizes the time taken. Show that
$$\frac{\sin \theta_{1}}{\sin \theta_{2}}=\frac{v_{1}}{v_{2}}$$
where $\theta_{1}$ (the angle of incidence) and $\theta_{2}$ (the angle of refraction) are as shown. This equation is known as Snell's Law.

Key Concepts

Snell's Law

Snell's Law quantitatively describes the refraction of light at an interface between two media with different propagation speeds. It demonstrates that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the light's velocities in the respective media, a relationship that is directly derived from the principle of least time.

Fermat's Principle

Fermat's Principle states that the path taken by light between two points is the one that requires the least time. This principle is fundamental in optics because it not only explains how light travels in a vacuum but also in media where its speed may vary, thereby laying the groundwork for understanding refraction and other optical phenomena.

Transcript

00:01 For this problem, we are going to be examining snell's law.

00:05 And for snell's law, we're going to be watching light.

00:09 And light's going to be coming off of some point a down toward the surface of the water.

00:18 We're told that this is going to be, i'm going to do my variables in green.

00:22 That angle is going to be theta 1.

00:25 And this object a is some height.

00:27 I'm going to just call this side a, that's some height above the ground.

00:31 It is traveling at a velocity of v1, and the light's coming that direction.

00:37 When it hits the water, the angle is going to change.

00:42 So we're going to be coming down here and it's coming to a point b underwater.

00:47 And that point where it hits the water, we're going to call that c.

00:52 B here has another angle.

00:54 It is probably not equal to angle theta 1, so we're going to call it theta 2.

00:59 And again, just so we can talk about it, we're going to say that b, point b, that distance from the water surface to b is lowercase b.

01:08 At this point, my light is traveling at speed v.

01:14 Sub 2.

01:15 What we want to show here is that we're going to be, it wants to minimize the time taken in the air.

01:23 And we want to show that snell's law is an effect in this case.

01:27 And that is that sine of theta 1 over sine of theta 2 equals v1.

01:36 Over v2.

01:37 This is what we want to show.

01:45 Let's see what we can do.

01:46 First of all, let's talk about the sides that we haven't labeled yet.

01:50 I want to have variables on everything so we can talk about all of our pieces, our angles and our sides.

01:56 So let's say that the distance from c, because c is somewhere along the water surface.

02:03 So let's call this whole distance d.

02:09 So if i call that piece x, this would be d minus x.

02:14 And how far is my light actually traveling? well, let's call this piece y that it travels in the air, and we'll call this z that it travels in the water.

02:24 So lots of variables here.

02:27 It's just a way that we can talk about them and not get too confused, hopefully.

02:31 Okay.

02:31 So i've got my angles.

02:32 I've got every side of every triangle marked.

02:35 So what do i want to do? we're told we want to minimize time.

02:45 So if i want to minimize time, the time is, if you think back to early algebra classes, distance equals rate times time.

02:56 So time is going to be distance divided by rate.

02:59 So we've got two different pieces in the air.

03:02 The distance that it travels is y, and its rate is v sub 1.

03:08 That's its velocity.

03:09 So y divided by v sub 1 is the time.

03:12 It takes for the light to travel in the air from a to c.

03:16 Now, in the water from c to b, that distance is z, and its speed or rate is v sub 2.

03:26 Well, this is great.

03:28 V1 and v2 are constants.

03:30 That's a constant velocity of light.

03:32 But i have two variables, y and z.

03:35 So i want to get both of those in terms of one variable so i can take the derivative.

03:40 So let's get both of them in terms of x.

03:45 Let's look at triangle number one.

03:47 We'll call that one and this one too.

03:50 If i look at triangle number one, the pythagorean theorem says, hypotenuse squared equals a squared plus x squared.

04:01 Or if i take the square root, y equals the square root of a squared plus x squared.

04:06 A is a constant because that object is a certain height above the water.

04:11 X is my variable.

04:13 So this is getting everything...

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