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 uravel 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. (FIGURE CANNOT COPY)

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 uravel 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.
(FIGURE CANNOT COPY)

Key Concepts

Fermat's Principle

Fermat's Principle states that the path taken by light between two points is the one that minimizes the travel time. This foundational concept in optics explains why light naturally follows a route that optimizes its passage through different media, leading to the observed phenomena of reflection and refraction.

Snell's Law

Snell's Law provides the mathematical relationship that governs the refraction of light at an interface between two media with different propagation speeds. It states that the ratio of the sines of the angles of incidence and refraction is equal to the ratio of the velocities of light in the two media, linking the geometry of the light path with its speed in different substances.

Calculus of Variations

The derivation of Snell's Law using Fermat's Principle involves minimizing the total travel time of light over a variable path. This process, which employs techniques from the calculus of variations, involves taking the derivative of the time with respect to the path parameters and setting it to zero, thereby ensuring that the optimal path corresponds to a minimum travel time.

Light Propagation in Different Media

The behavior of light as it travels through different media is fundamentally influenced by the variation in its speed. When light moves from one medium to another, such as from air to water, its speed changes due to the optical properties of the medium. This change in speed results in a change in the direction of the light ray, a phenomenon described by refraction and quantified by Snell's Law.

Transcript

00:02 This problem, they asked to demonstrate for ma's principle and show that it reduces to snow's law for light.

00:17 And so the idea is that the path light takes between two points will be the, be the, we'll minimize the time taken.

00:32 So if you have light that wants to go from a to b, it's going to minimize the time that it takes to get from a to b.

00:44 And so now if you just have a single medium where the speed of light is the same, then you're just going to have a straight line.

00:55 But if the light passes from one substance to another where things change, so then you, then you, have a different problem.

01:11 It's not going to necessarily be a straight line.

01:14 And so they want us to show that, again, that we can figure out where this point is here, given the speed of light in these two materials, such that the time it takes to get from here to here is minimized.

01:32 So first we need to look at the distance.

01:35 So we look at this distance here, a to c, and that's square root of a.

01:39 Squared plus x squared.

01:43 So the time that the light takes the light to go from a to c is this distance divided by the velocity of the light in that medium.

01:52 We can take the derivative of that with respect to x so that we can define the path that minimizes it.

02:00 And so with that we wind up with taking the derivative and then using ac again we wind up with the derivative of t with respect to x is x divided by v1 over the distance ac and x over the distance ac is just sign of this angle here so we get this is sign of theta 1 that divided by b1 now we can do the same thing for for the for the distance here and time it takes so that distance i call this y where y is now c this total distance, this total horizontal distance, c minus x.

02:42 So this distance here is squared or b squared plus y squared...

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