If light travels with speed v1 in one medium and speed v2 in...

If light travels with speed $v_{1}$ in one medium and speed $v_{2}$ in a second medium, and if the two media are separated by a plane interface, show that a ray of light passing from point $A$ in one medium to point $B$ in the other is bent at the interface in such a way that $$\frac{\sin i}{\sin r}=\frac{v_{1}}{v_{2}},$$ where $i$ and $r$ are the angles of incidence and refraction, as is shown in Figure 4.59. This is known as Snell's Law. Deduce it from the least-time principle stated in Exercise 39.

If light travels with speed $v_{1}$ in one medium and speed $v_{2}$ in a second medium, and if the two media are separated by a plane interface, show that a ray of light passing from point $A$ in one medium to point $B$ in the other is bent at the interface in such a way that
$$\frac{\sin i}{\sin r}=\frac{v_{1}}{v_{2}},$$
where $i$ and $r$ are the angles of incidence and refraction, as is shown in Figure 4.59. This is known as Snell's Law. Deduce it from the least-time principle stated in Exercise 39.

Key Concepts

Snell's Law

Snell's Law is a fundamental principle in optics that relates the angles of incidence and refraction when light passes between two media with different propagation speeds. It quantitatively describes how the path of a light ray is bent at an interface due to the change in speed, providing a direct relationship between the sine of the incident angle and the sine of the refracted angle, proportional to the speeds of light in the respective media.

Fermat's Principle (Principle of Least Time)

Fermat's Principle, also known as the Principle of Least Time, is the idea that the path taken by light between two points is the one that minimizes the travel time. This variational principle underlies many results in optics, including the derivation of Snell's Law, and provides a unifying framework for understanding how light propagates through different media by optimizing its path based on the speeds within those media.

Refraction

Refraction is the bending of a wave, such as light, when it passes from one medium to another due to a change in its speed. This behavior is observed at the boundary (or interface) between the two media and is directly related to the differences in optical properties, like the refractive index. Understanding refraction is essential for analyzing how light alters its course, as encapsulated by phenomena like Snell's Law.

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