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 path $ ACB $ that minimized the time taken. Show that $$ \dfrac{\sin \theta_1}{\sin \theta_2} = \dfrac{v_1}{v_2} $$ 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 path $ ACB $ that minimized the time taken. Show that
$$ \dfrac{\sin \theta_1}{\sin \theta_2} = \dfrac{v_1}{v_2} $$
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

Fermat's Principle

Fermat's Principle is a fundamental concept in optics stating that the path taken by a ray of light between two points is the path that minimizes (or makes stationary) the travel time. This principle explains why light bends when it moves between media with different speeds, as the light will naturally choose the fastest route—even if that path is not the shortest in distance.

Snell's Law

Snell's Law is the mathematical formulation that describes how light refracts, or changes direction, when it passes from one medium into another with a different speed of light. It states 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 velocities of light in the two media, establishing a clear quantitative relationship between the angles and the properties of the media.

Light Propagation in Media

Light propagation in media refers to the behavior of light as it travels through different substances, which affects its speed. Since the speed of light varies with the optical density of the medium, the change in speed at the interface between two media leads to bending of the light rays—a phenomenon known as refraction, which is governed by Snell's Law.

Calculus of Variations

Calculus of variations is a mathematical method used to find the functions that maximize or minimize a certain quantity, such as time, energy, or distance. In optics, this technique is applied to Fermat's Principle to derive the path of least time, thereby leading to the formulation of Snell's Law for the refraction of light.

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