Let v 1 be the velocity of light in air and v 2 the...

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

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.

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.

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