For the interface between air (refractive index 1 ) and a material with refractive index \(n\), show that the critical angle and the polarizing angle are related by \(\sin \theta_{c}=\cot \theta_{p}\)

Understanding Snell's Law is crucial when exploring the behavior of light as it travels between different media. In its simplest form, Snell's Law provides a way to calculate the change in direction, or refraction, of a light ray when it passes from one transparent medium into another with a different refractive index. It is expressed mathematically as:
\( n_1 \sin \theta_1 = n_2 \sin \theta_2 \).

Here, \( n_1 \) and \( n_2 \) are the refractive indices of the first and second medium, respectively, and \( \theta_1 \) and \( \theta_2 \) are the angles of incidence and refraction, respectively. This law can be used to determine the critical angle, which is the angle of incidence where the refracted light skims along the boundary instead of moving into the second medium.

When exploring the relationship between critical angle and polarizing angle, Brewster's Law becomes an indispensable tool. Brewster's Law defines the polarizing angle \( \theta_p \) as the angle of incidence at which light with a specific polarization will pass through a transparent medium without any reflection. At this angle, the reflected and transmitted rays are at ninety degrees to each other. The formulation of Brewster's Law states that:
\( \tan \theta_p = \frac{n_{2}}{n_{1}} \), where \( n_1 \) is the refractive index of the first medium and \( n_2 \) is the refractive index of the second medium. When light strikes the surface at Brewster's angle, it is perfectly polarized along the plane of incidence. This law interconnects with Snell's Law to provide a wider understanding of light behavior at material boundaries.

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables where both sides of the identity are defined. They serve as tools for rewriting trigonometric expressions in different forms. For instance, a trigonometric identity can relate the trigonometric functions of an angle to the refractive index of a medium. Particularly in our case, we use the identity: \( \cot \theta = \frac{1}{\tan \theta} \), to prove that the critical angle \( \theta_c \) and the polarizing angle \( \theta_p \) are related as \( \sin \theta_c = \cot \theta_p \). These identities allow us to manipulate and equate different expressions of Snell's Law and Brewster's Law, thereby deducing deeper relationships between angles in the context of optical phenomena.

The refractive index, typically denoted as \( n \), is a dimensionless number that describes how light propagates through a medium. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium: \( n = \frac{c}{v} \), where \( c \) is the speed of light in vacuum and \( v \) is the speed of light in the medium. A higher refractive index indicates that light travels slower in the medium compared to a lower refractive index. This concept is foundational when applying Snell's Law since different refractive indices between media are what cause light to bend or change direction, a phenomenon known as refraction. By understanding the refractive index, we can explain why the angles of incidence and refraction change at the boundary between two materials and how this leads to the familiar concepts of the critical and polarizing angles.