Problem 88 - Master Resource Book in JEE Main Physics

Question

What is the Brewster's angle for air to glass transition? (Refractive index of glass \(=1.5\) ) \([\) NCERT] (a) \(15^{\circ} \overline{27}\) (b) \(36^{\circ} 27^{\prime}\) (c) \(50^{\circ} 16^{\prime}\) (d) \(56^{\circ} 18^{\prime}\)

Step-by-Step Solution

Answer

The Brewster's angle for air to glass is approximately \(56^{\circ} 18'\), matching option (d).

Step 1: Understand the Brewster's Angle Formula

Brewster's angle is given by the formula \( \theta_B = \arctan\left(\frac{n_2}{n_1}\right) \), where \( n_2 \) is the refractive index of glass and \( n_1 \) is the refractive index of air, which is approximately 1.

Step 2: Substitute Known Values

Substitute \( n_1 = 1 \) and \( n_2 = 1.5 \) into the formula: \( \theta_B = \arctan\left(\frac{1.5}{1}\right) = \arctan(1.5) \).

Step 3: Calculate the Angle

Using a calculator, find \( \theta_B = \arctan(1.5) \). This gives an angle of approximately \( 56^{\circ} 18' \).

Step 4: Match the Result with Options

The calculated Brewster's angle \( 56^{\circ} 18' \) corresponds to option (d).

Key Concepts

Refractive Index

The refractive index is a dimensionless number that describes how light propagates through a medium. Essentially, it signifies how much the speed of light is reduced when passing through a substance compared to its speed in a vacuum. The refractive index () is calculated as follows:

• The speed of light in a vacuum is denoted as \( c \).
• The speed of light in the medium is denoted as \( v \).

Thus, the refractive index formula is expressed as \( n = \frac{c}{v} \). A higher refractive index means that light travels slower in the medium. For instance, if the refractive index of glass is 1.5, it means that light travels 1.5 times slower in glass than in a vacuum. Understanding refractive indices is essential when explaining phenomena like refraction and reflection, and it plays a critical role in optics.
Refractive index is crucial in calculating angles such as Brewster's angle, which occurs at a specific incidence angle when light is perfectly polarized upon reflection. The refractive index differences between media will influence how light bends or refracts, making it a fundamental concept in optics.

Snell's Law

Snell's Law, a fundamental principle in optics, provides a quantitative understanding of how light bends when transitioning between two media with different refractive indices. This law is crucial for explaining refraction, which is the bending of light as it passes from one medium to another.
Snell's Law can be represented by the equation:\[ n_1 \sin \theta_1 = n_2 \sin \theta_2\]Here:

• \( n_1 \) and \( n_2 \) are the refractive indices of the respective media.
• \( \theta_1 \) is the angle of incidence, and \( \theta_2 \) is the angle of refraction.

This law helps predict how light will behave at the boundary between media, such as when it travels from air to glass. Knowing Snell's Law enables us to determine the angles of incidence and refraction, allowing for precise control in applications like optics design and ray tracing. Understanding Snell's Law is a significant step towards grasping other related optical phenomena, such as Brewster's Angle and Total Internal Reflection.

Polarization of Light

Polarization refers to the orientation of light waves. In an unpolarized light wave, the electric field oscillates in all directions perpendicular to the direction of propagation. However, polarized light waves oscillate in particular directions only. Polarization can occur through various processes, such as reflection, refraction, or by passing through a polarizing filter.
Brewster's angle is a special case of light polarization through reflection. When light hits a medium at Brewster's angle (also known as the polarization angle), the reflected light is perfectly polarized. This occurs because the reflected and refracted light waves are perpendicular to each other.
The significance of Brewster's angle is especially noticeable in applications like reducing glare. Polarized sunglasses utilize this concept to block horizontally polarized light, which is the type most commonly reflected off horizontal surfaces like water or roads, thus reducing the glare to the observer. Additionally, understanding polarization is crucial in enhancing contrast in optical instruments and improving the clarity in photography and displays.