Problem 88 - College Physics

Question

\(\bullet\) The refractive index of a certain glass is \(1.66 .\) For what angle of incidence is light that is reflected from the surface of this glass completely polarized if the glass is immersed in (a) air or (b) water?

Step-by-Step Solution

Answer

(a) The angle of complete polarization in air is approximately 59.7°, and (b) in water is approximately 51.3°.

Step 1: Understand Brewster's Law

According to Brewster's Law, light is completely polarized when the reflected and refracted rays are at a right angle to each other. The angle of incidence at which this occurs is called Brewster's angle (\( \theta_B \)). It is given by the formula: \[ \tan \theta_B = \frac{n_2}{n_1} \] where \( n_1 \) is the refractive index of the medium from which the light is coming from and \( n_2 \) is the refractive index of the medium into which the light is entering.

Step 2: Calculate Brewster's Angle for Air-Glass Interface

For part (a), when the glass is immersed in air, the refractive index of air is approximately 1. Thus, \( n_1 = 1 \) and \( n_2 = 1.66 \). Using Brewster's Law: \[ \tan \theta_B = \frac{1.66}{1} = 1.66 \] To find \( \theta_B \), compute \( \theta_B = \tan^{-1}(1.66) \), which gives \( \theta_B \approx 59.7^\circ \).

Step 3: Calculate Brewster's Angle for Water-Glass Interface

For part (b), when the glass is immersed in water, the refractive index of water is approximately 1.33. Thus, \( n_1 = 1.33 \) and \( n_2 = 1.66 \). Using Brewster's Law: \[ \tan \theta_B = \frac{1.66}{1.33} \approx 1.248 \] To find \( \theta_B \), compute \( \theta_B = \tan^{-1}(1.248) \), which gives \( \theta_B \approx 51.3^\circ \).

Key Concepts

Refractive Index

The refractive index is a crucial concept in understanding how light behaves when moving between different media. It is a measure of how much the speed of light is reduced inside a material compared to the vacuum of space. The refractive index is denoted by the letter \(n\) and can be calculated using the formula:

• The refractive index \(n = \frac{c}{v}\)

where \(c\) is the speed of light in a vacuum, approximately \(3 \times 10^8\) meters per second, and \(v\) is the speed of light in the medium.
The higher the refractive index, the slower light travels in that medium. For a given angle of incidence, a higher refractive index means a greater bending of light. In this exercise, the refractive indices provided for glass, air, and water help determine at which angle light reflects off glass and becomes completely polarized.

Angle of Incidence

The angle of incidence is the angle between the incoming ray of light and the normal (an imaginary line perpendicular to the surface) at the point of contact on a reflective surface. Understanding the angle of incidence is key to utilizing Brewster's Law, which predicts when light reflecting off a surface becomes completely polarized.
In our exercise, we consider the angle of incidence called Brewster's angle, which is calculated using the formula:

• \(\theta_B = \tan^{-1}\left(\frac{n_2}{n_1}\right)\)

where \(n_1\) is the refractive index of the medium from which the light originates, and \(n_2\) is the refractive index of the medium it enters. By calculating Brewster's angle for different media combinations, you can predict the conditions necessary for light to be polarized through reflection.

Polarization

Polarization of light occurs when waves oscillate in one particular direction, rather than in all directions. This can happen naturally. For instance, when light reflects off a non-metallic surface, such as glass or water.
Brewster's Law provides a method to find the angle at which light will become completely polarized upon reflection. This polarization angle, or Brewster's angle, occurs when the angle of incidence and angle of refraction sum to \(90^\circ\). The computation involves the refractive indices of the two media interacting:

• For air-glass, \(\theta_B \approx 59.7^\circ\)
• For water-glass, \(\theta_B \approx 51.3^\circ\)

At these angles, reflected light is entirely polarized, which has practical applications in photography, sunglasses, and reducing glare.

Optics

Optics is the branch of physics that focuses on the study of light and how it interacts with different materials. It encompasses phenomena such as reflection, refraction, and polarization, which are all integral to analyzing light's behavior.
In the context of this exercise, optics provides the framework needed to understand how light reflects differently depending on the medium and angles involved. Key principles in optics include:

• Reflection: Light bouncing back from a surface.
• Refraction: Light bending as it passes through different media.
• Polarization: Light waves oscillating in a particular direction.

By mastering these principles, students can gain insight into various applications of optics, such as in cameras, eyeglasses, microscopes, and even in the design of optical instruments used throughout science and industry.