Problem 51 - College Physics

Question

A beam of light is incident on a glass plate \((n=1.62)\) in air and the reflected ray is completely polarized. What is the angle of refraction for the beam?

Step-by-Step Solution

Answer

The angle of refraction is approximately 31.8 degrees.

Step 1: Understand Brewster's Angle

When the reflected ray is completely polarized, the angle of incidence is called Brewster's angle. According to Brewster's Law, this happens when the angle between the reflected and refracted rays is 90 degrees.

Step 2: Use Brewster's Law Equation

Brewster's Law gives us the relationship \( \tan(\theta_B) = n \), where \( \theta_B \) is Brewster's angle and \( n \) is the refractive index of the medium. Here, \( n = 1.62 \).

Step 3: Calculate Brewster's Angle

Using \( \tan(\theta_B) = 1.62 \), find \( \theta_B = \tan^{-1}(1.62) \). This calculates to \( \theta_B \approx 58.2^{\circ} \).

Step 4: Determine the Angle of Refraction

Since the reflected and refracted rays are 90 degrees apart at Brewster's angle, use \( \theta_B + \theta_r = 90^{\circ} \) to find the angle of refraction \( \theta_r \). Thus, \( \theta_r = 90^{\circ} - 58.2^{\circ} = 31.8^{\circ} \).

Key Concepts

Angle of Refraction

The angle of refraction is a critical concept in understanding how light behaves when it passes from one medium to another. This angle is labeled as \( \theta_r \) in scientific equations. It is the angle between the refracted ray and a line perpendicular to the surface called the normal.

When a light beam enters a different medium, its speed changes, which in turn changes its direction. This bending is what we call refraction. According to Snell's Law, \( n_1 \sin(\theta_i) = n_2 \sin(\theta_r) \), where \( n_1 \) and \( n_2 \) are the refractive indices of the different media, and \( \theta_i \) and \( \theta_r \) are the angles of incidence and refraction, respectively.

In the case of Brewster's Angle, things are unique. The angle of reflection and refraction are 90 degrees apart. Hence, if you know Brewster's angle (\( \theta_B \)), you can find the angle of refraction simply by calculating \( 90^{\circ} - \theta_B \).

• Light changes direction upon entering a different medium.
• Angle of refraction is between the refracted ray and the normal.
• Use Brewster's angle for calculating refractive angles when light is polarized.

Refractive Index

The refractive index, denoted as \( n \), is a measure of how much a ray of light bends when it enters a different material. Technically, it's the ratio of the speed of light in a vacuum to that in the medium. A higher refractive index means that light travels more slowly in that medium.

For instance, with a refractive index of 1.62 for glass, this indicates that light slows down significantly when it passes through glass compared to air. The change in speed causes the light to bend, resulting in refraction.

Calculating Brewster's angle involves the refractive index, using the equation \( \tan(\theta_B) = n \). Understanding this helps us determine that at this specific angle, the reflected light is completely polarized. This specific angle makes significant use of the refractive index.

• Refractive index determines bending of light.
• Higher \( n \) means light travels slower in the medium.
• Integral for calculating Brewster's angle.

Polarized Light

Polarized light refers to light waves in which the vibrations occur in a single plane. This type of light is common in nature, for instance, when sunlight reflects off lakes or roads. Light can become polarized through reflection, refraction, or by passing through polarizing materials.

When a light beam reflects off a surface at a specific angle called Brewster's angle, the reflected beam is completely polarized. This phenomenon results because the reflected and refracted rays are at a 90-degree angle to each other. At this angle, only one plane of light is reflected, while others are absorbed by the refracted ray.

Understanding polarized light is crucial, especially in designing sunglasses and camera lenses, which minimize glare by blocking horizontally polarized light. This property is exploited in optical devices to enhance image quality or reduce reflections.

• Light vibrating in a single plane is polarized.
• Occurs at Brewster's angle through reflection.
• Used in sunglasses to reduce glare.