Problem 29 - University Physics with Modern Physics

Question

A parallel beam of unpolarized light in air is incident at an angle of 54.5\(^\circ\) (with respect to the normal) on a plane glass surface. The reflected beam is completely linearly polarized. (a) What is the refractive index of the glass? (b) What is the angle of refraction of the transmitted beam?

Step-by-Step Solution

Answer

(a) 1.376; (b) 36.1°.

Step 1: Understanding Brewster's Angle

Brewster's angle, at which light is completely polarized upon reflection, is defined as the angle of incidence where the reflected and refracted light rays are perpendicular to each other. According to Brewster's Law, this occurs when the tangent of the angle of incidence (Brewster's angle \(\theta_B\)) is equal to the refractive index \(n\) of the material: \(\tan(\theta_B) = n\).

Step 2: Calculating the Refractive Index

Given that \(\theta_B = 54.5^\circ\), we can find \(n\) by taking the tangent of this angle: \(n = \tan(54.5^\circ)\). Using a calculator, we find that \(\tan(54.5^\circ) \approx 1.376\). Therefore, the refractive index of the glass is approximately 1.376.

Step 3: Finding the Angle of Refraction

Using Snell's Law, \(n_{air} \cdot \sin(\theta_i) = n_{glass} \cdot \sin(\theta_r)\), where \(n_{air} = 1\), \(\theta_i = 54.5^\circ\), and \(n_{glass} = 1.376\). We can solve for \(\theta_r\): \(\sin(\theta_r) = \frac{\sin(54.5^\circ)}{1.376}\).\ Using a calculator, we find that \(\sin(54.5^\circ) \approx 0.8090\), then \(\sin(\theta_r) = \frac{0.8090}{1.376} \approx 0.588\). Thus, \(\theta_r \approx \sin^{-1}(0.588) \approx 36.1^\circ\).

Key Concepts

Refractive Index

The refractive index is a fundamental concept in physics that represents how light travels through different mediums. It's a measure of the bending of a ray of light when it passes from one medium to another. The refractive index, denoted by the symbol \( n \), is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium.

• If light slows down as it enters the medium, the refractive index is greater than 1.
• If light speeds up, the refractive index is less than 1. Though this occurs rarely for typical materials.

In our problem, light moves from air to glass. Air has a refractive index of approximately 1, meaning light travels at nearly its maximum speed. Glass has a higher refractive index, indicating light travels slower in glass than in air. This slowing down and bending are what causes light to refract or change direction when entering a new medium. The greater the difference in refractive index between two mediums, the more the light bends.

Snell's Law

Snell's Law is an important relationship that describes how light bends when it passes from one transparent medium to another, like from air to glass in our exercise. The law is stated as:\[ n_1 \cdot \sin(\theta_1) = n_2 \cdot \sin(\theta_2) \]where:

• \( n_1 \) and \( n_2 \) are the refractive indices of the first and second medium, respectively.
• \( \theta_1 \) is the angle of incidence (the angle the incoming light makes with the normal).
• \( \theta_2 \) is the angle of refraction (the angle the refracted light makes with the normal).

This equation helps us calculate the angle of refraction when the incident angle and refractive indices are known. In the exercise, using Snell's Law, we determined that the angle of refraction for the light transitioning from air to glass is approximately 36.1°.

Linear Polarization

Linear polarization refers to the property of waves that describes the direction in which the wave oscillates. For light, which has both electric and magnetic components, it usually means defining the direction of the electric field. When light is unpolarized, the electric field vibrates in multiple directions. However, when light becomes linearly polarized, all the vibrations align in just one direction. This can happen naturally when light reflects off of certain surfaces at Brewster's angle. At this angle, any reflected light is fully polarized in one plane, perpendicular to where the refracted ray would be.

• This means that sunglasses, which reduce glare, often make use of polarized lenses to block this type of light.
• In the exercise, the light is completely polarized upon reflection because it hits the glass at Brewster's angle.

Understanding how light behaves at Brewster's angle can help us control and use light more effectively in various technologies, from photography to optical devices.