Problem 31 - 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) The refractive index is approximately 1.376. (b) The angle of refraction is approximately 37.2°.

Step 1: Understanding Brewster's angle

When light is incident at the Brewster angle, the reflected light is completely polarized. Brewster's angle, \( \theta_p \), is given by the equation: \[ \tan(\theta_p) = n \] where \( n \) is the refractive index of the glass.

Step 2: Calculate the refractive index of the glass

Since the angle of incidence \( \theta_i = 54.5^{\circ} \) is the Brewster angle, we can use the equation: \[ \tan(54.5^{\circ}) = n \]. Calculating this gives: \[ n = \tan(54.5^{\circ}) \approx 1.376 \].

Step 3: Use Snell's Law to find the angle of refraction

To find the angle of refraction, \( \theta_t \), we use Snell's Law: \[ n_1 \sin(\theta_i) = n_2 \sin(\theta_t) \]. Here, \( n_1 = 1 \) for air and \( n_2 \approx 1.376 \) for glass. Thus, \[ \sin(54.5^{\circ}) = 1.376 \sin(\theta_t) \].

Step 4: Solve for the angle of refraction

Rearrange the equation to solve for \( \sin(\theta_t) \): \[ \sin(\theta_t) = \frac{\sin(54.5^{\circ})}{1.376} \]. Compute this to find \( \theta_t \): \[ \theta_t \approx \sin^{-1}(\frac{\sin(54.5^{\circ})}{1.376}) \approx 37.2^{\circ} \].

Key Concepts

Unpolarized Light

Unpolarized light is light that vibrates in multiple planes rather than just one. Essentially, the vibrations of the light waves are oriented in random directions. Examples in everyday life range from sunlight to the light emitted by most electric bulbs. When light is reflected off surfaces, it can become polarized, meaning the light waves align in a single plane.

In the context of our exercise, a beam of unpolarized light hits a glass surface at Brewster’s angle, leading to the reflected light being completely polarized. When Brewster's angle is achieved, certain rules about light behavior come into play, which helps us calculate values like the refractive index of the glass.

• Unpolarized light includes all orientations of light waves.
• Reflection off a surface can polarize unpolarized light.
• Brewster’s angle results in fully polarized reflected light.

This property is useful in optics, giving us a mechanism to study and exploit the behavior of light in various applications, especially in technologies like polarized sunglasses and camera lenses.

Refractive Index

The refractive index, denoted as 'n', is a measure of how much a medium reduces the speed of light. The slower the light travels through a medium, the higher its refractive index. Common substances have known indices; for instance, air has a refractive index close to 1.0, while glass ranges around 1.5 depending on its type.

Our exercise uses the refractive index to determine how light bends when passing through different media. At Brewster’s angle, the refractive index can be found using the formula \[ \tan(\theta_p) = n \]If light hits the glass at Brewster's angle, you can calculate 'n' directly. The formula relates the angle to the medium’s refractive property, highlighting a unique relationship between angle and refractive index.

• Refractive index indicates a medium’s light bending ability.
• The slower the light, the higher the refractive index.
• Refractive index is key to understanding light behavior across media.

This insight can be used not just for calculations, but also for understanding phenomena like mirages, lensing in telescopes, and even the sparkle of diamonds.

Snell's Law

Snell's Law is fundamental in optics, providing a relationship between the angles and the refractive indices of two media through which light passes. The law is mathematically expressed as:\[ n_1 \sin(\theta_i) = n_2 \sin(\theta_t) \]where:

• \( n_1 \) and \( n_2 \) are the refractive indices of the first and second medium, respectively.
• \( \theta_i \) and \( \theta_t \) are the angles of incidence and refraction.

This formula allows us to solve for unknowns like the angle of refraction or the refractive index, essential data when designing optical systems or studying light behavior in different environments.

In this exercise, Snell’s Law helps determine the angle of refraction when light transitions from air to glass. Starting with a known incident angle and refractive indices, we rearrange the formula to solve for the transmitted angle. Using basic algebra and trigonometry, we uncover how light bends, which is critical when crafting anything that needs precise control of light, like lenses and periscopes.

• Describes the light path between different materials.
• Essential for understanding and designing refractive optical systems.
• Demonstrates how to calculate angles of refraction.

Mastering Snell's Law equips one with tools for broader applications in optical technologies and scientific investigations.