Problem 64 - College Physics

Question

\(\bullet$$\bullet\) A 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 of the glass is approximately 1.376. (b) The angle of refraction is approximately 36.3°.

Step 1: Recognize Brewster's Angle Condition

When reflected light is completely polarized, the angle of incidence is known as Brewster's angle, \( \theta_B \). At Brewster's angle, the tangent of the angle is equal to the refractive index of the glass (provided the light is coming from air). This can be written as \( \tan \theta_B = n \), where \( n \) is the refractive index of the glass.

Step 2: Apply Brewster's Law to Find the Refractive Index

Given \( \theta_B = 54.5^{\circ} \), we can use the equation \( n = \tan(\theta_B) \) to find the refractive index. Thus, \( n = \tan(54.5^{\circ}) \). Calculate this using a calculator to get \( n \approx 1.376 \).

Step 3: Use Snell's Law to Find the Angle of Refraction

Snell's Law relates the angle of incidence \( \theta_i \) and the angle of refraction \( \theta_r \) to the indices of refraction: \( n_1 \sin \theta_i = n_2 \sin \theta_r \). Here, \( n_1 = 1 \) for air, \( n_2 = 1.376 \) for glass, and \( \theta_i = 54.5^{\circ} \). Substitute these values to get \( \sin(54.5^{\circ}) = 1.376 \sin \theta_r \).

Step 4: Solve for Angle of Refraction

Rearrange the equation from Snell's Law to find \( \theta_r \): \[ \sin \theta_r = \frac{\sin(54.5^{\circ})}{1.376} \]. Calculate this value to find \( \sin \theta_r \approx 0.593 \). Take the inverse sine (arcsin) to find \( \theta_r \approx 36.3^{\circ} \).

Key Concepts

Refractive Index

The refractive index is a crucial concept in optics, showing how light travels through different materials. It describes how much a material slows down light compared to its speed in a vacuum. Given by the symbol \( n \), it's calculated as the speed of light in a vacuum divided by the speed of light in the material, \( n = \frac{c}{v} \). A higher refractive index means light travels slower through the material.For example:

• In air, the refractive index is approximately 1.
• In water, it's about 1.33.
• Glass has a refractive index usually around 1.5, but it varies depending on the type of glass.

Understanding the refractive index helps in comprehending why light bends, or refracts, when moving between different media. This bending is critical in optics and affects everything from glasses to camera lenses. In our exercise, the refractive index of glass at Brewster's angle was calculated to be approximately 1.376 based on the tangent of the incidence angle.

Snell's Law

Snell's Law is fundamental in explaining how light bends when it moves from one medium to another. It relates the angles of incidence and refraction to the refractive indices of the involved media. Mathematically, it's represented as:\[ n_1 \sin \theta_i = n_2 \sin \theta_r \]Where:

• \( n_1 \) = refractive index of the first medium
• \( \theta_i \) = the angle of incidence
• \( n_2 \) = refractive index of the second medium
• \( \theta_r \) = the angle of refraction

Using Snell's Law, one can find how light bends at the boundary of two media. This principle is essential in designing lenses and understanding natural phenomena like rainbows.In our specific example, by knowing the refractive indices of air (≈1) and glass (≈1.376), and the angle of incidence (54.5°), we used Snell's Law to calculate the angle of refraction, which turned out to be approximately 36.3°.

Unpolarized Light

Unpolarized light consists of waves that vibrate in multiple planes perpendicular to the direction of propagation. Most natural light sources, such as sunlight or artificial lights, emit unpolarized light. This means the electric fields of these light waves are oriented randomly. When unpolarized light strikes a surface, some of it reflects and some refracts. Upon reflection, the light can become polarized, meaning the electric fields align in one direction. At Brewster's angle, the reflected light is completely polarized. This concept is particularly important in understanding how polarization works in everyday applications, such as:

• Reducing glare with polarized sunglasses.
• Improving screen visibility on devices.
• Photography, where polarized filters enhance image quality.

In our exercise, the incident light at 54.5° led to the reflected beam being linearly polarized at Brewster's angle, which is why the refractive index could be determined using the tangent of the angle.