Problem 48 - College Physics

Question

The angle of incidence and angle of refraction along a particular interface between two media are \(30^{\circ}\) and \(40^{\circ}\), respectively. What is the critical angle for the same interface?

Step-by-Step Solution

Answer

The critical angle is approximately \(53.13^{\circ}\).

Step 1: Identify the Known Quantities

We are given the angle of incidence, \(\theta_i = 30^{\circ}\), and the angle of refraction, \(\theta_r = 40^{\circ}\), for an interface between two media. We aim to find the critical angle.

Step 2: Apply Snell's Law

Snell's Law relates the angles of incidence and refraction to the refractive indices of the two media: \(n_1 \sin(\theta_i) = n_2 \sin(\theta_r)\).

Step 3: Calculate Unknowns Using Snell's Law

Solving Snell's Law for one side, we get \(\frac{n_1}{n_2} = \frac{\sin(\theta_r)}{\sin(\theta_i)} = \frac{\sin(40^{\circ})}{\sin(30^{\circ})}\). Calculate this value to determine the ratio of refractive indices \(\frac{n_1}{n_2}\).

Step 4: Determine the Critical Angle

The critical angle \(\theta_c\) occurs when the angle of refraction is \(90^{\circ}\). Thus, using Snell's Law for the critical angle: \(n_1 \sin(\theta_c) = n_2 \sin(90^{\circ})\). This simplifies to \(\sin(\theta_c) = \frac{n_2}{n_1}\). Calculating \(\sin(\theta_c)\) from \(\frac{n_1}{n_2}\) found earlier, we find the critical angle by taking \(\theta_c = \arcsin\left(\frac{n_2}{n_1}\right)\).

Step 5: Calculate the Critical Angle

Using the earlier value, \(\frac{n_1}{n_2} = \frac{\sin(40^{\circ})}{\sin(30^{\circ})}\), calculate \(\sin(\theta_c) = \frac{n_2}{n_1} = \frac{1}{\frac{n_1}{n_2}}\). Then, \(\theta_c = \arcsin\left(\sin(\theta_c)\right)\) to determine the critical angle.

Key Concepts

Snell's Law

Snell's Law is a fundamental principle that explains how light bends, or refracts, when it travels from one transparent medium into another. It describes the relationship between the angles of incidence (where the light enters) and refraction (where the light exits), as well as the refractive indices of the two materials. The formula for Snell's Law is given by:\[ n_1 \sin(\theta_i) = n_2 \sin(\theta_r) \]Here:

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

Snell's Law helps us predict how much the direction of light will change. It is especially useful in applications like lenses in glasses, cameras, and instruments used in medicine. It also helps in understanding phenomena such as rainbows and mirages.

Angle of Incidence

The angle of incidence is the angle formed by a ray or wave striking a surface, measured from a normal line that is perpendicular to that surface. In simpler terms, it is the angle at which the incoming light hits the boundary between two different media, like air and water.
Key points about the angle of incidence include:

• It is usually denoted as \( \theta_i \).
• In the given problem, the angle of incidence is \( 30^{\circ} \).
• This angle is crucial as it affects the behavior of the light at the boundary, dictating how much the light will bend or refract.

Understanding the angle of incidence allows us to analyze various optical phenomena, and it is essential in fields like optics and physics.

Angle of Refraction

The angle of refraction is the angle that a refracted wave or ray makes with the normal after passing through the interface between two media. Just like the angle of incidence is the angle at which light arrives, the angle of refraction is the angle at which light bends as it enters a second medium.
Important points regarding the angle of refraction:

• It is denoted by \( \theta_r \).
• In the example, the light refracts at an angle of \( 40^{\circ} \).
• This angle changes depending on the refractive indices of the two materials.

The angle of refraction helps us understand the light bending more quantitatively and is instrumental for designing optical instruments.

Refractive Indices

Refractive indices are numerical values that describe how much light will bend when it enters a different medium. Each material has a characteristic refractive index, which indicates how much slower light travels through the medium in comparison to a vacuum.
Key details about refractive indices include:

• The refractive index is denoted by \( n \).
• It affects how light refracts when moving between two media.
• In the problem, the ratio of the refractive indices \( \frac{n_1}{n_2} \) was used to determine the bending of light.

By using the refractive indices of materials, one can calculate important phenomena like the critical angle, which is the angle of incidence that causes the refracted angle to be \( 90^{\circ} \). This understanding is essential in the study of lenses and optical systems.