Problem 16 - College Physics

Question

The critical angle for a certain type of glass in air is \(41.8^{\circ} .\) What is the index of refraction of the glass?

Step-by-Step Solution

Answer

The index of refraction of the glass is approximately 1.50.

Step 1: Understanding the Critical Angle

The critical angle is the angle of incidence in a denser medium (glass) such that the angle of refraction in the less dense medium (air) is 90°. This means light leaves at the boundary and travels along the interface.

Step 2: Applying Snell's Law

Snell's Law is given by \( n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \). For critical angle, \( \theta_2 = 90° \). Hence, \( \sin(\theta_2) = 1 \) and Snell's Law simplifies to \( n_{glass} \sin(\theta_c) = n_{air} \sin(90°) \).

Step 3: Simplifying Snell's Law

Since \( n_{air} \approx 1 \), the equation becomes \( n_{glass} \sin(\theta_c) = 1 \). The critical angle \( \theta_c \) is given as \( 41.8° \), so substitute it into the equation.

Step 4: Solving for Glass Index of Refraction

Rearrange the equation to solve for \( n_{glass} \): \[ n_{glass} = \frac{1}{\sin(\theta_c)} \]. Substitute \( \theta_c = 41.8° \) and calculate \( \sin(41.8°) \).

Step 5: Calculating the Sine Value and Result

Using a calculator, find \( \sin(41.8°) \approx 0.6669 \). Then, calculate \( n_{glass} = \frac{1}{0.6669} \approx 1.50 \).

Key Concepts

Critical Angle

When light travels from a denser medium, like glass, to a less dense medium, such as air, it bends away from the normal. The critical angle is a special angle of incidence where the refracted light travels along the boundary, making the refraction angle exactly 90 degrees. At this angle, light doesn't actually pass into the air but rather propagates along the interface of the two media. This concept is crucial as it helps to determine the conditions needed for total internal reflection, which is important in many optical applications. Total internal reflection is when all the light is reflected back into the denser medium, such as in fiber optic cables.

Snell's Law

Snell's Law is the foundation of understanding how light refracts when moving between different mediums. It is expressed mathematically as \(n_1 \sin(\theta_1) = n_2 \sin(\theta_2)\), where \(n_1\) and \(n_2\) are the indices of refraction for the initial and second medium, respectively. Also, \(\theta_1\) and \(\theta_2\) are the corresponding angles of incidence and refraction. For critical angles, the refraction angle \(\theta_2\) is \(90^{\circ}\), so \(sin(\theta_2) = 1\).
In this scenario, Snell’s Law simplifies significantly, allowing us to easily find the index of refraction for the second medium (the glass in our problem). By knowing \(\theta_c\) for the glass, Snell's Law is modified to solve for the unknown, and it leads to the conclusion that \(n_{glass} = \frac{1}{\sin(\theta_c)}\). This relationship is very useful in determining the properties of the glass related to light behavior.

Optics Calculation

Understanding optics calculations often involves a series of steps using core principles like Snell's Law. In the given exercise, solving for the index of refraction begins by utilizing the critical angle, which is provided as \(41.8^{\circ}\).

The task is to determine \(n_{glass}\), the index of refraction for the glass. Using \(\sin(41.8^{\circ})\), we calculate its value to be approximately \(0.6669\). Hence, using the derived formula, \(n_{glass} = \frac{1}{0.6669}\), this results in around \(1.50\), which tells us about the glass's optical density. These calculations are standard in optics, providing insights into how glass will interact with light paths, essential in device engineering and scientific research involving optics.