Problem 39 - College Physics

Question

\(\bullet\) A beam of light in air makes an angle of \(47.5^{\circ}\) with the surface (not the normal) of a glass plate having a refractive index of 1.66 (a) What is the angle between the reflected part of the beam and the surface of the glass? (b) What is the angle between the refracted beam and the surface (not the normal) of the glass?

Step-by-Step Solution

Answer

(a) 47.5°; (b) 66.0°.

Step 1: Understanding the Problem

We are given a light beam hitting a glass plate at an angle of \( 47.5^{\circ} \) with the surface. The refractive index of the glass is 1.66. We need to find two angles: (a) the angle between the reflected beam and the surface, and (b) the angle between the refracted beam and the surface.

Step 1: Calculating the Incident Angle

The angle given, \(47.5^{\circ}\), is with the surface, not the normal. Hence, the incident angle (angle with the normal) \( \theta_i \) will be \( \theta_i = 90^{\circ} - 47.5^{\circ} \). Calculate \( \theta_i \): \[ \theta_i = 42.5^{\circ} \]

Step 2: Using the Law of Reflection

According to the law of reflection, the angle of incidence is equal to the angle of reflection. Therefore, the angle of reflection \( \theta_r \) with the normal is:\[ \theta_r = \theta_i = 42.5^{\circ} \] To find the angle with the surface, use \( 90^{\circ} - \theta_r \):\[ \angle_{reflection} = 90^{\circ} - 42.5^{\circ} = 47.5^{\circ} \]

Step 3: Applying Snell's Law for Refraction

Snell's Law is given by:\[ n_1 \sin(\theta_i) = n_2 \sin(\theta_t) \]where \( n_1 = 1 \) (air), \( n_2 = 1.66 \) (glass), and \( \theta_t \) is the angle of refraction.Substitute the known values:\[ 1 \cdot \sin(42.5^{\circ}) = 1.66 \cdot \sin(\theta_t) \]Solve for \( \sin(\theta_t) \):\[ \sin(\theta_t) = \frac{\sin(42.5^{\circ})}{1.66} \]

Step 4: Calculating the Angle of Refraction

Calculate \( \sin(42.5^{\circ}) \) which is approximately 0.674:\[ \sin(\theta_t) = \frac{0.674}{1.66} \approx 0.406 \]Now calculate \( \theta_t \) by finding the inverse sine:\[ \theta_t \approx \sin^{-1}(0.406) \approx 24.0^{\circ} \]

Step 5: Finding the Angle with the Surface

The angle between the refracted beam and the surface is:\[ \angle_{refraction} = 90^{\circ} - \theta_t = 90^{\circ} - 24.0^{\circ} = 66.0^{\circ} \]

Key Concepts

Law of Reflection

The Law of Reflection is a fundamental concept in physics that describes how light behaves when it encounters a surface. It states that the angle of incidence, which is the angle the incoming ray makes with the normal (an imaginary line perpendicular to the surface), is equal to the angle of reflection. This principle applies to smooth surfaces, causing reflections like those seen in mirrors.
For example, in our problem, the light strikes the glass plate at an angle of incidence of 42.5° with the normal. According to the Law of Reflection, the angle of reflection will also be 42.5°. This means the reflected light behaves predictably when it bounces off surfaces. This principle is crucial for understanding how light behaves in optics.

Refractive Index

The refractive index is a measure of how much a medium bends or refracts light. It's a dimensionless number that describes how fast light travels in a medium compared to a vacuum. For instance, in air, the refractive index is approximately 1, while in denser materials like glass, it is higher.
In the problem, the glass has a refractive index of 1.66, indicating that light travels slower in glass than in air. This difference in speed causes the light to change direction when it enters or leaves the glass. Understanding refractive index is crucial for fields like optics, where precision in light manipulation is key.

Angle of Incidence

The angle of incidence is the angle between the incoming light ray and the normal to the surface. It's integral in both reflecting and refracting light as it dictates the initial path of the light.
In our exercise, the angle given was with the surface, so it was necessary to convert this to an angle with the normal by subtracting from 90°. The original angle with the surface was 47.5°, leading to an angle of incidence of 42.5°. This angle is used to determine the angle of reflection owing to the Law of Reflection. Additionally, Snell's Law employs the angle of incidence to calculate the angle of refraction.

Angle of Refraction

The angle of refraction is the angle between the refracted ray and the normal inside the medium. It varies depending on the refractive indices of the two media involved, as described by Snell's Law.
Snell's Law is formulated as: \[ n_1 \sin(\theta_i) = n_2 \sin(\theta_t) \]where \( n_1 \) and \( n_2 \) are the refractive indices of the initial and second medium, and \( \theta_i \) and \( \theta_t \) are the angles of incidence and refraction respectively. In our case, solving Snell's Law gave an angle of refraction of approximately 24.0° with the normal. This showcases how light enters a denser medium, slowing down and bending towards the normal. Understanding this angle is key to comprehending light behavior as it moves between different substances.