Problem 19 - College Physics

Question

A beam of light in air is incident on the surface of a slab of fused quartz. Part of the beam is transmitted into the quartz at an angle of refraction of \(30^{\circ}\) relative to a normal to the surface, and part is reflected. What is the angle of reflection?

Step-by-Step Solution

Answer

The angle of reflection is equal to the angle of incidence.

Step 1: Understanding the Law of Reflection

The law of reflection states that the angle of reflection is equal to the angle of incidence. This means that the angle at which light approaches a surface (angle of incidence) is the same as the angle at which it reflects off the surface (angle of reflection).

Step 2: Determine the Angle of Incidence

Since a beam of light is incident on the surface, the angle of incidence can be determined by applying the fact that refraction and reflection phenomena are occurring simultaneously. The angle of incidence is measured with respect to the normal, similar to refraction.

Step 3: Application to the Question

The problem states that one part of the light is reflected. According to the law of reflection, the angle of reflection is equal to the angle of incidence. However, the angle of reflection does not depend on the refraction angle given. The reflection angle is calculated along with the appropriate reflection law.

Key Concepts

Law of Reflection

When light encounters a surface, it can either be reflected, refracted, or absorbed. The law of reflection is a fundamental principle that governs how light reflects off surfaces. According to this law, the angle of reflection is equal to the angle of incidence. This means if a light beam hits a surface at a certain angle (known as the angle of incidence), it will bounce off the surface at the same angle but on the opposite side of the normal.

• Normal Line: The normal is an imaginary line perpendicular to the surface at the point of incidence.
• Angles: The angle of incidence and angle of reflection are measured from this normal line.

This law is universal for all types of reflections, whether the surface is smooth, like a mirror, or rough, causing diffuse reflection. Understanding this basic law helps in predicting the path of light when designing optical devices.

Angle of Refraction

Light can change direction when passing from one medium to another, which is known as refraction. The angle of refraction is the angle between the refracted ray and the normal to the interface between the two media. Snell's Law mathematically describes this phenomenon using the equation \[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \] where:

• \( n_1 \) and \( n_2 \) are the refractive indices of the two media.
• \( \theta_1 \) is the angle of incidence, and \( \theta_2 \) is the angle of refraction.

In the context of our problem, when light enters the fused quartz from air, it bends and creates a new path inside the quartz. The given angle of refraction is \(30^{\circ}\), indicating the direction change due to the difference in speed of light between air and quartz.

Fused Quartz

Fused quartz is a type of glass made from silica in a non-crystalline (amorphous) form. It is renowned for its unique optical properties. Fused quartz is highly transparent, allowing efficient light passage. This characteristic makes it ideal for use in optics and photonic applications.

• High Refractive Index: Quartz has a refractive index of about 1.46. This affects how much light slows down and bends when entering or leaving the quartz.
• Durability: It's resistant to thermal shock and can withstand high temperatures, making fused quartz suitable for various industrial applications.

These properties are crucial in understanding why the refractive angle in the original exercise was given. The refractive index of a material defines how light interacts at its boundaries relative to air.

Angle of Incidence

The angle of incidence plays a crucial role in determining how light behaves when it strikes a surface. Measured from the normal (a line perpendicular to the surface), this angle helps us understand both reflective and refractive behaviors.

• Incidence and Reflection: For any reflective surface, the angle of incidence equals the angle of reflection, a principle dictated by the law of reflection.
• Impact on Refraction: When light transitions between various media, the incidence angle affects refraction as described by Snell's Law.

In the given problem, understanding the angle of incidence allows us to apply both the law of reflection to find the reflection angle and Snell's Law to comprehend refraction into fused quartz. Observing these angles helps predict light paths in numerous real-world optical applications.