Problem 83 - College Physics

Question

\(\bullet$$\bullet\) A beaker with a mirrored bottom is filled with a liquid whose index of refraction is \(1.63 .\) A light beam strikes the top surface of the liquid at an angle of \(42.5^{\circ}\) from the normal. At what angle from the normal will the beam exit from the liquid after traveling down through it, reflecting from the mirrored bottom, and returning to the surface?

Step-by-Step Solution

Answer

The beam exits at an angle of 42.5° from the normal.

Step 1: Understand the Concept of Refraction

When light travels from one medium to another, it bends at the interface. This bending is governed by Snell's Law, which relates the angles of incidence and refraction to the indices of refraction of the two media.

Step 2: Apply Snell's Law on Entry

According to Snell's Law, \( n_1 \sin \theta_1 = n_2 \sin \theta_2 \), where \( n_1 = 1 \) (the index of refraction of air) and \( n_2 = 1.63 \) (the index of refraction of the liquid). The angle of incidence \( \theta_1 \) is \( 42.5^\circ \). Calculate \( \theta_2 \) using:\[ \sin \theta_2 = \frac{n_1}{n_2} \sin \theta_1 = \frac{1}{1.63} \sin 42.5^\circ \]

Step 3: Calculate the Angle in the Liquid

Compute \( \sin \theta_2 \) and then find \( \theta_2 \) using inverse sine (\( \sin^{-1} \)).\[ \sin \theta_2 = \frac{1}{1.63} \times \sin 42.5^\circ \approx 0.4097 \]\[ \theta_2 = \sin^{-1}(0.4097) \approx 24.2^\circ \]

Step 4: Reflection at the Mirrored Bottom

Since the bottom is mirrored, the angle of incidence in the liquid \( \theta_2 \) becomes the angle of reflection. Thus, the light reflects at \( 24.2^\circ \) within the liquid.

Step 5: Apply Snell's Law on Exit

When the light exits the liquid, it again follows Snell's Law:\[ n_2 \sin \theta_2 = n_1 \sin \theta_3 \]Use \( \theta_2 = 24.2^\circ \) and \( n_2 = 1.63 \), \( n_1 = 1 \) to calculate \( \theta_3 \):\[ \sin \theta_3 = 1.63 \sin 24.2^\circ \]

Step 6: Calculate the Exit Angle

Compute \( \sin \theta_3 \) and find \( \theta_3 \):\[ \sin \theta_3 = 1.63 \times \sin 24.2^\circ \approx \sin 42.5^\circ \approx 0.6756 \]\[ \theta_3 = \sin^{-1}(0.6756) \approx 42.5^\circ \]

Key Concepts

Refraction

Refraction is the process where light changes direction when it enters a different medium. This change happens because light travels at different speeds in different materials. The phenomenon of refraction is what causes a straw in a glass of water to look bent. It's a fascinating principle that has practical applications in everyday objects like lenses, glasses, and cameras. When light enters a new medium, its speed changes, leading to bending of the light path. This is beautifully explained by Snell's Law, which allows us to predict the path of light as it moves between different materials. Understanding refraction is key to grasping more complex optical phenomena, making it a fundamental concept in physics.

Angle of Incidence

The angle of incidence is the angle between the incoming light ray and the normal, an imaginary line perpendicular to the surface at the point of contact. In simple terms, it's the angle at which light hits a surface. Knowing the angle of incidence is important because it affects the amount of bending the light ray experiences. For example, in the beaker exercise, the angle of incidence is given as 42.5°, meaning the light beam is hitting the liquid surface at this angle from the normal. Correctly identifying and measuring this angle is essential in applying Snell's Law to find out how the light will behave as it enters a new medium.

Angle of Refraction

The angle of refraction is the angle between the refracted light ray and the normal. It tells us how much the light ray has bent after entering the new medium. To find this angle, we use Snell's Law, which relates it to the angle of incidence and the indices of refraction of the two media. In our example, after knowing the angle of incidence is 42.5° and the index of refraction for the liquid is 1.63, we calculate the angle of refraction to be approximately 24.2°. This calculation shows how the light's path has changed direction upon entering the liquid.

Index of Refraction

The index of refraction is a number that describes how fast light travels in a particular medium compared to the speed of light in a vacuum. It is denoted by the letter "n." The higher the index, the slower the light moves through that substance. This property is crucial for realizing how much light will bend when entering or exiting a material. The exercise mentions an index of refraction of 1.63 for the liquid, meaning light travels slower in this liquid than in air, which has an index of approximately 1. Understanding the index of refraction helps us predict the degree of bending light will undergo, making it an essential parameter in applying Snell's Law.