Problem 18 - University Physics with Modern Physics

Question

A beam of light is traveling inside a solid glass cube that has index of refraction 1.62. It strikes the surface of the cube from the inside. (a) If the cube is in air, at what minimum angle with the normal inside the glass will this light \(not\) enter the air at this surface? (b) What would be the minimum angle in part (a) if the cube were immersed in water?

Step-by-Step Solution

Answer

(a) The minimum angle with the normal is approximately 38.49°. (b) In water, it's approximately 55.42°.

Step 1: Understand the Critical Angle Concept

To solve the problem, we need to determine the critical angle at which total internal reflection occurs. This is when the angle of refraction would be 90 degrees, meaning the light does not pass into the other medium.

Step 2: Use Snell's Law for the Air Condition

For the situation where the cube is in the air, we will apply Snell's Law, which is given by: \(n_1 \sin \theta_1 = n_2 \sin \theta_2\). Here, \(n_1 = 1.62\) for glass and \(n_2 = 1.00\) for air, and \(\theta_2 = 90^\circ\). Thus, the equation becomes: \(1.62 \sin \theta_c = 1 \cdot \sin 90^\circ\).

Step 3: Solve for the Critical Angle in Air

Since \(\sin 90^\circ = 1\), rearrange and solve for the critical angle \(\theta_c\): \(\sin \theta_c = \frac{1}{1.62}\), which gives \(\theta_c = \sin^{-1}\left(\frac{1}{1.62}\right)\). Calculate this value to find \(\theta_c\).

Step 4: Use Snell's Law for the Water Condition

Now, consider the cube immersed in water. Here \(n_2 = 1.33\) for water. Apply Snell's Law again: \(1.62 \sin \theta_c' = 1.33 \cdot \sin 90^\circ\).

Step 5: Solve for the Critical Angle in Water

Rearranging the equation gives \(\sin \theta_c' = \frac{1.33}{1.62}\), which results in \(\theta_c' = \sin^{-1}\left(\frac{1.33}{1.62}\right)\). Calculate this to find \(\theta_c'\).

Key Concepts

Snell's Law

Snell's Law is a fundamental principle in optics that helps us understand how light behaves when it passes from one medium to another. The law is mathematically expressed as \( n_1 \sin \theta_1 = n_2 \sin \theta_2 \), where:

• \( n_1 \) and \( n_2 \) are the indices of refraction for the first and second medium, respectively.
• \( \theta_1 \) is the angle of incidence, and \( \theta_2 \) is the angle of refraction.

Snell's Law shows how the path of a light beam changes direction at the interface between two different materials, such as glass and air or water.
The index of refraction determines how much the speed of light changes in the new medium. Light bends towards the normal when moving to a material with a higher index and away from the normal when moving to one with a lower index. This behavior is due to the change in light speed depending on the medium.

critical angle

The critical angle is a key concept in understanding total internal reflection, a phenomenon that occurs when light cannot pass through the surface of a medium into a less dense medium.
The critical angle is the specific angle of incidence at which the angle of refraction reaches 90 degrees, meaning the light travels along the boundary and cannot escape.

• Only occurs when light attempts to move from a denser medium (higher index of refraction) to a less dense one (lower index of refraction).
• If the angle of incidence is larger than the critical angle, all of the light is reflected back inside the medium, leading to total internal reflection.

In the exercise, calculating the critical angle allows us to determine the minimum angle for which light will remain trapped inside the glass cube rather than passing into air or water.

index of refraction

The index of refraction is a measure of how much a medium can bend light, represented by the symbol \( n \). It describes how fast light travels in that medium compared to the speed of light in a vacuum, which is the fastest it can travel.

• An index of refraction greater than 1 indicates the medium is denser than a vacuum.
• Glass has an index of refraction of 1.62, meaning light travels slower in glass than in air or vacuum.

The index of refraction not only affects how much light bends as it enters a new medium but is also crucial in the calculation of the critical angle and the application of Snell's Law. Greater indices mean more bending when light passes through material boundaries.

optics in physics

Optics is a branch of physics that explores the behavior and properties of light. It encompasses various phenomena and principles that describe how light interacts with different materials and boundaries.
The study of optics includes concepts such as reflection, refraction, diffraction, and interference, and it applies to numerous technologies and natural phenomena.

• Understanding total internal reflection, as in the exercise, is vital for fiber optics and other technologies that rely on guiding light efficiently.
• Optics helps us design lenses and mirrors used in everyday items like glasses, cameras, and telescopes.

By mastering optics concepts, we can exploit light's properties for practical applications, from telecommunications to improving visual capabilities with optical instruments.