Problem 29 - Physics

Question

A layer of liquid \(B\) floats on liquid \(A\). A ray of light begins in liquid \(A\) and undergoes total internal reflection at the interface between the liquids when the angle of incidence exceeds \(36.5^{\circ} .\) When liquid \(B\) is replaced with liquid \(C,\) total internal reflection occurs for angles of incidence greater than \(47.0^{\circ} .\) Find the ratio \(n_{B} / n_{C}\) of the refractive indices of liquids \(B\) and \(C\).

Step-by-Step Solution

Answer

The ratio \( n_B / n_C \) is approximately \( 0.809 \).

Step 1: Understand Total Internal Reflection

Total internal reflection occurs when a light ray traveling from a medium of higher refractive index to a medium of lower refractive index reflects completely, without any refraction. This phenomenon occurs when the angle of incidence exceeds the critical angle.

Step 2: Find Critical Angles

For liquid \(B\): the critical angle \( \theta_B \) is \(36.5^{\circ}\). For liquid \(C\): the critical angle \( \theta_C \) is \(47.0^{\circ}\).

Step 3: Use Snell's Law for Critical Angle

The critical angle \( \theta_c \) is found using Snell's Law: \( n_A \sin(\theta_c) = n_B \sin(90^\circ) \). Since \( \sin(90^\circ) = 1 \), the equation becomes \( n_A \sin(\theta_c) = n_B \), so \( n_B = n_A \sin(\theta_B) \). Similarly, for liquid \(C\), \( n_C = n_A \sin(\theta_C) \).

Step 4: Derive the Ratio of Refractive Indices

The refractive index ratio \( n_B / n_C \) is given by \( \frac{n_B}{n_C} = \frac{n_A \sin(\theta_B)}{n_A \sin(\theta_C)} \) because \( n_A \) cancels out. Thus, the ratio becomes \( \frac{\sin(36.5^\circ)}{\sin(47.0^\circ)} \).

Step 5: Calculate the Ratio

Calculate \( \frac{\sin(36.5^\circ)}{\sin(47.0^\circ)} \). Use a calculator to find \( \sin(36.5^\circ) \) and \( \sin(47.0^\circ) \). The result is approximately \( 0.809 \).

Key Concepts

Total Internal Reflection

Total internal reflection is an interesting phenomenon that can happen with light. It occurs when a light ray travels from a denser medium, which has a higher refractive index, to a less dense medium. In this situation, if the angle at which the light hits the boundary is larger than a particular angle known as the critical angle, the light will not pass through into the second medium. Instead, it gets completely reflected back into the first medium.

• Occurs when moving from a higher to a lower refractive index.
• Happens only if incident angle exceeds the critical angle.
• Results in 100% reflection without any transmission.

Understanding this concept is essential for technologies like fiber optics, where light needs to stay within the fiber without escaping and is entirely reliant on total internal reflection.

Critical Angle

The critical angle is a specific angle of incidence beyond which total internal reflection occurs. It's a threshold value that depends on the refractive indices of the two involved media. When a light ray travels from one medium (say, glass) to another (like air), where the latter one has a lower refractive index, the critical angle determines if the light will entirely reflect back instead of refracting

• Applicable when moving light from a denser to a less dense medium.
• At critical angle, the refracted ray will travel along the boundary.
• If the incident angle is greater than the critical angle, no refraction occurs.

The formula to find the critical angle (\( \theta_c \)) involves the refractive indices of the two media (\(n_1\) and \(n_2\)): \[\theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right)\]where \(n_2\) is the refractive index of the less dense medium.

Snell's Law

Snell's Law is a fundamental principle that describes how light bends when it travels from one medium to another. It states that the ratio of the sine of the angle of incidence (\(\theta_i\)) to the sine of the angle of refraction (\(\theta_r\)) is constant and equivalent to the ratio of the refractive indices of the two media (\(n_1\) and \(n_2\)):\[n_1 \sin(\theta_i) = n_2 \sin(\theta_r)\]When considering a situation with a critical angle, Snell's Law helps to express the condition mathematically so that the refractive index of a medium could be calculated using the critical angle. It helps us in many practical applications, from understanding optical lenses to perfectly designing glasses organization, leveraging how light bends.

Refractive Index

The refractive index is a measure of how much light bends when it enters a material. It's a dimensionless number that tells us the speed of light in a medium compared to the speed of light in a vacuum. A higher refractive index means light travels slower in the material compared to air or vacuum.

• Calculated as the ratio of the speed of light in vacuum to the speed in the medium.
• Helps determine bending behavior using Snell's Law.
• Essential for figuring out the critical angle.

In the original problem, the refractive index helps determine the behavior of light at the interface between two liquids. Different liquids have distinct refractive indices. Thus, their effect on the light's path varies, influencing angles like the critical angle and behaviors like total internal reflection. It can be calculated experimentally using the equation:\[n = \frac{c}{v}\]where \(c\) is the speed of light in vacuum and \(v\) is the speed of light in the material.