Problem 21 - Arihant AIEEE Physics

Question

A vessel contains a slab of glass \(8 \mathrm{~cm}\) thick and of refractive index 1.6. Over the slab, the vessel is filled by oil of refractive index \(\mu\) upto height \(4.5 \mathrm{~cm}\) and also by another liquid i.e., water of refractive index \(4 / 3\) and height \(6 \mathrm{~cm}\) as shown in figure. An observer lookingdown from above, observes that, a mark at the bottom of the glass slab appears to be raised up to position \(6 \mathrm{~cm}\) from the bottom of the slab. The refractive index of oil \((\mu)\) is: (a) \(1.5\) (b) \(2.5\) (c) \(0.5\) (d) \(1.2\)

Step-by-Step Solution

Answer

The refractive index of oil is approximately 1.2 (option d).

Step 1: Understand the Problem Statement

The problem involves calculating the apparent height of a mark viewed through multiple layers of different media. Relevant for this problem are the refractive indices and the thicknesses of the glass, oil, and water layers.

Step 2: Definition of Apparent Depth

The apparent depth is calculated using the formula \( h' = \frac{h}{\mu} \), where \( h' \) is the apparent depth, \( h \) is the real depth, and \( \mu \) is the refractive index of the medium. The apparent depth of the entire system can be used to find the refractive index of the oil.

Step 3: Calculate Apparent Depth of Glass

For the glass, the real depth is \( 8 \mathrm{~cm} \) and its refractive index is 1.6. Thus, the apparent depth of the glass is \( h' = \frac{8}{1.6} = 5 \mathrm{~cm} \).

Step 4: Understand Total Apparent Movement of the Mark

According to the observer, the mark seems to move up to a position \( 6 \mathrm{~cm} \) from the bottom. Therefore, the overall apparent depth as seen through all three substances combined is \( 6 \mathrm{~cm} \).

Step 5: Calculate Apparent Depth of Water

The refractive index of water is \( \frac{4}{3} \), and its height is \( 6 \mathrm{~cm} \). Hence, the apparent depth of water is \( \frac{6}{\frac{4}{3}} = 4.5 \mathrm{~cm} \).

Step 6: Set up Equation for Oil's Apparent Depth

Let the real depth of the oil layer be \( 4.5 \mathrm{~cm} \) and its refractive index \( \mu \). The apparent depth of oil, glass, and water must add up to \( 6 \mathrm{~cm} \): \( 5 + \frac{4.5}{\mu} + 4.5 = 6 \).

Step 7: Solve for the Refractive Index of Oil

Rearrange and solve the equation: \[ 9.5 - \frac{4.5}{\mu} = 6 \] which simplifies to \[ \frac{4.5}{\mu} = 3.5 \] hence \[ \mu = \frac{4.5}{3.5} = \frac{9}{7} \approx 1.2857 \]. The closest option provided is (d) 1.2.

Key Concepts

Refractive Index

The refractive index, often denoted by the Greek letter \( \mu \), is a measure of how much the speed of light is reduced inside a medium compared to its speed in a vacuum.
The concept is crucial in physics, particularly in optics, as it helps understand how light behaves when it travels from one medium to another.The value of the refractive index is calculated by the formula:\[ \mu = \frac{c}{v} \]where \( c \) is the speed of light in a vacuum, and \( v \) is the speed of light in the material.
A higher refractive index means the light slows down more when entering that medium. In the exercise provided, different materials (glass, oil, water) each have their own refractive indices, affecting how light travels through them and determines the apparent position of objects viewed through these layers.

Apparent Depth

Apparent depth is a fascinating optical effect that explains why objects under a liquid surface appear closer to the surface than they actually are.
This phenomenon occurs due to the bending or refraction of light as it travels from a denser medium (like water or glass) to a less dense one (like air).The formula used to calculate the apparent depth \( h' \) is given by:\[ h' = \frac{h}{\mu} \]where \( h \) is the real depth of the object in the medium, and \( \mu \) is the medium's refractive index.In this exercise, the apparent depth is used to calculate the position where an observer perceives a mark on the glass slab when looking through the combined layers of glass, oil, and water. This helps in determining the actual refractive index of the oil.

Snell's Law

Snell's Law is a fundamental principle in optics that describes how light rays change direction or "refract" when passing from one medium to another. The law is given by:\[ n_1 \sin \theta_1 = n_2 \sin \theta_2 \]Where:

• \( n_1 \) and \( n_2 \) are the refractive indices of the first and second media, respectively.
• \( \theta_1 \) and \( \theta_2 \) are the angles of incidence and refraction, respectively.

However, Snell's Law does not need to be explicitly used in the solution to the original exercise; its principle underlies the entire refraction process as it quantifies the bending of light at the medium boundaries. Understanding this law can provide deeper insights into why light refracts and how the path changes influence the perception of submerged objects.

Light Refraction

Light refraction is the bending of a light wave as it passes at an angle between two mediums with different refractive indices.
This principle explains many optical illusions and everyday phenomena, such as a straw appearing bent in a glass of water. Light bending occurs because light travels at different speeds in different materials.

• It moves slower in denser materials like glass and faster in less dense materials like air.
• The extent to which light bends is determined by the refractive index of the materials.

In the provided exercise, refraction is crucial in understanding the apparent depth changes occurring in each layer: glass, oil, and water.
The cumulative effect defines how deeply embedded the mark seems to an observer viewing from above.