Problem 33 - Master Resource Book in JEE Main Physics

Question

In determination of refractive index of glass slab using travelling microscope, first of all we take a reading when the microscope is focused on a mark. This reading comes out to be \(s_{1}\), then we place a glass slab on the surface covering the mark. Now, the microscope is re-adjusted to focus the mark through the slab and this time reading comes out to be \(s_{2}\). Then, we place an opaque object on the glass slab and adjust the microscope to focus on opaque object, this time the reading of microscope is \(s_{3} .\) The refractive index of the glass slab is (a) \(\frac{s_{3}-s_{1}}{s_{2}-s_{1}}\) (b) \(\frac{s_{3}-s_{2}}{s_{2}-s_{1}}\) (c) \(\frac{s_{3}-s_{1}}{s_{3}-s_{2}}\) (d) \(\frac{s_{3}}{s_{3}-s_{2}}\)

Step-by-Step Solution

Answer

The correct option is (c) \( \frac{s_3-s_1}{s_3-s_2} \).

Step 1: Identify the Problem

We are determining the refractive index of a glass slab using readings obtained from a travelling microscope. The challenge is to select the correct formula from the given options for calculating this refractive index.

Step 2: Understand the Variables

Identify the specific readings from the problem: - Take the first reading when the microscope is focused on the mark, denoted as \(s_1\).- Adjust the microscope to focus on the mark through the slab for the second reading, \(s_2\).- Adjust the microscope to focus on an opaque object placed on the slab for the third reading, \(s_3\).

Step 3: Recall the Formula for Refractive Index

The formula for the refractive index \( n \) of a glass slab when using a microscope involves using the readings \(s_1\), \(s_2\), and \(s_3\). It is based on the ratio of the real depth to the apparent depth, which usually can be calculated as follows:\[ n = \frac{s_3 - s_1}{s_3 - s_2} \]

Step 4: Select the Correct Option

Compare our derived formula \( \frac{s_3 - s_1}{s_3 - s_2} \) to the given options:- (a) \( \frac{s_3-s_1}{s_2-s_1} \)- (b) \( \frac{s_3-s_2}{s_2-s_1} \)- (c) \( \frac{s_3-s_1}{s_3-s_2} \) - (d) \( \frac{s_3}{s_3-s_2} \)The correct choice matches option (c).

Key Concepts

Travelling Microscope

A travelling microscope is an essential tool in the field of optics for making precise measurements. It consists of a microscope mounted on a horizontal scale, allowing it to move across the scale to take readings. This mobility helps in focusing the microscope on the object of interest by aligning it accurately along the scale.In the context of determining the refractive index of a glass slab, the travelling microscope is used to take precise measurements at different reference points. Initially, it focuses directly on the mark, which gives the first reading, denoted as \(s_1\). Then, after placing the glass slab, the microscope is readjusted to focus through the slab to get the second reading \(s_2\). Finally, an opaque object is placed on the slab and focused to obtain the third reading \(s_3\). These readings are crucial for calculating the refractive index by allowing us to determine real and apparent depths relative to each other.

Glass Slab Experiment

The glass slab experiment is a classical method for determining the refractive index of transparent materials. It involves measuring how much the path of light bends or refracts when it passes through a medium.In this experiment:

• Begin with a direct focus on a reference mark on the base surface using a travelling microscope, which gives reading \(s_1\).
• Next, place the glass slab over the mark and adjust the microscope to focus through the slab. This gives the apparent reading \(s_2\).
• An opaque object is then placed on top of the slab, and the microscope is adjusted again to get the reading \(s_3\). This reading represents the refracted light path's exit point.

Using these readings, the refractive index can be calculated, as the experiment cleverly uses the change in focus introduced by the glass slab to measure the shift in light's path.

Apparent Depth and Real Depth

Understanding apparent depth and real depth is key in optics, especially in the calculation of refractive index. When you look through a medium like a glass slab, objects appear at different depths than they are actually located - this is known as apparent depth.

• The real depth is the actual vertical distance from the object on the bottom (without the slab) to the observer's eye, recorded by \(s_1\).
• The apparent depth is the perceived depth when looking through the slab, given by \(s_2\).

The refractive index \(n\) is determined by the ratio of real depth to apparent depth. Mathematically, it can be represented as:\[ n = \frac{s_3 - s_1}{s_3 - s_2} \]This equation utilizes the difference in each of the depths scaled by the measurement taken when the opaque object \(s_3\) is focused, showing its critical role in determining the refractive index.

Optical Instruments Analysis

Analyzing optical instruments involves understanding how they manipulate light to help us see and measure with greater accuracy. In this context, the travelling microscope and the glass slab are particular tools designed for this purpose. Optical instruments like the travelling microscope focus light to give precise visual readings, which aid in accurately determining properties like refractive index. The analysis comprises not only focusing and adjusting the instrument but also interpreting the readings into meaningful data. By examining how effectively the glass slab experiments use these readings, one can reveal the inherent properties of the glass, such as its refractive index. This experiment demonstrates the microscope's capability to study intricate details in optics, highlighting the intrinsic relationship between measurement and analysis in determining materials' properties.