Problem 43 - Master Resource Book in JEE Main Physics

Question

A glass slab of thickness \(8 \mathrm{~cm}\) contains the same number of waves as \(10 \mathrm{~cm}\) of water when both are traversed by the same monochromatic light. If the refractive index of water is \(4 / 3\), the refractive index of glass is (a) \(5 / 4\) (b) \(3 / 2\) (c) \(5 / 3\) (d) \(16 / 15\)

Step-by-Step Solution

Answer

The refractive index of glass is \( \frac{5}{3} \).

Step 1: Understanding the Problem

We are given that a glass slab of thickness 8 cm contains the same number of light waves as a 10 cm slab of water. We need to find the refractive index of the glass, given that the refractive index of water is \( \frac{4}{3} \).

Step 2: Formula for Number of Waves

The number of waves in a medium is given by multiplying the refractive index \(n\) of the medium with its thickness \(d\), so it becomes \( n \times d \).

Step 3: Equating Number of Waves

Since the number of waves is the same in both cases, we equate:\[ n_{\text{glass}} \times 8 = \frac{4}{3} \times 10 \]

Step 4: Solving for Refractive Index of Glass

Divide both sides by 8 to solve for \( n_{\text{glass}} \):\[ n_{\text{glass}} = \frac{4}{3} \times \frac{10}{8} = \frac{5}{3} \].

Step 5: Verification

To verify our solution, calculate the number of waves for both mediums: \( \frac{5}{3} \times 8 = \frac{40}{3} \) waves and \( \frac{4}{3} \times 10 = \frac{40}{3} \) waves, confirming that both numbers are equal.

Key Concepts

Glass Slab

A glass slab is a solid piece of glass with parallel surfaces that is often used in experiments involving light, such as refractive index calculations. This optical device helps demonstrate how light behaves when passing through different mediums. The thick, clear glass provides essential insights into properties like refraction, which is the bending of light as it passes from one medium to another. By understanding the effects of a glass slab on light waves, you can explore key physics principles that govern optical phenomena.

Glass slabs are particularly useful for measuring how light slows down in a medium compared to a vacuum. This slowing down is due to the glass slab's refractive index, which is a measure of how much slower light travels in the slab compared to air or a vacuum. In many experiments, calculating the number of waves contained within the slab can reveal much about its refractive properties.

Number of Waves

The number of waves in a medium, like a glass slab or water, refers to how many complete wave cycles can fit within a given distance. This concept is crucial in determining a material's refractive properties because light behaves differently depending on the medium it traverses.

The formula to calculate the number of waves is given by the product of the medium's refractive index \(n\) and its thickness \(d\). Mathematically, this is represented as:

• Number of Waves = \(n \times d\)

Equating the number of waves in two different mediums helps compare their refractive indices. In the provided example with the glass slab and water, this principle is used to find the refractive index of glass by equating the number of waves it contains to that in water.

This mathematical comparison highlights the importance of understanding how wave cycles occur differently in different materials.

Monochromatic Light

Monochromatic light is light that consists of a single wavelength or color. This type of light is essential in studying the behavior of light in various mediums, like glass or water, without the complicating factors of multiple wavelengths.

Using monochromatic light, often achieved with lasers or filters, ensures accurate experiments and calculations. It provides clarity in results because variations typically caused by different wavelengths, such as dispersion, are eliminated.

In calculations for experiments like the one presented, using monochromatic light simplifies the understanding of how the refractive index of materials like glass and water affects the light's path. When light enters a different medium, it experiences bending, the degree of which depends on the refractive index, and can be more easily observed and measured with monochromatic light.

Medium Thickness

The thickness of a medium, such as the glass slab or water in this exercise, greatly influences the behavior of light as it travels through. Thickness contributes to the total number of waves present in that medium, alongside its refractive index.

To measure this, you multiply the thickness by the refractive index, giving a clear idea of how many wave cycles can be contained within a specific distance:

• Number of Waves = Thickness \(\times\) Refractive Index

In glass and water, different thicknesses mean the same number of waves might result from different refractive indices. By analyzing this relationship, you can solve problems related to the properties of materials, such as finding out how they compare in allowing light to pass through them.

Understanding the influence of thickness on light's behavior is pivotal in optics, where precise calculations help determine properties like speed within different substances and their refractive indices.