Problem 15 - Master Resource Book in JEE Main Physics

Question

Find the thickness of a plate which will produce a change in optical path equal to half the wavelength \(\lambda\) of the light passing through it normally. The refractive index of the plate \(\mu\) is equal to (a) \(\frac{\lambda}{4(\mu-1)}\) (b) \(\frac{2 \lambda}{4(\mu-1)}\) (c) \(\frac{\lambda}{(\mu-1)}\) (d) \(\frac{\lambda}{2(\mu-1)}\)

Step-by-Step Solution

Answer

The thickness is \( t = \frac{\lambda}{2(\mu - 1)} \).

Step 1: Understand the Problem

We need to find the thickness of the plate that causes an optical path difference of half a wavelength when light passes through it. The formula to calculate optical path difference is given by the product of the refractive index difference and the thickness of the plate.

Step 2: Optical Path Difference Formula

The optical path difference (OPD) is given by the formula: \( OPD = (n - 1) imes t \), where \( n \) is the refractive index of the plate, and \( t \) is the thickness of the plate. In our case, \( OPD = \frac{\lambda}{2} \).

Step 3: Set Up the Equation

Substitute the given optical path difference into the formula: \( (\mu - 1) \times t = \frac{\lambda}{2} \). Here, \( \mu \) is the refractive index, and the OPD is half the wavelength \( \lambda \).

Step 4: Solve for Thickness

Rearrange the equation to solve for the thickness \( t \): \[ t = \frac{\lambda}{2(\mu - 1)} \]. This formula gives us the thickness of the plate that produces the desired optical path difference.

Key Concepts

Wavelength

The term "wavelength" refers to the distance between two consecutive points that are in phase on a wave, such as from crest to crest or trough to trough. It is usually represented by the Greek letter lambda (\( \lambda \)). Wavelength is a critical factor in determining the nature of a wave, including its color in the spectrum of visible light. For example, blue light has a shorter wavelength than red light. Understanding the wavelength of light is essential for this exercise because it helps define the change in optical path that the plate causes. When we talk about a change in optical path equal to half the wavelength, we are essentially talking about how much the medium alters the propagation route of light compared to what it would be in a vacuum. Think of it as the shrink or expansion of the wave's journey through a material.

Refractive Index

The refractive index, usually denoted as \( \mu \) or \( n \), is a number that describes how light or any other radiation propagates through a medium. The refractive index is a dimensionless number that is often greater than 1.

• If \( \mu = 1 \), light travels through the medium at the same speed as it does in a vacuum.
• Values of \( \mu \) greater than 1 indicate slowing of light in the medium.

The refractive index is crucial in understanding how much light bends, or refracts, when entering a new material. In our exercise, the refractive index tells us how much the plate slows down the light, which directly affects the optical path difference when light travels through the thickness of the plate. This index changes with different materials, influencing how much the light path alters and thus helping predict outcomes in optical systems.

Light Transmission

Light transmission refers to the passage of light through a medium, which can either be transparent, translucent, or opaque. The efficiency and the amount of light transmitted depend on the medium's properties and its thickness.

• In a transparent medium, most of the light passes through with little absorption or scattering.
• For a translucent material, only some light passes through, causing the light to scatter, while opaque materials completely block the light.

Understanding how light transmits through a material aids in designing systems where control over light paths is desired, such as lenses or optical filters. In the context of our problem, we care about how much the light path is extended or shortened when passing through a medium, a concept captured by the optical path difference.

Optical Thickness

Optical thickness is a concept that combines the physical thickness of a medium and its refractive index. It's important to distinguish this from just physical thickness—optical thickness accounts for how much a medium interferes with the light path due to its properties. The equation \( OPD = (\mu - 1) \times t \) illustrates how optical thickness is calculated. Here, the refractive index (\( \mu \)) along with the physical thickness (\( t \)) determines how different the optical path is compared to a vacuum. Therefore, optical thickness explains how much delay or advancement occurs in the light's journey through the plate, affecting how the light exits the material. This is why we are interested in calculating the thickness of the plate that will achieve an optical path difference equivalent to half a wavelength, adjusting its optical thickness.