Problem 25 - Master Resource Book in JEE Main Physics

Question

In Young's double slit experiment, if \(d, D\) and \(\lambda\) represent, the distance between the slits, the distance of the screen from the slits and wavelength of light used respectively, then the bandwidth is inversely proportional to (a) \(\underline{\lambda}\) (b) \(d\) (c) \(D\) (d) \(\lambda^{2}\) (e) \(D^{2}\)

Step-by-Step Solution

Answer

Bandwidth is inversely proportional to (b) \(d\).

Step 1: Understanding Young's Double Slit Experiment

Young's double slit experiment involves light passing through two closely spaced slits and forming an interference pattern on a screen. The interference pattern is characterized by bright and dark fringes, where the separation between consecutive bright or dark fringes is called the fringe width or bandwidth.

Step 2: Expressing the Bandwidth Formula

The formula for the fringe width (or bandwidth) in Young's double slit experiment is given by: \[ \beta = \frac{\lambda D}{d} \]where \( \beta \) is the fringe width, \( \lambda \) is the wavelength of light, \( D \) is the distance from the slits to the screen, and \( d \) is the distance between the slits.

Step 3: Identifying Inverse Proportionality

From the formula \( \beta = \frac{\lambda D}{d} \), we see that bandwidth \( \beta \) is inversely proportional to the distance between the slits \( d \). This is observed because \( \beta \) decreases as \( d \) increases, indicating that they have an inverse relationship.

Step 4: Verifying with Options

Analyzing the given options:(a) \( \lambda \) - Bandwidth is not inversely proportional to \( \lambda \), but directly proportional, since \( \lambda \) is in the numerator.(b) \( d \) - Bandwidth \( \beta \) is inversely proportional to \( d \) since \( d \) is in the denominator.(c) \( D \) - Bandwidth is not inversely proportional to \( D \), but directly proportional, since \( D \) is in the numerator.(d) \( \lambda^2 \) - Bandwidth is not inversely proportional to \( \lambda^2 \), as there is no such term.(e) \( D^2 \) - Bandwidth is not inversely proportional to \( D^2 \), as there is no such term.

Key Concepts

Interference Pattern

The interference pattern is an essential aspect of Young's Double Slit Experiment. Essentially, it represents the pattern of alternating light and dark bands, or 'fringes', that appear on a screen after light passes through two closely placed slits. The light waves from each slit overlap, causing regions of constructive and destructive interference.

• Constructive interference occurs when the waves from both slits meet in phase, resulting in a bright fringe.
• Destructive interference occurs when the waves meet out of phase, cancelling each other out and creating a dark fringe.

This pattern arises due to the wave nature of light and reflects the difference in path lengths the light waves travel from the slits to any point on the screen. The interference pattern gives us insights into the wave properties of light, such as wavelength and coherence, which are crucial for calculating fringe details.

Fringe Width

Fringe width, also referred to as fringe spacing or bandwidth, is a key concept in understanding the distribution of interference patterns in Young's Double Slit Experiment. It is the distance between two successive bright (or dark) fringes on the interference pattern. The expression for fringe width \( \beta \) is derived as follows: \[ \beta = \frac{\lambda D}{d} \]Where:- \( \beta \) is the fringe width,- \( \lambda \) stands for the wavelength of the light,- \( D \) represents the distance from the slits to the screen, and- \( d \) is the distance between the slits.This formula shows that the fringe width is directly proportional to both the wavelength \( \lambda \) and the screen distance \( D \). Conversely, it is inversely proportional to the slit separation distance \( d \). Hence, larger wavelengths or more extended distances to the screen will produce wider fringes, while increasing the slit separation decreases the fringe width.

Inverse Proportionality

In the context of Young's Double Slit Experiment, understanding inverse proportionality is crucial for predicting how changes in variables affect the fringe width. According to the formula \( \beta = \frac{\lambda D}{d} \), the fringe width \( \beta \) and the distance between the slits \( d \) are inversely related. Here’s why that happens:- When the distance \( d \) between the slits increases, it leads to a decrease in the fringe width \( \beta \), because more light is needed to cover the increased gap. This results in tighter fringes.- Conversely, by decreasing the slit distance \( d \), the fringe width \( \beta \) increases because light waves spread out more, widening the fringes.Therefore, if you are experimenting with setups of various slit distances, expect narrower fringes as the slits move further apart. This principle highlights how changes in just one aspect of the setup can significantly impact the observed interference pattern.