Problem 24 - 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

The bandwidth is inversely proportional to (b) \( d \).

Step 1: Recall Bandwidth in Young's Double Slit Experiment

The bandwidth (also known as fringe width) in Young's double-slit experiment is given by the formula \( \beta = \frac{\lambda D}{d} \). Here, \( d \) is the distance between the slits, \( D \) is the distance from the slits to the screen, and \( \lambda \) is the wavelength of the light used.

Step 2: Identify Inverse Proportionality

We need to determine which variable the bandwidth \( \beta \) is inversely proportional to. From the formula \( \beta = \frac{\lambda D}{d} \), the bandwidth is inversely proportional to \( d \), the distance between the slits. This means as \( d \) increases, \( \beta \) decreases.

Step 3: Review Other Options

Confirm that \( \beta \) is not inversely proportional to any other variables or squared forms. In the formula \( \beta = \frac{\lambda D}{d} \), \( \beta \) is directly proportional to both \( D \) and \( \lambda \), not inversely. It's also not proportional to \( \lambda^2 \) or \( D^2 \).

Key Concepts

Bandwidth in Young's Double Slit Experiment

In Young's Double-Slit Experiment, the term 'bandwidth' refers to the spacing between two consecutive bright or dark fringes on the interference pattern. This pattern is created by the overlapping of waves passing through two slits. To calculate the bandwidth (\( \beta \)), we use the formula \( \beta = \frac{\lambda D}{d} \).

This formula shows that bandwidth depends on three main factors:

• Wavelength (\( \lambda \)): The frequency of the light source used.
• Distance to the screen (\( D \)): The length from the slits to the screen where the pattern is projected.
• Distance between the slits (\( d \)): The separation between the two slits from which the light emerges.

This formula indicates that bandwidth is:

• Directly proportional to both wavelength and distance to the screen.
• Inversely proportional to the distance between the slits. Thus, if \( d \) increases, the bandwidth (\( \beta \)) becomes smaller, narrowing the space between fringes.

Wavelength of Light

Wavelength (\( \lambda \)) is a critical factor that influences many optical phenomena, including the bandwidth of Young's Double Slit Experiment. It is the distance between successive peaks or troughs of a wave and determines the color of light. In everyday terms, you can visualize it as the 'pink' and 'yellowish' colors we get from the rainbow.

Wavelength is directly proportional to the bandwidth. This means:

• Longer Wavelength: Results in a wider fringe bandwidth, making the fringes further apart.
• Shorter Wavelength: Leads to a narrower bandwidth with fringes closer together.

In practical applications, changing the wavelength by using light of different colors will visibly alter the interference pattern. This showcases its tangible impact and helps us see how wavelength precisely interacts with other elements in the formula \( \beta = \frac{\lambda D}{d} \). It underscores an important aspect of wave behavior in optics.

Distance Between the Slits

The distance between the slits (\( d \)) in Young's Double-Slit Experiment is a vital variable influencing the outcome of the interference pattern. It represents the separation between the two openings that light passes through. Adjusting \( d \) changes the way the light waves overlap, thus altering the fringe pattern observed on the screen.

Inverse Proportionality: The formula \( \beta = \frac{\lambda D}{d} \) makes it clear that the bandwidth is inversely proportional to \( d \):

• Decreasing \( d \): Widens the bandwidth as the fringes become wider apart.
• Increasing \( d \): Narrows the bandwidth, bringing the fringes closer together.

The distance between the slits can be controlled in experimental setups to achieve desired patterns. This concept illustrates the precision and predictability of wave interference, and why even simple changes in \( d \) can have significant effects on the observed bandwidth.