Problem 42 - Master Resource Book in JEE Main Physics

Question

Light of wavelength \(\lambda\) is incident on a slit width \(d\). The resulting diffraction pattern is observed on a screen at a distance \(D\). The linear width of the principal maximum is equal to the width of the slit, if \(D\) equals (a) \(\frac{d^{2}}{2 \lambda}\) (b) \(\frac{d}{\lambda}\) (c) \(\frac{2 \lambda^{2}}{d}\) (d) \(\frac{2 \lambda}{d}\)

Step-by-Step Solution

Answer

(a) \(\frac{d^2}{2\lambda}\) is the correct answer.

Step 1: Understand the Problem

We are given a situation involving single-slit diffraction where light of wavelength \(\lambda\) passes through a slit of width \(d\). It creates a diffraction pattern on a screen located at a distance \(D\) from the slit. We need to find out for which value of \(D\) the linear width of the principal maximum equals the width of the slit.

Step 2: Recall the Single-Slit Diffraction Formula

The angular width of the principal maximum in a single-slit diffraction pattern is given by \(2\theta\), where \(\sin \theta = \frac{\lambda}{d}\). Using small angle approximation, \(\sin \theta \approx \tan \theta \approx \theta\), the linear width of the principal maximum on the screen is \(2D\theta\).

Step 3: Find the Linear Width of the Principal Maximum

Using \(\theta = \frac{\lambda}{d}\), the linear width \(W\) of the principal maximum on the screen is given by:\[ W = 2D\theta = 2D\left(\frac{\lambda}{d}\right) = \frac{2D\lambda}{d} \]

Step 4: Set Linear Width Equal to Slit Width

We need \(W = d\). Set the expression for \(W\) equal to \(d\):\[ \frac{2D\lambda}{d} = d \]

Step 5: Solve for D

Rearrange the equation to solve for \(D\):\[ \frac{2D\lambda}{d} = d \Rightarrow 2D\lambda = d^2 \Rightarrow D = \frac{d^2}{2\lambda} \]

Step 6: Check Choices

The correct expression for \(D\) is \(\frac{d^2}{2\lambda}\), which matches option (a).

Key Concepts

Diffraction Pattern

When light waves encounter an obstacle or aperture, such as a slit, they bend around this obstacle and create a pattern of light and dark regions called a diffraction pattern. This phenomenon occurs due to the interference of light waves.
Understanding diffraction patterns is crucial because they reveal how waves propagate in different media. In the context of a single-slit diffraction experiment, the pattern observed on a screen consists of a central bright region, known as the principal maximum, and alternating dark and light fringes on either side.

• The central, brightest region is the principal maximum.
• Secondary maxima are less intense and located further from the center.
• Dark fringes signify destructive interference at those locations.

Appreciating these patterns helps in understanding the wave nature of light and its interaction with matter.

Wavelength

The term wavelength refers to the distance between consecutive peaks (or troughs) of a wave. It is denoted by the Greek letter \( \lambda \) and is often measured in nanometers (nm) or meters (m).
Wavelength is a critical factor in diffraction as it determines the extent to which light waves spread after passing through a slit or around an obstacle. In essence, the longer the wavelength, the more pronounced the diffraction and vice versa.

• Shorter wavelengths result in sharper and more narrowly spaced fringes.
• Longer wavelengths create broader and more widely spaced diffraction patterns.
• The relationship between wavelength and angle of diffraction is given by the equation \( \sin \theta = \frac{\lambda}{d} \).

Understanding wavelength helps in predicting and analyzing diffraction patterns effectively.

Principal Maximum

The principal maximum is the brightest spot in a single-slit diffraction pattern, located at the center of the screen where the light is most intense. This central maximum occurs when the path difference between light from different parts of the slit is zero or an integer multiple of the wavelength.
To find the width of this principal maximum on a screen, we use the angle \( \theta \) (calculated as \( \theta = \frac{\lambda}{d} \)) and the distance to the screen \( D \). The linear width \((W)\) is calculated using the formula:
\[ W = 2D\left(\frac{\lambda}{d}\right) = \frac{2D\lambda}{d} \\]

• The linear width gives the actual size of the maximum as observed on the screen.
• It is influenced by the wavelength, slit width, and distance to the observing screen.
• Understanding the principal maximum is key to deciphering patterns in experiments like the single-slit diffraction.

Slit Width

The slit width, denoted by \( d \), is the physical measurement of the gap through which light passes in a diffraction experiment. The size of the slit width is pivotal in determining the characteristics of the resulting diffraction pattern.
In single-slit diffraction, the slit width influences:

• The angular width of the diffraction pattern (how far the light spreads).
• The intensity and sharpness of the principal maximum and the subsequent secondary maxima.
• The relationship \( \sin \theta = \frac{\lambda}{d} \), which shows how slit width and wavelength together control the diffraction angle.

Choosing the right slit width is vital in experiments to control the diffraction pattern for precise observations and measurements.