Question

Light of wave-length \(\lambda\) is incident on a slit of width \(\mathrm{d}\). The resulting diffraction pattern is observed on a screen placed at a distance \(\mathrm{D}\). The linear width of the principal maximum is equal to the width of the slit, then \(\mathrm{D}=\) (A) \(\left(\mathrm{d}^{2} / 2 \lambda\right)\) (B) \(\left(2 \lambda^{2} / \mathrm{d}\right)\) (C) \((\mathrm{d} / \lambda)\) (D) \((2 \lambda / \mathrm{d})\)

Short answer

The correct answer is (A), where the distance D is approximately equal to \(\dfrac{d^2}{2\lambda}\), with d being the width of the slit and λ being the wavelength of the incident light.

Step-by-step solution

Determine the angular width of the principal maximum

In the case of single-slit diffraction (using Fraunhofer diffraction), the angular width of the principal maximum can be determined using the formula: \(\sin \theta = \dfrac{\lambda}{d}\). This formula is derived from the Huygens principle and the interference of light waves entering through the slit.

Convert angular width to linear width

To convert the angular width of the principal maximum to linear width (Δx), we can use the formula: \(\Delta x = D \tan \theta\). In this problem, Δx is given as equal to d. However, since the angle θ is relatively small, we can utilize the small angle approximation: \(\tan \theta \approx \sin \theta\), thus the expression simplifies to: \(\Delta x \approx D \sin \theta\).

Replace sin(θ) in the equation and solve for D

Now, we replace \( \sin \theta \) using the previously mentioned relation: \(\sin \theta = \dfrac{\lambda}{d}\). The equation for Δx becomes: \(\Delta x \approx D \dfrac{\lambda}{d}\). We are given that the linear width of the principal maximum is equal to the width of the slit, thus we can write: \(d \approx D \dfrac{\lambda}{d}\). To solve for D, we can multiply both sides with d and divide both sides by λ: \[D \approx \dfrac{d^2}{\lambda}\]

Compare the derived formula for D with the given options

Comparing the derived expression for D with the options given, we find that it matches with option (A): \[D \approx \dfrac{d^2}{2\lambda}\] So, the correct answer is (A).

Key concepts

Diffraction Pattern

When light encounters an obstacle such as a slit, it bends around the corners, creating an interference pattern on the other side. This is known as a diffraction pattern. A single-slit diffraction pattern consists of a central bright region known as the principal maximum, flanked by alternating dark and bright fringes. The central maximum is the most intense and widest part of the pattern.
Typically, the intensity gradually diminishes as you move away from this central maximum. The distribution and intensity of these fringes are determined by factors like the wavelength of light (\(\lambda\)), the width of the slit (\(d\)), and the distance from the slit to the screen (\(D\)).
These factors influence both the size and location of the peaks and troughs observed on the viewing screen.

Fraunhofer Diffraction

Fraunhofer diffraction is a simple and common case of diffraction that occurs when both the source and the viewing screen are at infinite distances from the diffracting object. In practical terms, this means using parallel beams of light.
In lab setups, lenses are typically used to simulate these infinite distances, allowing us to observe Fraunhofer diffraction patterns on a convenient screen.
This type of diffraction is characterized by sharp and distinct interference patterns, unlike Fresnel diffraction, which occurs at closer distances. In single-slit experiments, using Fraunhofer diffraction allows us to derive formulas like \(\sin \theta = \frac{\lambda}{d}\), which relate angles, wavelengths, and slit widths in creating diffraction patterns.

Huygens Principle

Huygens Principle is a fundamental concept that explains how light waves propagate. According to this principle, every point on a wavefront can be considered a source of secondary spherical wavelets. As these wavelets spread out, their sum forms a new wavefront.
When applied to single-slit diffraction, Huygens Principle helps us understand how different parts of a wavefront passing through the slit interfere with each other, resulting in the observed diffraction pattern.
This principle is an essential tool for predicting the behavior of waves, not just for light, but also for sound and water waves. It provides the theoretical foundation for equations used in studying diffraction, like those used to calculate angular widths of bright regions in a pattern.

Angular Width

Angular width refers to the angle over which a particular feature of the diffraction pattern, such as the central maximum, extends. For single-slit diffraction, this width is determined by the angle at which the first minimum occurs on either side of the central maximum.
In the solution provided, the angular width is calculated using the formula \(\sin \theta = \frac{\lambda}{d}\). This implies the position of the first minimum can be predicted based on the wavelength and slit width.
Understanding angular width is crucial for converting it to linear width (\(\Delta x\)) on the screen, using approximations like \(\tan \theta \approx \sin \theta\) under small angle conditions. This relationship helps link the theoretical models of diffraction with observable and measurable results.