Question

The first diffraction minima due to a single slit diffraction is at $\theta = {30^\circ }$ for a light of wavelength 5000 ${A^\circ }$ . What is the width of the slit?
(A) $5 \times {10^{ - 5}}$cm
(B) $10 \times {10^{ - 5}}$cm
(C) $2.5 \times {10^{ - 5}}$cm
(D) $1.25 \times {10^{ - 5}}$cm

Answer 1

Hint:
Diffraction patterns are observed when light passes through a series of regularly placed slits. These patterns are observable when the width of the slit is comparable to the wavelength of the incident light.
Formula used: $d = \dfrac{{n\lambda }}{{\sin \theta }}$, where d is the width of the slit that forms the diffraction pattern, n is the position on the pattern corresponding to incident light of angle $\theta $, and $\lambda $ is the wavelength of the light.

Complete step by step answer:
When a wave encounters an obstacle or an opening, diffraction occurs. It is the bending of waves around the corners of an obstacle into the region of geometrical shadow of the obstacle.
In this question, we know the position of the diffraction pattern and are asked to find the width of the slit based on this information. The data provided to us is:
Angle at which the light is incident $\theta = {30^\circ }$
Wavelength of light $\lambda = 5000{A^\circ } = 5000 \times {10^{ - 10}}$m
We remember to keep the units in their standard form. We are also given that the pattern is considered at the first minima, this implies that $n = 1$.
We know that the width of the slit depends on the wavelength of incident light as:
$\Rightarrow d = \dfrac{{n\lambda }}{{\sin \theta }}$
Directly substituting the known values in to this equation gives us:
$\Rightarrow d = \dfrac{{1 \times 5000 \times {{10}^{ - 10}}}}{{\sin 30}}$
On solving further we get:
$\Rightarrow d = \dfrac{{5000 \times {{10}^{ - 10}}}}{{0.5}}$ [As $\sin 30 = 0.5$]
$\Rightarrow d = 10000 \times {10^{ - 10}} = 10 \times {10^{ - 7}}$m
Converting it into cm as per our question [1 m = 100 cm]:
$\Rightarrow d = 10 \times {10^{ - 5}}$cm
Hence, the correct answer is option (B).

Note:
Diffraction and interference are two properties of electromagnetic radiation that are often confused with each other. Diffraction is produced by secondary wavelets that occur from the same wave but different parts of it, whereas interference pattern is originated by waves coming from two different coherent sources.