Question

A parallel beam of monochromatic light of wavelength $5000{\AA}$ is incident on a single narrow slit of width $0.001mm$ . The light is focused by a convex lens on a screen placed on the focal plane. The first minimum will be formed for the angle of diffraction equal to:
A. ${20^ \circ }$
B. ${15^ \circ }$
C. ${30^ \circ }$
D. ${50^ \circ }$

Answer 1

Hint : For solving this question, it is necessary to remember the formula of destructive interference for a single slit. It is also important to understand the basics of single slit diffraction, so that values can be assumed correctly, and the formula generates the correct answer.

Complete Step By Step Answer:
According to the question, the light beam is monochromatic. There is a single, narrow slit, and the light is focused by a convex lens on a screen placed on the focal plane. We need to find the angle of diffraction for the first minimum.
The following values are given in the question:
Wavelength: $\lambda = 5000{\AA} = 0.5 \times {10^{ - 6}}m$
Width of Slit: $d = 0.001mm = {10^{ - 6}}m$
Order of the minimum: $m = 1$
Now, the formula for destructive interference of a single slit:
d\sin \theta = m\lambda \\\ \theta = {\sin ^{ - 1}}\left( {\dfrac{{m\lambda }}{d}} \right) \\\
Placing the values in the formula, we have,
\theta = {\sin ^{ - 1}}\left( {\dfrac{{1 \times 0.5 \times {{10}^{ - 6}}}}{{{{10}^{ - 6}}}}} \right) \\\ \theta = {\sin ^{ - 1}}\left( {0.5} \right) = {30^ \circ } \\\
The first minimum will be formed for the angle of diffraction equal to 30°.
Hence the correct option is C.

Note :
Diffraction is the phenomenon in which light bends around the corner of an object in its path. The reflected light produces light, dark or colored fringes. There are two types of diffraction: Fraunhofer Diffraction and Fresnel Diffraction. A detailed pattern is formed if the source is coherent and the width of the slit is comparable to the wavelength of light.