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# Light of wavelength 550nm falls normally on a slit of width $22.0 \times {10^{ - 5}}cm$. The angular position of the second minima from the central maximum will be (in radians)-A. $\dfrac{\pi }{8}$B. $\dfrac{\pi }{{12}}$C. $\dfrac{\pi }{4}$D. $\dfrac{\pi }{6}$

Last updated date: 15th Jun 2024
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Hint: Diffraction is defined as the phenomenon where the light wave bends around a sharp edge or corner, which can be visualised as intrusion of light in the region of the shadow of the object. There are several orders of diffraction wherein the intensity of the light decreases as we proceed away from the first order of diffraction, which is also known as the central maximum. The second minima are the second in order from the central maximum, on either side.

The angular position of nth order diffraction is calculated from the following equation-
$n\lambda = d\sin \theta$
Where,
$\lambda$=wavelength of light ,
d=slit width,
$\theta$= angular position
And n= order of diffraction, for minima $n = \pm 1, \pm 2, \pm 3,.......$ and for maxima $n = 0, \pm \dfrac{3}{2}, \pm \dfrac{5}{2},.............$

Step1:
From the equation of angular position of nth order diffraction we have,
$n\lambda = d\sin \theta$ …………..(1)
Where,
$\lambda$=wavelength of light ,
d=slit width,
$\theta$= angular position
And n= order of diffraction, for minima $n = \pm 1, \pm 2, \pm 3,.......$ and for maxima $n = 0, \pm \dfrac{3}{2}, \pm \dfrac{5}{2},.............$
Since it is given that we have to find angular position of second minima ,
Therefore n=2
Also given that wavelength $\lambda = 550nm = 550 \times {10^{ - 9}}m$
Slit width $d = 22.0 \times {10^{ - 5}}cm = 22 \times {10^{ - 7}}m$

Step2:
Substitute all the values in equation (1) we get,
$2 \times 550 \times {10^{ - 9}} = 22 \times {10^{ - 7}} \times \sin \theta$
$\Rightarrow \sin \theta = \dfrac{{2 \times 550 \times {{10}^{ - 9}}}}{{22 \times {{10}^{ - 7}}}}$
$\Rightarrow \sin \theta = \dfrac{1}{2}$
$\therefore \theta = {\sin ^{ - 1}}\left( {\dfrac{1}{2}} \right) = \dfrac{\pi }{6}$

Hence, the correct answer is option (D).