Question

A screen is placed $50cm$from a single slit which is illuminated with light of wavelength$6000{{A}^{0}}$. If the distance between the first and third minima in the diffraction pattern is$3.0mm$, the width of the slit is
(A) $0.1mm$
(B) $0.2mm$
(C) $0.3mm$
(D) $0.4mm$

Answer 1

Hint : In this question, we have to find the width of the slit. We have given the distance between the first and third minima in the diffraction pattern in the single-slit diffraction and the wavelength of the light so we use this formula to find out the width of the slit is given ${{x}_{2}}-{{x}_{1}}=(\dfrac{2\lambda D}{d})$.

Complete step by step answer:
Given: We have given the following terms in this question are
 ${{x}_{2}}-{{x}_{1}}=3mm$ Here we convert $mm$ into $m$ then we get,
 $\Delta x=3\times {{10}^{-3}}m$
 $D=50cm$ Here we convert $cm$ into $m$ then we get,
 $D=0.5m$
 $\lambda =6000{{A}^{0}}$ Here we convert ${{A}^{0}}$into $m$then we get,
 $\lambda =6\times {{10}^{-10}}m$
Here is the formula for single slit diffraction is given as,
 ${{x}_{2}}-{{x}_{1}}=\dfrac{(3\lambda -\lambda )D}{d}$
We simplify this formula as per the given conditions,
$\Delta x=\dfrac{2\lambda D}{d}$
Here,
 $\Delta x$= The distance between the first and third minima in the diffraction pattern.
 $D$= The distance of a screen from a single slit.
 $\lambda $= Wavelength of light.
 $d$= Width of the slit.
Now we substitute the given values in the formula,
 $d=\dfrac{2\lambda D}{\Delta x}$
After substituting the values equation is written as,
 $d=\dfrac{2\times 6000\times {{10}^{-10}}\times 0.5}{3\times {{10}^{-3}}}$
After simplifying this we get,
 $d=2\times {{10}^{-4}}m$
We have to calculate the width of the slit $mm$.
 $d=0.2mm$

So option B is the correct answer.

Note: To solve this question we have to study the diffraction of light and the single slit diffraction experiment. In this experiment, we can observe the bending phenomenon of light or diffraction that causes light from a coherent source to interfere with itself and produce a distinctive pattern on the screen called the diffraction pattern. Using this experiment we can solve questions based on the single slit experiment easily.