Question

A parallel beam of light of wavelength \[6000A{}^\circ \] gets diffracted by a single slit of width 0.3mm. The angular position of first minima of diffracted light is-
$\begin{align}
  & \text{A}\text{. 2 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{-3}}}\text{rad} \\
 & \text{B}\text{. 3 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{-3}}}\text{rad} \\
 & \text{C}\text{. 1}\text{.8 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{-3}}}\text{rad} \\
 & \text{D}\text{. 6 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{-3}}}\text{rad} \\
\end{align}$

Answer 1

Hint: Diffraction is the phenomenon of bending of light around the corners of sharp objects. The angular position is the angle subtended by either maxima or minima measured from the slit. The general formula for a minima in a single slit diffraction is given by the expression $a\sin \theta =n\lambda $ in which $\lambda $ is the wavelength of light. By solving for $\theta $ in this equation, we will get the angular position.

Complete step-by-step answer:
Given,
Wavelength of beam of light, \[\lambda =6000\text{ }A{}^\circ \] .
\[\begin{align}
  & 1\text{ }A{}^\circ =\text{ }{{10}^{-10}}m \\
 & \Rightarrow \lambda =\text{6000 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{-10}}}\text{m} \\
\end{align}\]
Width of single slit, a = 0.3 mm
\[\begin{align}
  & 1\text{ }mm={{10}^{-3}}m \\
 & a=0.3\text{ }x\text{ }{{10}^{-3}}m \\
\end{align}\]
Now, we know that for the first minima the value of n will be 1.
Therefore, using the formula for single slit is given by,
$a\sin \theta =n\lambda $
Rearranging the above formula,
$\sin \theta =n\dfrac{\lambda }{a}$
Here, in diffraction the \[\sin \theta \simeq \theta \] as $\theta $ is very small. Therefore,
$\theta =n\dfrac{\lambda }{a}$
Putting the given values from question in above formula, we get
\[\theta =1\times \dfrac{6000\times {{10}^{-10}}}{0.3\times {{10}^{-3}}}=2\times {{10}^{-3}}\text{rad}\].
Hence, the angular position of the first minima is found to be \[2\times {{10}^{-3}}\text{rad}\].

So, the correct answer is “Option A”.

Additional Information: This phenomenon of diffraction of light was first carefully observed by Francesco Maria Grimaldi which means “break into pieces”. This can be observed in daily life examples like bending of light around the edges of doors, splitting of colors on the surface of CD and DVD etc. Also, this phenomenon gave rise to different other instruments like spectrometers, diffraction grating etc.

Note: Students should keep in mind that the bending of light is negligible to notice therefore we assumed θ to be very small and also the answer which is given by the sine function will be in radians instead of degrees.