Assuming that the human pupil has a radius of $0.\mathrm{25 }\mathrm{cm}$ and a comfortable viewing distance of $25 \mathrm{cm}$. The minimum separation between two point objects that the human eye can resolve for the light of wavelength $500 \mathrm{nm}$ is

• A

  $300\mathrm{ }\mathrm{\mu m}$
• B

  $1\mathrm{ }\mathrm{\mu m}$
• Correct answer
  C

  $30\mathrm{ }\mathrm{\mu m}$
• D

  $100\mathrm{ }\mathrm{\mu m}$

Solution

By Fraunhofer diffraction through a circular aperture, $\mathrm{\theta }=\frac{1.22\mathrm{\lambda }}{d}$

$d=$ diameter of pupil $=2\times \text{0.25}=\text{0.5}\mathrm{ }\mathrm{cm}$
$\mathrm{\lambda }=500\mathrm{ }\mathrm{nm}$. The first dark ring is formed by the light diffracted from the hole at an angle $\mathrm{\theta }$ with the axis. Viewing distance, $D=25 \mathrm{cm}$$\therefore$ The minimum separation between $2$ objects $=D\theta$$=\frac{25\times {10}^{-2}\times 1.22\times 500\times {10}^{-9}}{5\times {10}^{-3}} \mathrm{m}$
$D\theta =30\times {10}^{-6} \mathrm{m}$$\Rightarrow D\theta =30\mathrm{ }\mathrm{\mu m}$.

Hence, the correct answer is option: 'C'.