Question

Assuming human pupil to have radius of $0.25\text{ cm}$ and comfortable viewing distance of $25\text{ cm}$ , the minimum separation between the two objects that human eye can resolve at $500\text{ nm}$ wavelength is nearly}

• A. $330\mu\text{ m}$
• B. $30\mu\text{ m}$
• C. $1\mu\text{ m}$
• D. $100\mu\text{ m}$

Hint:

For resolution problems: \[ s=L\left(1.22\frac{\lambda}{D}\right) \] Use diameter, not radius, in the diffraction formula.

Answer 1

The Correct Option is B

Concept:
For a circular aperture, angular resolution is given by Rayleigh criterion: \[ \theta = 1.22\frac{\lambda}{D} \] The minimum linear separation at distance \(L\) is: \[ s=L\theta \] ip

Step 1: Write the given values in SI units.
Pupil radius: \[ r=0.25\text{ cm}=2.5\times 10^{-3}\text{ m} \] So diameter: \[ D=2r=5\times10^{-3}\text{ m} \] Wavelength: \[ \lambda=500\text{ nm}=5\times10^{-7}\text{ m} \] Viewing distance: \[ L=25\text{ cm}=0.25\text{ m} \] ip

Step 2: Find the angular resolution.
\[ \theta=1.22\frac{5\times10^{-7}}{5\times10^{-3}} \] \[ \theta=1.22\times10^{-4}\text{ rad} \] ip

Step 3: Find minimum linear separation.
\[ s=L\theta=0.25\times 1.22\times10^{-4} \] \[ s=3.05\times10^{-5}\text{ m} \] \[ s\approx 30\times10^{-6}\text{ m}=30\mu\text{m} \] ip Hence, the correct answer is:
\[ \boxed{(B)\ 30\mu\text{m}} \]