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

Step 1: Understanding the Concept:
The resolution of an optical instrument is limited by diffraction. The minimum angular separation is given by Rayleigh's criterion.
Step 2: Key Formula or Approach:
Angular resolution: $\Delta \theta = \frac{1.22 \lambda}{D}$.
Linear resolution: $y = L \cdot \Delta \theta$.
Step 3: Detailed Explanation:
Radius $= 0.25\text{ cm} \implies$ Diameter $D = 0.5\text{ cm} = 5 \times 10^{-3}\text{ m}$.
Distance $L = 25\text{ cm} = 0.25\text{ m}$.
Wavelength $\lambda = 500 \times 10^{-9}\text{ m}$.
\[ \Delta \theta = \frac{1.22 \times 500 \times 10^{-9}}{5 \times 10^{-3}} = 1.22 \times 10^{-4}\text{ rad} \]
\[ y = 0.25 \times 1.22 \times 10^{-4} = 0.305 \times 10^{-4}\text{ m} \]
\[ y = 30.5 \times 10^{-6}\text{ m} \approx 30\mu\text{m} \]
Step 4: Final Answer:
The minimum separation is nearly $30\mu\text{m}$.