Question

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

• A. $30\, \mu m$
• B. $1\, \mu m$
• C. $100 \,\mu m$
• D. $300 \,\mu m$

Answer 1

The Correct Option is A

To solve this question, we start by applying the Rayleigh criterion for the minimum angular resolution of a circular aperture, which is given by:

\(\theta = 1.22 \frac{\lambda}{D}\)

where:

• \(\theta\) is the angular resolution in radians.
• \(\lambda\) is the wavelength of light, given as \(500\ nm = 500 \times 10^{-9}\ m\).
• \(D\) is the diameter of the pupil.

The pupil diameter is twice the radius, so:

\(D = 2 \times 0.25\, cm = 0.5\, cm = 0.005\, m\)

Substitute the values into the formula:

\(\theta = 1.22 \times \frac{500 \times 10^{-9}\ m}{0.005\ m}\)

= 1.22 \times 10^{-4} \ rad

The minimum resolvable distance \(d\) at a comfortable viewing distance is given by:

d = \theta \times \text{viewing distance}

Substitute the values:

d = 1.22 \times 10^{-4} \times 25 \, cm = 1.22 \times 10^{-4} \times 0.25 \, m = 3.05 \times 10^{-5} \, m

Convert this to micrometers:

d = 3.05 \times 10^{-5} \, m = 30.5 \, \mu m

Rounding gives approximately 30 \, \mu m, which matches the correct answer.

Therefore, the minimum separation that the human eye can resolve at a wavelength of 500 \, nm is 30 \mu m.