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 than human eye can resolve at 500nm wavelength is :

Options

A. `30 mum`
B. `100 mum`
C. `300 mum`
D. `1 mum`

Answer

To solve the problem of determining the minimum separation between two objects that the human eye can resolve at a wavelength of 500 nm, we will use the formula derived from the Rayleigh criterion for resolution:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Radius of the pupil, \( r = 0.25 \, \text{cm} = 0.0025 \, \text{m} \)
- Diameter of the pupil, \( D = 2r = 0.005 \, \text{m} \)
- Comfortable viewing distance, \( L = 25 \, \text{cm} = 0.25 \, \text{m} \)
- Wavelength of light, \( \lambda = 500 \, \text{nm} = 500 \times 10^{-9} \, \text{m} \)

2. **Use the Rayleigh Criterion Formula:**
The formula for the minimum resolvable separation \( d \) is given by:
\[ d = \frac{1.22 \lambda L}{D} \]

3. **Substitute the Values into the Formula:**
\[ d = \frac{1.22 \times (500 \times 10^{-9}) \times 0.25}{0.005} \]

4. **Calculate the Numerator:**
- First, calculate \( 1.22 \times 500 \times 10^{-9} \):
\[ 1.22 \times 500 = 610 \times 10^{-9} \, \text{m} \]
- Now multiply by \( 0.25 \):
\[ 610 \times 10^{-9} \times 0.25 = 152.5 \times 10^{-9} \, \text{m} \]

5. **Calculate the Minimum Separation \( d \):**
- Now divide by \( D = 0.005 \):
\[ d = \frac{152.5 \times 10^{-9}}{0.005} = 30.5 \times 10^{-6} \, \text{m} = 30.5 \, \mu m \]

6. **Final Result:**
The minimum separation between two objects that the human eye can resolve at a wavelength of 500 nm is approximately:
\[ d \approx 30 \, \mu m \]

### Conclusion:
The correct answer is approximately **30 micrometers**.

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