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Step-by-Step Solution
Step 1: Identify the Concept
The problem concerns the resolving power of the human eye, which is governed by the diffraction limit. According to the Rayleigh criterion, two points can just be resolved if the angular separation between them is at least
$1.22 \frac{\lambda}{d}$,
where
$\lambda$
is the wavelength of light and
$d$
is the diameter of the aperture (in this case, the pupil of the eye).
Step 2: Write Down the Known Quantities
Separation between two dots, $y = 1\,\text{mm} = 1 \times 10^{-3}\,\text{m}$
Pupil diameter, $d = 3\,\text{mm} = 3 \times 10^{-3}\,\text{m}$
Wavelength of light, $\lambda = 500\,\text{nm} = 5 \times 10^{-7}\,\text{m}$
Step 3: Apply the Rayleigh Criterion
If $D$ is the maximum distance at which the two dots can be resolved, the condition for resolution is:
$\frac{y}{D} \geq 1.22 \frac{\lambda}{d}$
Rearranging this gives:
$D \leq \frac{y \, d}{1.22 \, \lambda}$
Step 4: Substitute the Values
Substitute
$y = 1 \times 10^{-3}\,\text{m}$,
$d = 3 \times 10^{-3}\,\text{m}$,
and
$\lambda = 5 \times 10^{-7}\,\text{m}$
into the expression:
$D \leq \frac{(1 \times 10^{-3}) \times (3 \times 10^{-3})}{1.22 \times (5 \times 10^{-7})}$
$D \leq \frac{3 \times 10^{-6}}{6.1 \times 10^{-7}}$
$D \approx 4.9 \,\text{m} \approx 5 \,\text{m}$
Step 5: Conclude the Answer
Therefore, the maximum distance at which the two white dots can be resolved by the human eye (with a pupil diameter of $3\,\text{mm}$) is about
$\boxed{5\,\text{m}}$.
