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Step-by-Step Solution
Step 1: Understand the Physical Situation
The human eye can be approximated as a circular aperture (the pupil) with a given radius. When two objects are at a certain minimum separation, the diffraction of light passing through the pupil limits the eye’s ability to distinguish them as separate. This limit is often estimated using the Rayleigh criterion for resolution.
Step 2: Identify the Relevant Parameters
Pupil radius, $r = 0.25\,\text{cm}$.
Comfortable viewing distance, $L = 25\,\text{cm}$.
Wavelength of light, $\lambda = 500\,\text{nm} = 500\times 10^{-9}\,\text{m}$.
We also note that $\sin\theta$ can be approximated for small angles by the ratio of the pupil radius to the viewing distance, i.e. $\sin \theta = \frac{r}{L}$.
Step 3: Apply the Approximation for Small Angles
For small angles,
$$
\sin\theta \approx \theta = \frac{r}{L}.
$$
Substituting the given values:
$$
\sin\theta = \frac{0.25\,\text{cm}}{25\,\text{cm}} = \frac{1}{100}.
$$
Step 4: Use the Rayleigh Criterion Formula
The minimum resolvable angle for a circular aperture (assuming the Rayleigh criterion) is often given by:
$$
\theta_\mathrm{min} = 1.22 \frac{\lambda}{D},
$$
where $D$ is the diameter of the aperture. However, another form uses aperture radius $\mu$ and $\sin \theta$ directly. According to the solution reference:
$$
d = \frac{1.22\,\lambda}{2\mu \,\sin\theta}.
$$
Here,
$$
\mu = r = 0.25\,\text{cm}, \quad \sin\theta = \frac{1}{100}.
$$
Step 5: Substitute the Numerical Values
• Convert the pupil radius to meters if needed, but as long as units are consistent in ratio form, we can keep in cm.
• Convert the wavelength to cm if needed or keep track in standard SI units carefully.
The key ratio form given in the provided solution yields:
$$
d = \frac{1.22\times 500\times 10^{-9}\,\text{m}}{2 \times 0.25\times 10^{-2}\,\text{m}\,\times \frac{1}{100}}
\approx 30 \,\mu\text{m}.
$$
Step 6: Conclusion
Hence, the minimum separation that the human eye can resolve under these conditions is about $30\,\mu\text{m}$. This corresponds to Option 4 in the given choices.
Reference Image from Provided Solution
