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Question #43593

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The Question

Two point objects are placed at a distance $u = 10 \text{ cm}$ in front of a thin convex lens of focal length $f = 20 \text{ cm}$ and a large aperture. An observer's eye is placed at a distance $x = 30 \text{ cm}$ behind the lens to view the virtual images of the objects. The pupil of the eye has a diameter $d = 3.0 \text{ mm}$ and acts as the limiting aperture of the optical system. If the wavelength of light used is $\lambda = 600 \text{ nm}$ , what is the minimum separation $y$ between the two objects such that their images are just resolved by the eye? (Assume the paraxial approximation holds for the position of the images).

Options

A.
$24.4 \; \mu \mathrm{m}$
B.
$97.6 \; \mu \mathrm{m}$
C.
$61.0 \; \mu \mathrm{m}$
D.
$122.0 \; \mu \mathrm{m}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Angular Resolution Limit|Virtual Object Trap|Sign Convention Trap

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Rayleigh criterion in a compound optical system where the limiting aperture (the eye pupil) is not located at the objective lens.

A common trap is blindly applying the standard resolution formula $y = 1.22\lambda u / d$ (Option A), incorrectly assuming the eye pupil limits the entrance cone directly from the object distance $u$ .

Another trap is calculating the correct angular resolution of the eye but applying it to the direct object-to-eye distance $u+x$ (Option B), ignoring the lens entirely.

A third trap is finding the correct virtual image separation $y' = 122 \; \mu\mathrm{m}$ but forgetting to divide by the transverse magnification $m=2$ to find the actual object separation (Option D).

The correct physical approach requires evaluating the angular separation of the magnified virtual images at the actual position of the eye, effectively accounting for the system's entrance pupil.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Angular Resolution Limit
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free