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Question #43559

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The Question

An astronomical telescope in normal adjustment is used to observe two distant point sources emitting light of wavelength $\lambda = 500 \text{ nm}$ . The angular separation between the sources is $\theta_s = 1.22 \times 10^{-6} \text{ rad}$ . The human eye observing through the telescope has a pupil diameter of $d_e = 2.5 \text{ mm}$ and its angular resolution is strictly diffraction-limited by its pupil. The objective lens of the telescope has a focal length $f_o = 200 \text{ cm}$ . To just resolve the two sources, what is the minimum required diameter $D$ of the objective lens and the maximum allowed focal length $f_e$ of the eyepiece?

Options

A.
$D = 50 \text{ cm}, f_e = 1.0 \text{ cm}$
B.
$D = 100 \text{ cm}, f_e = 0.5 \text{ cm}$
C.
$D = 50 \text{ cm}, f_e = 1.22 \text{ cm}$
D.
$D = 25 \text{ cm}, f_e = 2.0 \text{ cm}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Telescope Normal Adjustment|Angular Resolution Limit|Diffraction vs Interference

Why This Question Is Tricky

Common Mistakes

This question tests the integration of wave optics (diffraction limits) with geometrical optics (telescope magnification).

A common trap is misapplying the Rayleigh criterion by using the radius instead of the diameter ( $\theta = 1.22\lambda/R$ ), leading to exactly double the required aperture and half the eyepiece focal length (Option B).

Another lethal mistake is forgetting the $1.22$ factor for the eye's resolution while keeping it for the objective, which mathematically yields an erroneous $f_e = 1.22 \text{ cm}$ (Option C).

Students must recognize that the minimum useful magnification is fundamentally the ratio of the objective diameter to the eye pupil diameter ( $M = D/d_e$ ) when both are diffraction-limited.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Telescope Normal Adjustment
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free