Before You Begin

Question #43231

Difficulty: 8/10 | 🟢 Free

The Question

An astronomical telescope in normal adjustment is used to observe a binary star system emitting light of wavelength $\lambda = 500 \text{ nm}$ . The objective lens has a diameter $D = 12.2 \text{ cm}$ and a focal length $f_o = 150 \text{ cm}$ . The human eye has an effective pupil diameter $d_e = 2.44 \text{ mm}$ . To fully utilize the resolving power of the telescope, the magnified angular separation of the stars must be at least equal to the diffraction limit of the human eye. What is the maximum focal length of the eyepiece $f_e$ that satisfies this condition, and what is the minimum angular separation $\Delta \theta$ of the binary stars that can be resolved by this setup?

Options

A.
$f_e = 3.0 \text{ cm}, \Delta \theta = 5.0 \times 10^{-6} \text{ rad}$
B.
$f_e = 3.0 \text{ cm}, \Delta \theta = 2.5 \times 10^{-4} \text{ rad}$
C.
$f_e = 6.0 \text{ cm}, \Delta \theta = 5.0 \times 10^{-6} \text{ rad}$
D.
$f_e = 0.06 \text{ cm}, \Delta \theta = 2.5 \times 10^{-4} \text{ rad}$

Check Your Answer

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|Resolving Power Limits|Diffraction Width

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of geometric optics (telescope magnification) and physical optics (diffraction limits).

Students often memorize the resolving power of a telescope as a standalone formula without considering the observer's eye as the final optical instrument in the system.

The trap lies in assuming the eye can automatically resolve whatever the objective resolves.

By forcing students to equate the angular magnification to the ratio of the Rayleigh criteria of the eye and the objective, the problem highlights that 'useful magnification' is physically bounded by the exit pupil matching the eye's pupil.

A common mistake is calculating the resolution limit based solely on the eye's pupil (Option B) or misapplying the magnification formula.

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