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Question #43141

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The Question

A spectrometer is designed to resolve a spectral doublet of mean wavelength $\lambda = 500 \text{ nm}$ and wavelength separation $\Delta \lambda = 0.05 \text{ nm}$ . The instrument consists of a dispersive element with a rectangular limiting aperture of width $a = 2.0 \text{ cm}$ , and a focusing lens of focal length $f = 50.0 \text{ cm}$ that projects the spectrum onto a digital sensor. The geometric resolution limit of the sensor requires a minimum spatial separation of $\Delta y = 10.0 \text{ \mu m}$ between the centers of the spectral lines on the focal plane. Taking into account both the geometric resolution limit of the sensor and the fundamental diffraction limit of the aperture (Rayleigh criterion), what is the minimum angular dispersion $D = \frac{d\theta}{d\lambda}$ required from the dispersive element to successfully resolve the doublet?

Options

A.
$4.0 \times 10^{-4} \text{ rad/nm}$
B.
$5.0 \times 10^{-4} \text{ rad/nm}$
C.
$6.1 \times 10^{-4} \text{ rad/nm}$
D.
$9.0 \times 10^{-4} \text{ rad/nm}$

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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

Dispersion vs Deviation|Least Count and Resolution Limits|Small Angle Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to reconcile geometric optics with wave optics.

A common trap is to calculate the required dispersion based solely on the geometric constraints of the detector (pixel size and focal length), completely ignoring the fundamental diffraction limit imposed by the finite aperture of the dispersive element.

By setting the diffraction limit ( $\Delta \theta = \lambda/a$ ) higher than the geometric limit ( $\Delta \theta = \Delta y/f$ ), the problem forces a 'lethal collision' where geometric ray-tracing suggests the lines are resolved, but wave optics dictates they are still blurred together.

Students who fail to check the Rayleigh criterion will incorrectly choose the lower dispersion value (Option A).

Additionally, using the circular aperture formula ( $1.22 \lambda/a$ ) instead of the rectangular one leads to Option C.

Quick Facts About This Question

Question Information

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