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Question #43592

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The Question

A circular aperture of a telescope is formed by a smart material membrane. Due to a heavily damped boundary condition, the effective diameter of the aperture decreases exponentially with time as $D(t) = D_0 e^{-bt}$ , where $D_0$ is the initial diameter in meters ( $\text{m}$ ) at $t=0$ and $b$ is a positive damping constant in inverse seconds ( $\text{s}^{-1}$ ). The telescope is used to resolve two distant point sources emitting light of wavelength $\lambda$ (in $\text{m}$ ). Assuming the standard Rayleigh criterion applies instantaneously, what is the time-averaged angular resolution limit $\langle \theta \rangle$ of the telescope over the time interval from $t=0$ to $t=1/b$ ?

Options

A.
$\frac{1.22 \lambda}{D_0} (e - 1)$
B.
$\frac{1.22 \lambda}{D_0} \left( \frac{e}{e - 1} \right)$
C.
$\frac{1.22 \lambda}{D_0} \left( 1 - \frac{1}{e} \right)$
D.
$\frac{1.22 \lambda}{D_0} e$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Angular Resolution Limit|Boundary Condition Reflection|Damped Oscillation Equation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply time-averaging to a non-linear, time-dependent physical limit.

The primary trap lies in the mathematical order of operations regarding averages.

Students often incorrectly assume that the average of a function's reciprocal is the reciprocal of the function's average (i.e., $\langle 1/D \rangle = 1/\langle D \rangle$ ).

If a student calculates the time-averaged diameter $\langle D \rangle$ first and then substitutes it into the Rayleigh criterion, they will arrive at the incorrect Option B.

The correct physical approach requires recognizing that the resolution limit itself is the quantity being averaged, necessitating the integration of $e^{bt}$ rather than $e^{-bt}$ .

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