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Question #43222

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The Question

A microscope tube of mechanical length $L$ is completely filled with a compressible liquid. The liquid has a mean refractive index $n_0$ , mean density $\rho_0$ , bulk modulus $B$ , and dispersive power $\omega_d$ . The tube is rotated in a horizontal plane with a constant angular velocity $\omega$ about a vertical axis passing through the objective lens (at $r=0$ ). Assuming the Gladstone-Dale relation ( $\frac{n-1}{\rho} = \text{constant}$ ) holds for all wavelengths and the density change is small ( $\Delta \rho \ll \rho_0$ ), what is the change in the longitudinal chromatic aberration (defined here as the difference in the optical path lengths for violet and red light) of the liquid column due to the rotation?

Options

A.
$\frac{\omega_d (n_0 - 1) \rho_0 \omega^2 L^3}{6B}$
B.
$\frac{\omega_d (n_0 - 1) \rho_0 \omega^2 L^3}{2B}$
C.
$\frac{\omega_d n_0 \rho_0 \omega^2 L^3}{6B}$
D.
$\frac{\omega_d (n_0 - 1) \rho_0 \omega^2 L^3}{3B}$

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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

Chromatic Aberration|Microscope Tube Length|Accelerated Frame Hydrostatics

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of fluid statics in non-inertial frames with wave optics principles.

The primary trap is failing to integrate the optical path length over the varying refractive index field.

Students often calculate the maximum pressure at the end of the tube ( $r=L$ ), find the corresponding maximum change in refractive index $\Delta n_{max}$ , and simply multiply by $L$ to find the change in optical path length ( $\Delta \text{OPL} = \Delta n_{max} L$ ), leading to Option B.

Another trap is misapplying the Gladstone-Dale relation as $n \propto \rho$ instead of $(n-1) \propto \rho$ , leading to Option C.

Finally, forgetting the factor of $1/2$ in the integration of the pressure gradient leads to Option D.

Quick Facts About This Question

Question Information

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