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Question #43207

Difficulty: 8/10 | 🟢 Free

The Question

A thin biconvex lens is manufactured from a transparent material whose refractive index varies radially due to cohesive force gradients during cooling. The refractive index profile is given by $\mu(r) = \mu_0 - \alpha r^2$ , where $r$ is the perpendicular distance from the optical axis. The lens has a central thickness $t_0 = 4 \text{ mm}$ and symmetric radii of curvature $R_1 = R_2 = 20 \text{ cm}$ . Given $\mu_0 = 1.50$ and $\alpha = 625 \text{ m}^{-2}$ , calculate the effective focal length $f$ of this lens for paraxial rays. Assume the lens is placed in air ( $\mu_{\text{air}} = 1$ ).

Options

A.
$10 \text{ cm}$
B.
$20 \text{ cm}$
C.
$13.3 \text{ cm}$
D.
$\infty$

Check Your Answer

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

Lens Maker’s Formula Constraints|Refractive Index Relativity|Molecular Forces of Cohesion

Why This Question Is Tricky

Common Mistakes

The question tests the student's fundamental understanding of how a lens focuses light via Optical Path Length (OPL) differences, rather than rote application of the Lens Maker's formula.

The common trap is assuming the standard Lens Maker's formula is universally valid and simply plugging in the central refractive index $\mu_0$ , completely ignoring the phase delay introduced by the radial gradient (the Wood lens effect).

Students fail to recognize that a varying refractive index $\mu(r)$ contributes an additional parabolic phase shift, effectively acting as an independent thin lens in series.

By forcing them to expand the OPL to the $r^2$ term, the problem filters out those who memorize formulas from those who understand Fermat's Principle and paraxial phase transformations.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Lens Maker’s Formula Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free