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Question #93384

Difficulty: 5/10 | 🟢 Free

The Question

A transparent solid sphere of radius $R$ is placed in a vacuum. The refractive index of the material of the sphere varies radially as $n(r) = n_0 \left(\frac{R}{r}\right)^k$ , where $n_0 > 1$ and $0 < k < 1$ are constants, and $r$ is the distance from the center of the sphere. A light ray from the vacuum is incident on the surface of the sphere at an angle of incidence $\theta$ . What is the total polar angle $\Delta \Theta$ swept by the position vector of the light ray from its point of entry to its point of exit from the sphere?

Options

A.
$\frac{2}{1-k} \arccos\left(\frac{\sin\theta}{n_0}\right)$
B.
$\frac{1}{1-k} \arccos\left(\frac{\sin\theta}{n_0}\right)$
C.
$\frac{2}{1+k} \arccos\left(\frac{\sin\theta}{n_0}\right)$
D.
$\frac{2}{1-k} \arcsin\left(\frac{\sin\theta}{n_0}\right)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Spherical Surface Refraction|Medium Dependency|Integration

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the application of Fermat's Principle in a spherically symmetric medium with a radial refractive index gradient.

The primary trap is the misapplication of the standard paraxial Snell's Law across a continuously varying medium.

Students must recognize that the generalized invariant is Bouguer's formula ( $r n(r) \sin\phi = \text{constant}$ ).

A secondary trap lies in the geometric setup of the differential equation for the ray trajectory; students often incorrectly define the path element or fail to account for the symmetry of the trajectory (missing the factor of 2 for the inbound and outbound legs).

Finally, recognizing the turning point boundary condition ( $\sin\phi = 1$ ) is required to correctly evaluate the integral.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Spherical Surface Refraction
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free