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Question #43191

Difficulty: 8/10 | 🟢 Free

The Question

A light ray travels through a transparent medium with a spherically symmetric refractive index given by $n(r) = \frac{n_0}{1 + (r/a)^2}$ , where $n_0$ is a dimensionless constant, $r$ is the radial distance from the origin, and $a = 2.0 \text{ m}$ . At a specific point $P$ located at a radial distance $r = 2.0 \text{ m}$ , the light ray's velocity vector makes an angle of $30^\circ$ with the outward radial vector. What is the radius of curvature of the light ray's trajectory at point $P$ ?

Options

A.
$2.0 \text{ m}$
B.
$4.0 \text{ m}$
C.
$\frac{4.0}{\sqrt{3}} \text{ m}$
D.
$8.0 \text{ m}$

Check Your Answer

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

Snell’s Law Vector Form|Huygens Principle Wavefronts|Gradient Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply the vector form of Snell's Law (the eikonal equation) to find the local geometric properties of a ray in a gradient-index medium.

The primary trap is 'math bloat': students often attempt to use Bouguer's formula ( $n r \sin\theta = \text{constant}$ ) to find the full trajectory equation $r(\theta)$ , which leads to an intractable differential equation for the radius of curvature.

The conceptual leap required is recognizing that the vector eikonal equation directly relates the local bending (curvature) of the wavefront normal to the component of the refractive index gradient perpendicular to the ray.

Secondary traps include ignoring the cross-product (using cosine instead of sine) or forgetting to divide the gradient magnitude by the local refractive index $n(r)$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Snell’s Law Vector Form
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free