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Question #43188

Difficulty: 8/10 | 🟢 Free

The Question

In a calm atmosphere with uniform pressure, the average translational kinetic energy of air molecules varies with height $y$ as $K(y) = K_0 (1 + \alpha y)$ , where $K_0$ and $\alpha$ are positive constants. At $y=0$ , the refractive index of air is $n_0$ . The refractive index $n$ and density $\rho$ of air are related by the Gladstone-Dale relation, $n - 1 = c\rho$ , where $c$ is a positive constant. A laser beam is emitted horizontally at a height $y = H$ . What is the radius of curvature $R$ of the laser beam's trajectory at the instant it is emitted?

Options

A.
$\frac{(n_0 + \alpha H)(1 + \alpha H)}{\alpha(n_0 - 1)}$
B.
$\frac{n_0 - \alpha H(n_0 - 1)}{\alpha(n_0 - 1)}$
C.
$\frac{n_0(1 + \alpha H)^2}{\alpha(n_0 - 1)}$
D.
$\frac{1 + \alpha H}{\alpha}$

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

Refractive Index Relativity|Temperature and Kinetic Energy|Binomial Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of Snell's Law in continuous media alongside thermodynamic density variations.

The primary trap is the premature use of the binomial approximation.

Students often approximate $n(y) \approx n_0 - \alpha(n_0 - 1)y$ before differentiation, which yields a constant gradient $-\alpha(n_0 - 1)$ and an incorrect local refractive index, leading directly to Option B.

Another trap is ignoring the '-1' in the Gladstone-Dale relation (incorrectly assuming $n \propto \rho$ ), which leads to Option D.

The problem forces students to maintain exact rational functions until the final evaluation.

Quick Facts About This Question

Question Information

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