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Question #42023

Difficulty: 8/10 | 🟢 Free

The Question

A thin converging lens of focal length $f$ (in air) is placed at the origin $x = 0$ . The region $x < 0$ is filled with air (refractive index $\mu = 1$ ). The region $x > 0$ is filled with a transparent medium whose refractive index varies along the x-axis as $\mu(x) = 1 + \alpha x$ , where $\alpha > 0$ is a constant. Let $F_1$ be the first focal point (the object position on the principal axis for which the image is formed at infinity) and $F_2$ be the second focal point (the image position for an object at infinity). If a point object is placed at a distance $x_1$ to the left of $F_1$ , its image is formed at a distance $x_2$ to the right of $F_2$ . Which of the following represents the modified Newton's lens equation for this system?

Options

A.
$x_1 \ln\left(1 + \alpha x_2 e^{-\alpha f}\right) = \alpha f^2$
B.
$x_1 x_2 = f^2 e^{\alpha f}$
C.
$x_1 \ln(1 + \alpha x_2) = \alpha f^2$
D.
$x_1 x_2 = \frac{f^2}{\alpha} \ln(1 + \alpha f)$

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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

The Directional Sign Convention|Newton’s Form of Lens Equation|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental origin of Newton's lens equation ( $x_1 x_2 = f^2$ ) by placing the system in a medium with a variable refractive index gradient.

Students typically memorize the algebraic Newton's formula and attempt to heuristically modify the focal length (leading to incorrect options like B or D).

However, a continuous refractive index gradient requires integrating the optical path via Snell's law for stratified media.

This integration reveals that the focal length is a local function of the gradient, transforming the standard algebraic relationship into a logarithmic one, while still preserving the core directional sign convention relative to the focal points.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Straight Line - Class 11 Part 1
Concept	The Directional Sign Convention
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free