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Question #43182

Difficulty: 8/10 | 🟢 Free

The Question

A small object is placed at the bottom of a transparent liquid of depth $H$ (in $\text{m}$ ). The refractive index of the liquid varies with depth $y$ (in $\text{m}$ ) from the surface ( $y=0$ ) as $n(y) = \sqrt{n_0^2 + \alpha y}$ , where $n_0$ is a dimensionless positive constant and $\alpha$ is a positive constant in $\text{m}^{-1}$ . An observer in air ( $n_{air} = 1$ ) looks at the object such that the emergent ray entering their eye makes an angle $\theta_0$ with the normal to the surface. The apparent depth $H'$ is defined as the depth at which the backward-extended emergent ray intersects the vertical line passing through the actual object. Which of the following expressions gives $H'$ ?

Options

A.
$\frac{2\cos\theta_0}{\alpha} \left( \sqrt{n_0^2 - \sin^2\theta_0 + \alpha H} - \sqrt{n_0^2 - \sin^2\theta_0} \right)$
B.
$\frac{H \cos\theta_0}{\sqrt{n_0^2 - \sin^2\theta_0}}$
C.
$\frac{\cos\theta_0}{\alpha} \left( \sqrt{n_0^2 - \sin^2\theta_0 + \alpha H} - \sqrt{n_0^2 - \sin^2\theta_0} \right)$
D.
$\frac{H \cos\theta_0}{\sqrt{n_0^2 + \alpha H - \sin^2\theta_0}}$

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

Medium Dependency|Apparent Depth/Shift|Vector Decomposition Ambiguity|Approximations

Why This Question Is Tricky

Common Mistakes

Students often incorrectly assume that apparent depth formulas derived for constant refractive indices can be applied to gradient-index media by simply substituting the boundary refractive index values.

This trap tests the misapplication of Snell's Law as a linear, single-interface phenomenon rather than a continuous path integral.

By assuming the ray travels in a straight line and using either the surface index $n_0$ or the bottom index $\sqrt{n_0^2+\alpha H}$ , students fail to account for the continuous curvature of the ray path.

The correct approach requires applying the generalized Snell's Law to find the local slope of the ray, integrating this slope over the depth to find the true lateral displacement, and then using geometry to find the apparent depth.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Medium Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free