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Question #43139

Difficulty: 8/10 | 🟢 Free

The Question

A transparent optical slab of thickness $d$ occupies the region $0 \le y \le d$ . The slab contains a uniform negative volume charge density $-\rho_0$ (where $\rho_0 > 0$ ). The boundary surfaces at $y=0$ and $y=d$ are maintained at a constant positive electrostatic potential $V_0$ . The refractive index $n(y)$ of the slab material is directly proportional to the local electrostatic potential $V(y)$ , such that $n(y) = \alpha V(y)$ where $\alpha$ is a positive constant. A light ray enters the slab at the origin $(0,0)$ from a medium of refractive index $n_0 = \alpha V_0$ , making an angle $\theta_0$ with the $y$ -axis. Which of the following expressions gives the minimum value of $\rho_0$ required so that the light ray does not cross the mid-plane ( $y=d/2$ ) of the slab?

Options

A.
$\frac{8\epsilon_0 V_0}{d^2} (1 - \sin\theta_0)$
B.
$\frac{4\epsilon_0 V_0}{d^2} (1 - \sin\theta_0)$
C.
$\frac{8\epsilon_0 V_0}{d^2} (1 - \cos\theta_0)$
D.
$\frac{2\epsilon_0 V_0}{d^2} (1 - \sin\theta_0)$

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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

Prism Minimum Deviation|Continuous Charge Distribution|Calculus

Why This Question Is Tricky

Common Mistakes

This problem tests the synthesis of electrostatics (Poisson's equation for continuous charge distributions) and optics (Fermat's principle in a gradient-index medium).

The continuous variation of the refractive index mimics the effective potential in a gravitational field (time dilation analogy), where light rays bend towards regions of higher refractive index.

A common trap is misapplying Gauss's law to find the potential difference, incorrectly assuming a uniform electric field or forgetting the spatial integration constants, which leads to an incorrect factor of 4 instead of 8 in the potential well depth (Option B).

Furthermore, students often fail to recognize that in a continuously stratified medium, Snell's law generalizes to $n(y)\sin\theta(y) = \text{constant}$ , and instead attempt to use discrete boundary formulas or confuse the normal angle with the grazing angle (Option C).

Quick Facts About This Question

Question Information

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