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Question #42234

Difficulty: 8/10 | 🟢 Free

The Question

A solid dielectric block of mass $M$ , height $L$ , base area $A$ , and dielectric constant $K$ is partially submerged in a non-conducting, unpolarized liquid of density $\rho$ . The liquid surface lies at $y=0$ , with the $y$ -axis pointing vertically downwards. An external non-uniform electric field $\vec{E} = E_0 \sqrt{\frac{y}{L}} \hat{j}$ exists in the region $y \ge 0$ , while $\vec{E} = \vec{0}$ for $y < 0$ . If the block is in vertical equilibrium, find its submerged depth $x$ .

Options

A.
$\frac{2 M g L}{2 \rho A g L - \epsilon_0 (K-1) A E_0^2}$
B.
$\frac{2 M g L}{2 \rho A g L + \epsilon_0 (K-1) A E_0^2}$
C.
$\frac{M g L}{\rho A g L - \epsilon_0 (K-1) A E_0^2}$
D.
$\frac{M g L}{\rho A g L + \epsilon_0 (K-1) A E_0^2}$

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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

Archimedes' Buoyancy Dynamics|Dielectric Constant (K) Shift|Calculus: Integration

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of Archimedes' principle with electrostriction (polarization force) in a non-uniform electric field.

Students often mistakenly assume that the electric force only acts on the surfaces or fail to correctly integrate the force density $\vec{f} = (\vec{P} \cdot \nabla)\vec{E} = \frac{1}{2}\epsilon_0(K-1)\nabla(E^2)$ .

A common trap is missing the factor of $1/2$ from the gradient of $E^2$ , or incorrectly assigning the direction of the polarization force.

Since the field intensity increases downwards, the dielectric is pulled downwards, effectively increasing its apparent weight and requiring a greater submerged depth to balance the forces.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Fluids - Class 11 Part 2
Concept	Archimedes' Buoyancy Dynamics
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free