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Question #42944

Difficulty: 8/10 | 🟢 Free

The Question

A large rectangular tank is completely filled with an uncharged, non-viscous liquid of density $\rho_0$ . A rigid, smooth, straight insulating rod of length $L$ is fixed inside the tank. The tank is subjected to a constant horizontal acceleration $a = g\sqrt{3}$ to the right. A uniform horizontal electric field of magnitude $E$ is established in the space, pointing to the right. Ignore any dielectric effects of the liquid.

A small bead of mass $m$ , volume $V$ , charge $q > 0$ , and density $2\rho_0$ is threaded on the rod. The rod is oriented such that it lies entirely within the vertical plane of acceleration and is parallel to the isobaric surfaces of the liquid, with its higher end on the left.

The bead is released from rest from the higher end of the rod. Given that $qE = \sqrt{3}mg$ (where $g$ is the acceleration due to gravity), what is the kinetic energy of the bead when it reaches the lower end of the rod?

Options

A.
$\frac{\sqrt{3}}{2}mgL$
B.
$\sqrt{3}mgL$
C.
$\frac{\sqrt{3}}{4}mgL$
D.
$\frac{3\sqrt{3}}{4}mgL$

Check Your Answer

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

Potential Gradient Direction|Conservative Field Path Independence|Accelerated Frame Hydrostatics

Why This Question Is Tricky

Common Mistakes

This question tests the superposition of inertial, gravitational, and electrostatic fields, specifically focusing on the boundary condition mismatch in an accelerated fluid.

Students often fall into the trap of either ignoring the pseudo force on the bead itself (focusing only on the fluid's tilt) or calculating buoyancy using standard gravity ( $V\rho_0 g$ ) instead of the effective gravity field ( $\vec{g}_{eff} = \vec{g} - \vec{a}$ ).

The correct approach requires treating the potential gradient as a vector sum and projecting the true net force onto the modified equipotential (isobaric) surface of the fluid.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Potential Gradient Direction
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free