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Question #42444

Difficulty: 8/10 | 🟢 Free

The Question

Two identical small spheres, each of mass $m$ and charge $q$ , are suspended from a common point by two insulating, inextensible strings of length $L$ . The system is placed inside a closed container completely filled with a dielectric liquid of dielectric constant $K$ . The density of the material of the spheres is $\rho$ and the density of the liquid is $\rho/2$ . The container is given a constant horizontal acceleration $a$ . Assuming the angle between the strings remains very small, what is the steady-state equilibrium separation $x$ between the two spheres?

Options

A.
$\left( \frac{q^2 L}{\pi\epsilon_0 K m \sqrt{g^2 + a^2}} \right)^{1/3}$
B.
$\left( \frac{q^2 L}{2\pi\epsilon_0 K m \sqrt{a^2 + g^2/4}} \right)^{1/3}$
C.
$\left( \frac{q^2 L}{2\pi\epsilon_0 K m \sqrt{g^2 + a^2}} \right)^{1/3}$
D.
$\left( \frac{q^2 L}{\pi\epsilon_0 K m g} \right)^{1/3}$

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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

Newton’s Third Law Pairing|Coulomb's Law Vector Form|Accelerated Frame Hydrostatics

Why This Question Is Tricky

Common Mistakes

This question tests the understanding of buoyancy and equilibrium in non-inertial reference frames.

A common lethal mistake is assuming that the buoyant force remains strictly vertical ( $V\rho g$ ) even when the container accelerates.

In reality, the acceleration of the container alters the fluid's pressure gradient, causing the buoyant force to align anti-parallel to the effective gravity vector ( $\vec{g}_{\text{eff}} = \vec{g} - \vec{a}$ ).

Failing to recognize this leads to an asymmetric force distribution and an incorrect effective weight magnitude, specifically trapping students into calculating a skewed vector sum of a horizontal pseudo-force and a vertical buoyancy (Option B).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Newton’s Third Law Pairing
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free