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Question #42772

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor consists of a fixed horizontal lower metal plate and a movable upper solid metal block of mass $M$ and base area $A$ . The upper block is suspended by an ideal spring of stiffness $k_s$ such that its lower face is parallel to the fixed plate. The entire setup is placed in a large tank filled with a dielectric liquid of density $ho$ and dielectric constant $K$ . The liquid level is maintained at a constant height $H$ above the fixed lower plate. In equilibrium, the lower face of the suspended block is at a distance $d$ ( $d < H$ ) from the fixed plate, and the block is partially submerged. Assume the block is sufficiently tall such that its top face always remains above the liquid surface. The capacitor is connected to a constant DC voltage source $V$ . Neglecting fringing effects, what is the critical voltage $V_c$ beyond which the upper block's equilibrium becomes unstable and snaps to the lower plate?

Options

A.
$\sqrt{\frac{(k_s + \rho A g) d^3}{\epsilon_0 K A}}$
B.
$\sqrt{\frac{k_s d^3}{\epsilon_0 K A}}$
C.
$\sqrt{\frac{(k_s - \rho A g) d^3}{\epsilon_0 K A}}$
D.
$\sqrt{\frac{2(k_s + \rho A g) d^3}{\epsilon_0 K A}}$

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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

Capacitance Independence|Dimensional Independence vs. Scalar/Vector Nature|Archimedes' Buoyancy Dynamics|Approximation

Why This Question Is Tricky

Common Mistakes

This question tests a severe compartmentalization trap in JEE Advanced physics.

Students typically memorize the electrostatic instability (pull-in voltage) formula $V_c \propto \sqrt{k_s d^3}$ and apply it blindly.

They recognize Archimedes' buoyancy as a force that shifts the static equilibrium position, but fail to recognize that because the buoyant force is depth-dependent ( $F_B = \rho A (H-d) g$ ), it has a non-zero spatial derivative.

Thus, the fluid dynamically couples to the system's stiffness, acting as a parallel mechanical spring of constant $k_B = \rho A g$ .

Ignoring the position-dependent nature of buoyancy leads to missing the $\rho A g$ term in the effective stiffness, resulting in the classic trap option B.

Quick Facts About This Question

Question Information

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