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Question #42695

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor with plates of length $L$ , width $b$ , and separation $d$ is fixed horizontally. The lower plate is rough with a coefficient of kinetic friction $\mu$ . A dielectric slab of mass $m$ , dielectric constant $\kappa$ , and dimensions $L \times b \times d$ is placed such that it can slide between the plates. The slab is connected to a fixed wall by a horizontal ideal spring of spring constant $k$ . Initially, the spring is relaxed and the slab is just at the edge of the capacitor ( $x=0$ ). A constant voltage $V$ is applied across the plates. Assume the normal force exerted by the lower plate on the slab is $mg$ , the electrostatic force is sufficient to pull the slab in, and the maximum displacement $x_{max} < L$ . What is the total work done by the battery from the instant the slab is released until it reaches its maximum displacement?

Options

A.
$\frac{\epsilon_0 b (\kappa - 1) V^2}{k d} \left( \frac{\epsilon_0 b (\kappa - 1) V^2}{d} - 2\mu m g \right)$
B.
$\frac{\epsilon_0 b (\kappa - 1) V^2}{2k d} \left( \frac{\epsilon_0 b (\kappa - 1) V^2}{d} - 2\mu m g \right)$
C.
$\frac{\epsilon_0 b (\kappa - 1) V^2}{4k d} \left( \frac{\epsilon_0 b (\kappa - 1) V^2}{d} - 2\mu m g \right)$
D.
$\frac{\epsilon_0 b (\kappa - 1) V^2}{k d} \left( \frac{\epsilon_0 b (\kappa - 1) V^2}{d} + 2\mu m g \right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Dielectric Slab Displacement|Kinetic Friction Transition|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of the Work-Energy Theorem in an electromechanical system.

A common trap is equating the electrostatic force to the restoring forces ( $F_e = kx + \mu mg$ ) to find the maximum displacement, which actually yields the equilibrium position (where velocity is maximum), missing a factor of 2.

Another critical trap is misunderstanding the energy supplied by the battery; students often incorrectly assume $W_{batt} = \Delta U_{cap}$ instead of $W_{batt} = 2\Delta U_{cap}$ .

Both of these misconceptions independently lead to the exact same incorrect distractor (Option B), making it a highly effective trap for students who rely on partial intuition rather than a strict energy balance.

Quick Facts About This Question

Question Information

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