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Question #42820

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor with plates of length $L$ , width $w$ , and separation $d$ is fixed horizontally. It is connected to a constant voltage source $V$ . A dielectric slab of mass $M$ , dielectric constant $K$ , and dimensions $L \times w \times d$ is initially placed such that its front edge just enters the capacitor ( $x=0$ ). The slab is connected to a massless, inextensible string that passes over a rough solid cylindrical pulley of mass $m_p$ and radius $R$ . The other end of the string is attached to a hanging block of mass $m$ . The system is released from rest. Assuming no friction between the slab and the capacitor plates, which of the following expressions gives the speed $v$ of the hanging block after the slab has been inserted a distance $x$ into the capacitor?

Options

A.
$\sqrt{ \frac{2x \left( mg + \frac{\epsilon_0 w (K-1) V^2}{2d} \right)}{M + m + \frac{m_p}{2}} }$
B.
$\sqrt{ \frac{2x \left( mg - \frac{\epsilon_0 w (K-1) V^2}{2d} \right)}{M + m + \frac{m_p}{2}} }$
C.
$\sqrt{ \frac{2x \left( mg + \frac{\epsilon_0 w (K-1) V^2}{2d} \right)}{M + m} }$
D.
$\sqrt{ \frac{2x \left( mg - \frac{\epsilon_0 w (K-1) V^2}{2d} \right)}{M + m} }$

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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

Dielectric Constant (K) Shift|Rigid Body Constraints|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Work-Energy Theorem in an electromechanical system, specifically targeting a lethal trap regarding the thermodynamic energy balance of a capacitor connected to a constant voltage source.

Students frequently misapply the conservation of mechanical energy by writing $\Delta K + \Delta U_g + \Delta U_E = 0$ .

Because the electrostatic potential energy $U_E$ increases as the dielectric is drawn in, this flawed equation suggests the electrostatic force opposes the motion, leading to a negative sign in the numerator (Option B).

In reality, the system is not isolated; the battery does work $W_b = 2\Delta U_E$ to maintain constant voltage.

Half of this work increases the stored electrostatic energy, and the other half manifests as mechanical work done on the dielectric slab, assisting gravity.

Furthermore, the question enforces rigid body constraints by including the pulley's moment of inertia, penalizing students who blindly use standard particle kinematics (Options C and D).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Dielectric Constant (K) Shift
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free