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Question #43083

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor has a plate area $A$ and an initial plate separation $d_0$ at $t=0$ . An external agent pulls one of the plates outwards at a constant velocity $v$ , such that the separation becomes $x(t) = d_0 + vt$ . A time-varying voltage source $V(t) = \alpha t$ (where $\alpha$ is a positive constant) is connected across the capacitor plates. What is the average electrical power delivered by the voltage source during the time interval from $t=0$ to $t=d_0/v$ ? (Assume ideal conditions and let $\epsilon_0$ be the permittivity of free space).

Options

A.
$\frac{\epsilon_0 A \alpha^2}{v} \left( \ln 2 - \frac{1}{2} \right)$
B.
$\frac{\epsilon_0 A \alpha^2}{4v}$
C.
$\frac{\epsilon_0 A \alpha^2}{v} \ln 2$
D.
$\frac{\epsilon_0 A \alpha^2}{2v} \left( \frac{3}{2} - 2 \ln 2 \right)$

Check Your Answer

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

Average Power over Cycle|Displacement Current Source|Variable Force Integration

Why This Question Is Tricky

Common Mistakes

This question tests the misapplication of the Work-Energy theorem in an open electro-mechanical system.

Students often incorrectly assume that the average power delivered by the voltage source is simply the rate of change of the electrostatic potential energy ( $\Delta U / \Delta t$ ), leading to Option B.

This fundamentally ignores the mechanical work done by the external agent moving the plates against the electrostatic force of attraction.

The correct approach requires calculating the instantaneous displacement current $i_d = dQ/dt$ and integrating the instantaneous electrical power $P(t) = V(t) i_d(t)$ .

The time-varying geometry coupled with a time-varying voltage creates a non-trivial displacement current that cannot be bypassed using simple energy state differences.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Average Power over Cycle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free