Before You Begin

Question #95374

Difficulty: 5/10 | 🟢 Free

The Question

An LCR series circuit is connected to an AC voltage source $V(t) = V_0 \sin(\omega t)$ . The circuit consists of a resistor of resistance $R$ , a capacitor of capacitance $C$ , and an inductor with a removable soft-iron core. Initially, the core is fully inserted, and the circuit is in exact electrical resonance. An external agent quasi-statically pulls the core out of the inductor, continuously decreasing its inductance. The process is stopped when the inductive reactance has decreased by exactly $R$ . Assuming the ohmic resistance of the inductor is negligible and the process is slow enough that steady-state AC conditions apply at each instant, what is the total mechanical work done by the external agent?

Options

A.
$\frac{\pi V_0^2}{16 \omega R}$
B.
$\frac{\pi V_0^2}{8 \omega R}$
C.
$\frac{V_0^2}{4 \omega R}$
D.
$\frac{V_0^2}{8 \omega R}$

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

Power Factor Definition|L-C Circuit Analogy|Energy Stored in Inductor|Integration

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the student's ability to bridge steady-state AC circuit analysis with the fundamental work-energy theorem in electromagnetism.

A common trap is attempting to use energy conservation by simply calculating the change in average stored magnetic energy ( $\Delta U_{avg}$ ), completely ignoring the electrical work done by the AC source against the motional back-EMF ( $I \frac{dL}{dt}$ ) generated during the core's removal.

Another trap is assuming the RMS current remains constant at its initial resonance value or final value, failing to recognize that the mechanical work integral must account for the continuous phase shift and current attenuation as the circuit moves out of resonance.

The correct approach requires integrating the time-averaged magnetic force over the changing inductance.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Power Factor Definition
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free