Before You Begin

Question #43346

Difficulty: 8/10 | 🟢 Free

The Question

An $LCR$ series circuit with resistance $R = 30 \text{ } \Omega$ , inductance $L = 0.2 \text{ H}$ , and capacitance $C = 400 \text{ } \mu\text{F}$ is driven by an alternating voltage source $V(t) = 100 \sin(50 t)$ , where $V$ is in volts and $t$ is in seconds. In an analogy to thermodynamic cycles, the external voltage $V(t)$ acts as an intensive variable (analogous to pressure) and the instantaneous charge $q(t)$ on the capacitor acts as an extensive variable (analogous to volume). After reaching steady state, the system traces a closed elliptical loop in the $V-q$ plane. What is the area enclosed by this loop?

Options

A.
$2.4\pi \text{ J}$
B.
$4.0\pi \text{ J}$
C.
$3.2\pi \text{ J}$
D.
0 \text{ J}

Check Your Answer

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

LC Oscillations Energy Cycle|Cyclic Process Geometry|State Function vs Path Function

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to map thermodynamic concepts (PV diagrams, work as area) to electromagnetism.

The primary trap lies in treating the $V-q$ loop area as zero (Option D), assuming that since $L$ and $C$ are conservative elements (state functions for energy), the net 'work' is zero.

Students fail to recognize that the external voltage $V(t)$ must overcome the non-conservative ohmic dissipation ( $I^2R$ ), making the process path-dependent and entropy-generating.

Another lethal trap (Option B) is calculating the area of the ellipse geometrically as $\pi V_0 q_0$ without accounting for the phase difference $\delta$ between $V(t)$ and $q(t)$ , which is analogous to ignoring the power factor in AC circuits.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	LC Oscillations Energy Cycle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free