Before You Begin

Question #43075

Difficulty: 8/10 | 🟢 Free

The Question

A series LCR circuit with resistance $R$ (in $\Omega$ ), inductance $L$ (in $\text{H}$ ), and capacitance $C$ (in $\text{F}$ ) is driven by an AC voltage source $V(t) = V_0 \sin(\omega t)$ (in $\text{V}$ ). The circuit operates in a steady state with an impedance $Z$ (in $\Omega$ ) and a phase angle $\phi$ (in $\text{rad}$ ) between the voltage and the current. Let $P_S(t)$ be the instantaneous power delivered by the source, and $P_R(t)$ be the instantaneous power dissipated in the resistor. If $\Delta P_{max}$ is the maximum value of the difference $P_S(t) - P_R(t)$ during one complete cycle, which of the following expressions correctly represents $\Delta P_{max}$ ?

Options

A.
$\frac{V_0^2}{2Z} |\sin \phi|\ \text{W}$
B.
$\frac{V_0^2}{2Z} \cos \phi\ \text{W}$
C.
$\frac{V_0^2}{Z} |\sin \phi|\ \text{W}$
D.
$\frac{V_0^2}{Z} |\tan \phi|\ \text{W}$

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|Series LCR Impedance Vector|Approximation (Small Angle)

Why This Question Is Tricky

Common Mistakes

This question tests the deep conceptual distinction between real power dissipation and reactive energy oscillation in an LCR circuit.

Students often memorize average power formulas ( $P_{avg} = V_{rms} I_{rms} \cos \phi$ ) and incorrectly assume that the maximum difference between source power and dissipated power relates to this average or to the apparent power $V_0 I_0$ .

The trap exploits the failure to apply the work-energy theorem to instantaneous circuit dynamics.

By recognizing that $P_S(t) - P_R(t)$ is exactly the instantaneous rate of change of energy stored in the reactive components ( $\frac{dU}{dt}$ ), students must perform a complex vector impedance analysis to find that this energy exchange rate oscillates with an amplitude proportional to the reactive impedance $|X_L - X_C|$ , governed by the phase angle $\phi$ .

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