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Question #42615

Difficulty: 8/10 | 🟢 Free

The Question

A non-sinusoidal periodic voltage $V(t) = 100 \sin(\omega t) + 80 \sin(2\omega t)$ (in Volts) is applied across a series LCR circuit. At the fundamental angular frequency $\omega$ , the circuit components have the following values: Resistance $R = 8 \, \Omega$ , Inductive reactance $X_L = 2 \, \Omega$ , and Capacitive reactance $X_C = 8 \, \Omega$ . Determine the root-mean-square current $I_{\text{rms}}$ flowing through the circuit and the total average power $P$ dissipated.

Options

A.
$10 \, \text{A}, \, 800 \, \text{W}$
B.
$10\sqrt{2} \, \text{A}, \, 1600 \, \text{W}$
C.
$\sqrt{82} \, \text{A}, \, 656 \, \text{W}$
D.
$10 \, \text{A}, \, 1000 \, \text{W}$

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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

Mean vs. RMS Values|Series LCR Impedance Vector|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle non-sinusoidal periodic waveforms in AC circuits.

The primary trap is the instinctive calculation of a single 'effective' impedance (usually at the fundamental frequency) and applying it to the total RMS voltage ( $I_{\text{rms}} = V_{\text{rms}}/Z$ ).

Because inductive and capacitive reactances are frequency-dependent ( $X_L \propto \omega$ , $X_C \propto 1/\omega$ ), the superposition principle must be used.

Students must perform a Fourier decomposition, calculate the impedance and current for each harmonic independently, and then find the total RMS current using the quadrature sum $\sqrt{\sum I_{n,\text{rms}}^2}$ .

Failing to recognize the frequency dependence of the impedance for different harmonics leads to incorrect current and power values.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Mean vs. RMS Values
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free