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Question #43333

Difficulty: 8/10 | 🟢 Free

The Question

An AC voltage source $V(t) = V_0 \sin(\omega t)$ with peak voltage $V_0 = 20 \ \text{V}$ and angular frequency $\omega = 1000 \ \text{rad/s}$ has an internal impedance consisting of a resistor $R_1 = 5 \ \Omega$ and an inductor $L_1 = 10 \ \text{mH}$ in series. The source is connected to a load consisting of a variable resistor $R_2$ , a fixed inductor $L_2 = 40 \ \text{mH}$ , and a variable capacitor $C$ all connected in series. The values of $R_2$ and $C$ are adjusted such that the average power transferred to the load is maximized. Under this condition, determine the required capacitance $C$ and the peak voltage $V_{C0}$ across the capacitor.

Options

A.
$20 \ \mu\text{F}, 100 \ \text{V}$
B.
$25 \ \mu\text{F}, 40\sqrt{2} \ \text{V}$
C.
$20 \ \mu\text{F}, 200 \ \text{V}$
D.
$25 \ \mu\text{F}, 80 \ \text{V}$

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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

Voltage Magnification in Resonance|Maximum Power Transfer|Integration

Why This Question Is Tricky

Common Mistakes

The core trap lies in the misapplication of resonance and the Maximum Power Transfer Theorem (MPTT) in AC circuits.

Students often incorrectly assume that resonance in the load ( $X_{L2} = X_C$ ) is sufficient, ignoring the reactive component of the source's internal impedance.

By applying DC-like logic ( $R_{\text{load}} = R_{\text{internal}}$ ) without complex conjugation ( $Z_{\text{load}} = Z_{\text{source}}^*$ ), they fail to realize the load must provide a capacitive reactance that cancels both the load's and the source's inductive reactances.

Furthermore, when calculating the voltage magnification, students frequently forget to include the source's internal resistance in the total damping of the circuit, leading to an incorrect peak current and capacitor voltage.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Voltage Magnification in Resonance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free