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Question #43081

Difficulty: 8/10 | 🟢 Free

The Question

A series circuit consists of an ideal capacitor of capacitance $C$ , an inductor of inductance $L$ , and a radiation resistance $R_{rad}$ (accounting for electromagnetic energy radiated by the system). The circuit is driven by an alternating voltage source $V(t) = V_0 \sin(\omega t)$ . The driving frequency is set to $\omega = 2\omega_0$ , where $\omega_0 = \frac{1}{\sqrt{LC}}$ is the natural undamped frequency of the $LC$ combination. At this driving frequency, the steady-state current lags the source voltage by a phase angle of $\frac{\pi}{3}$ rad. If the peak voltage across the capacitor is measured to be $V_C$ , what is the average power $P_{avg}$ dissipated as radiation?

Options

A.
$\sqrt{3} V_C^2 \sqrt{\frac{C}{L}}$
B.
$2\sqrt{3} V_C^2 \sqrt{\frac{C}{L}}$
C.
$\frac{\sqrt{3}}{3} V_C^2 \sqrt{\frac{C}{L}}$
D.
$\frac{\sqrt{3}}{4} V_C^2 \sqrt{\frac{C}{L}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Average Power over Cycle|LC Oscillations Energy Cycle|Bandwidth in RLC

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from ideal LC circuits to non-conservative driven RLC systems, specifically focusing on radiation damping.

The primary trap lies in the calculation of average power.

Students often incorrectly assume $P_{avg} = V_{rms} I_{rms}$ without accounting for the power factor ( $\cos\phi$ ), which leads to Option B.

Alternatively, they might incorrectly apply the capacitor's voltage directly to the resistor (Option C) or fail to use the driven frequency $\omega$ instead of the natural frequency $\omega_0$ when relating $V_C$ to $I_0$ (Option D).

The correct approach requires determining the hidden radiation resistance via the phase lag and using the local current-voltage relationship at the capacitor to find the true dissipated power.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Average Power over Cycle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free