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Question #42700

Difficulty: 8/10 | 🟢 Free

The Question

An ideal LC oscillator, acting as a Simple Harmonic Motion system for charge, generates an AC voltage $V(t) = V_0 \sin(\omega t)$ , where $V_0$ is the peak voltage in Volts and $t$ is time in seconds. This voltage is applied across the input terminals of an infinite ladder network. The ladder consists of series capacitors, each of capacitance $C$ (in Farads), and shunt inductors, each of inductance $L$ (in Henrys). The angular frequency of the oscillator is tuned to $\omega = \frac{1}{\sqrt{LC}} \text{ rad/s}$ . Find the peak instantaneous power $P_{peak}$ delivered by the source to the infinite ladder network.

Options

A.
$\frac{V_0^2}{4} \sqrt{\frac{C}{L}} (2 + \sqrt{3}) \text{ W}$
B.
$\frac{\sqrt{3} V_0^2}{4} \sqrt{\frac{C}{L}} \text{ W}$
C.
$V_0^2 \sqrt{\frac{C}{L}} \text{ W}$
D.
$\frac{V_0^2}{4} \sqrt{\frac{C}{L}} (2 - \sqrt{3}) \text{ W}$

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

Infinite Series Capacitance|Capacitive Reactance Dependency|SHM via Differential Equation

Why This Question Is Tricky

Common Mistakes

This problem tests the synthesis of recursive AC circuit analysis with continuous time-domain power equations.

The primary trap lies in confusing average (RMS) power with peak instantaneous power, or ignoring the frequency-dependent phase shift entirely.

Students often incorrectly assume $P_{peak} = V_0 I_0$ (Option C) by neglecting the $\pi/6$ phase difference caused by the complex impedance of the infinite ladder.

Alternatively, they calculate the average power $P_{avg} = V_{rms} I_{rms} \cos \phi$ (Option B).

The correct approach requires setting up the recursive complex fraction, finding the exact phase shift, and maximizing the time-dependent power function $P(t) = V(t)I(t)$ using trigonometric identities.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Infinite Series Capacitance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free