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Question #43070

Difficulty: 8/10 | 🟢 Free

The Question

A long single-layer solenoid is used in a high-frequency circuit where its continuous charge distribution cannot be ignored. It is modeled as an ideal lossless distributed-parameter transmission line of length $l$ , with a total low-frequency inductance $L_0$ and a total distributed capacitance to ground $C_0$ . One end of the solenoid is directly grounded. An ideal AC voltage source of variable angular frequency $\omega$ is connected between the other end of the solenoid and ground. What is the lowest non-zero angular frequency $\omega_1$ at which the amplitude of the current drawn from the AC source becomes zero?

Options

A.
$\frac{1}{\sqrt{L_0 C_0}}$
B.
$\frac{\pi}{2\sqrt{L_0 C_0}}$
C.
$\frac{\pi}{\sqrt{L_0 C_0}}$
D.
$\frac{2}{\pi\sqrt{L_0 C_0}}$

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

Bandwidth in RLC|Energy Stored in Inductor|Continuous Charge Distribution

Why This Question Is Tricky

Common Mistakes

Students commonly misapply the lumped-parameter LC resonance formula ( $\omega = 1/\sqrt{LC}$ ) to high-frequency inductors, ignoring the continuous spatial distribution of charge and magnetic flux.

By treating the solenoid's self-capacitance and inductance as discrete components in parallel, they incorrectly predict the first self-resonant frequency as $1/\sqrt{L_0 C_0}$ .

This question tests the understanding that at high frequencies, the distributed nature of the electric and magnetic fields causes the solenoid to behave as a transmission line resonator, introducing a spatial phase delay that shifts the resonance condition by a factor of $\pi/2$ for a quarter-wave mode.

Quick Facts About This Question

Question Information

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