Before You Begin

Question #43344

Difficulty: 8/10 | 🟢 Free

The Question

An ideal LC circuit consists of a capacitor of capacitance $C$ and a long ideal solenoid of length $l$ , radius $b$ , and $n$ turns per unit length. A solid, non-magnetic conducting cylinder of radius $a$ ( $a < b$ ), length $l$ , and uniform electrical conductivity $\sigma$ is placed coaxially inside the solenoid. The capacitor is initially charged to $Q_0$ and the circuit is closed at $t=0$ . Due to the time-varying magnetic field, eddy currents are induced in the conducting cylinder, leading to energy dissipation. Assuming the damping is very small (quality factor $Q \gg 1$ ) and ignoring the skin effect, what is the effective series resistance $R_{\text{eff}}$ of the circuit?

Options

A.
$\frac{\sigma \mu_0 a^4}{8 b^2 C}$
B.
$\frac{\sigma \mu_0 a^2}{8 C}$
C.
$\frac{\sigma \mu_0 a^4}{4 b^2 C}$
D.
$\frac{\sigma \mu_0 b^2}{8 C}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

LC Oscillations Energy Cycle|Faraday’s Law Vector Form|Energy Density Locality

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to bridge lumped-element circuit theory with local electromagnetic field theory.

A common trap is treating the LC circuit as an isolated conservative system or incorrectly modeling the eddy current dissipation as a standard constant resistance $R = \rho l/A$ .

Students must recognize that the localized energy dissipation rate (power) via eddy currents scales with $(dI/dt)^2$ , not $I^2$ .

By applying Faraday's Law to find the induced azimuthal electric field and integrating the local power density $J^2/\sigma$ over the cylinder's volume, they obtain a non-standard damping term.

Equating the cycle-averaged power to an effective series resistance $R_{\text{eff}}$ requires substituting the unperturbed natural frequency $\omega_0 = 1/\sqrt{LC}$ , revealing that the effective resistance is inversely proportional to the capacitance.

Distractor options specifically target errors in the volume integration limits or incorrectly assuming the solenoid's inductance is confined only to the conducting cylinder's volume.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	LC Oscillations Energy Cycle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free