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Question #43334

Difficulty: 8/10 | 🟢 Free

The Question

A series LCR circuit with resistance $R$ , inductance $L$ , and capacitance $C$ is driven by an AC voltage source $V(t) = V_0 \sin(\omega t)$ . The circuit is tuned to resonance. An electron (rest mass $m_0$ , charge $e$ ) is accelerated from rest through a static potential difference equal to the peak voltage across the capacitor. After acceleration, the electron enters a region of uniform magnetic field $B$ perpendicular to its velocity, moving in a circular path. Which of the following represents the magnetic dipole moment of the revolving electron? (Let $c$ be the speed of light in vacuum).

Options

A.
$\frac{e V_0 \sqrt{L} \left( e V_0 \sqrt{L} + 2 R \sqrt{C} m_0 c^2 \right)}{2 B R \sqrt{C} \left( e V_0 \sqrt{L} + R \sqrt{C} m_0 c^2 \right)}$
B.
$\frac{e V_0 \sqrt{L}}{B R \sqrt{C}}$
C.
$\frac{e V_0 \sqrt{L} \left( e V_0 \sqrt{L} + 2 R \sqrt{C} m_0 c^2 \right)}{2 B R^2 C m_0 c^2}$
D.
$\frac{e V_0 \left( e V_0 + 2 m_0 c^2 \right)}{2 B \left( e V_0 + m_0 c^2 \right)}$

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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|Magnetic Dipole Moment of Revolving Electron|Vector Analysis

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of AC transient magnification with relativistic electrodynamics.

The primary trap lies in the misapplication of the classical gyromagnetic ratio.

Students often blindly use the non-relativistic relation $\mu = K/B$ (Option B) or assume the classical proportionality $\mu = \frac{e}{2m_0}L$ holds at high energies, leading to an incorrect relativistic momentum substitution (Option C).

Furthermore, students might ignore the Q-factor voltage magnification of the LCR circuit and use the source voltage $V_0$ instead (Option D).

The correct approach requires recognizing that relativistic mass variation alters the current loop's effective dipole moment, necessitating a derivation from fundamental principles ( $\mu = IA$ ) using the relativistic energy-momentum invariant.

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