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Question #43085

Difficulty: 8/10 | 🟢 Free

The Question

An ideal LC circuit consists of an inductor of inductance $L$ and a parallel plate capacitor with circular plates of radius $R$ and separation $d$ . The space between the capacitor plates is filled with a non-uniform dielectric material whose relative permittivity varies radially as $\epsilon_r(r) = \alpha r^2$ , where $\alpha$ is a positive constant and $r$ is the radial distance from the central axis of the plates. The circuit sustains undamped electromagnetic oscillations. If the maximum electrical energy stored in the capacitor during the oscillation cycle is $U_0$ , what is the maximum magnitude of the induced magnetic field $B_{max}$ at a radial distance $r = R/2$ from the central axis, inside the capacitor? (Assume edge effects are negligible and all quantities are in SI units)

Options

A.
$\frac{\mu_0}{16 \pi R} \sqrt{\frac{2 U_0}{L}}$
B.
$\frac{\mu_0}{4 \pi R} \sqrt{\frac{2 U_0}{L}}$
C.
$\frac{\mu_0}{16 \pi R} \sqrt{\frac{U_0}{L}}$
D.
$\frac{\mu_0}{4 \pi R} \sqrt{\frac{U_0}{L}}$

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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

Displacement Current Source|LC Oscillations Energy Cycle|Phasor Rotation Convention

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to link the macroscopic energy cycle of an LC circuit to the microscopic Maxwell-Ampere law, bypassing standard kinematic formulas like $\omega = 1/\sqrt{LC}$ .

A common trap is assuming the displacement current density $J_d$ is uniform, which incorrectly yields $I_{enc} = I_{max}/4$ and leads to option B.

Furthermore, students often attempt to calculate the equivalent capacitance and peak voltage to find the current, which becomes algebraically dense and prone to error.

The robust physical path requires recognizing that $J_d \propto r^2$ due to the radially varying dielectric, and applying energy conservation ( $U_{E,max} = U_{B,max}$ ) to directly find the peak total current.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Displacement Current Source
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free