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Question #43090

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The Question

Consider an ideal LC circuit consisting of an inductor of inductance $L$ and a parallel-plate capacitor of capacitance $C$ with plate separation $d$ . The physical dimensions of the circuit are much smaller than the wavelength of the emitted electromagnetic waves. Due to the time-varying displacement current between the capacitor plates, the system acts as a Hertzian dipole and radiates electromagnetic energy into space. The power radiated by a dipole of moment $p$ is given by Larmor's formula $P = \frac{\mu_0 \ddot{p}^2}{6 \pi c}$ , where $c$ is the speed of light in vacuum and $\mu_0$ is the vacuum permeability. Assuming the radiative damping is very small, what is the time $t$ taken for the total electromagnetic energy stored in the LC circuit to decrease to $1/e$ of its initial value?

Options

A.
$\frac{6 \pi c L^2 C}{\mu_0 d^2}$
B.
$\frac{3 \pi c L^2 C}{\mu_0 d^2}$
C.
$\frac{12 \pi c L^2 C}{\mu_0 d^2}$
D.
$\frac{6 \pi c L C^2}{\mu_0 d^2}$

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

Maxwell's Correction to Ampere's Law|Energy Stored in Inductor|Integration

Why This Question Is Tricky

Common Mistakes

Students often treat ideal LC circuits as purely conservative systems, assuming energy oscillates indefinitely between the electric and magnetic fields.

This question tests the Maxwell correction: the time-varying electric field (displacement current) between the capacitor plates implies the system acts as a radiating dipole antenna.

The trap lies in failing to recognize that radiation introduces a non-conservative energy flux (Poynting vector), acting as an effective radiation resistance.

Furthermore, students who correctly identify the power loss often misapply the energy averaging, confusing the amplitude decay time constant ( $2\tau$ ) with the energy decay time constant ( $\tau$ ), leading to a missing factor of 2.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Maxwell's Correction to Ampere's Law
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free