Before You Begin

Question #43343

Difficulty: 8/10 | 🟢 Free

The Question

An ideal $LC$ circuit consists of an inductor of inductance $L$ and a capacitor of capacitance $C$ . It undergoes undamped electrical oscillations with an initial total energy $E_0$ . The state of the system is represented in a phase space by plotting the magnetic flux linkage $\Phi = Li$ against the charge $q$ on the capacitor. The initial area enclosed by the phase space trajectory is $A_0$ .

A dielectric slab of dielectric constant $K = 4$ is suddenly inserted to completely fill the space between the capacitor plates. Let $A_1$ be the new area of the phase space trajectory if the slab is inserted at the exact instant the current in the circuit is maximum. Let $A_2$ be the new area of the phase space trajectory if the slab is inserted at the exact instant the charge on the capacitor is maximum.

Which of the following correctly gives $A_1$ and $A_2$ in terms of $A_0$ ?

Options

A.
$A_1 = 2A_0, \quad A_2 = \frac{A_0}{2}$
B.
$A_1 = A_0, \quad A_2 = \frac{A_0}{4}$
C.
$A_1 = 4A_0, \quad A_2 = A_0$
D.
$A_1 = 2A_0, \quad A_2 = A_0$

Check Your Answer

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

LC Oscillations Energy Cycle|Phase Space Trajectory|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the analogy between LC circuits and mechanical oscillators in phase space, specifically focusing on how sudden topological changes (dielectric insertion) affect the system's energy depending on its phase.

Students often incorrectly assume energy is always conserved ( $E_1 = E_2 = E_0$ ) or that the phase space area (an adiabatic invariant) remains constant during sudden changes.

By mapping flux linkage $\Phi$ to momentum $p$ and charge $q$ to position $x$ , the phase space area is $A = 2\pi E \sqrt{LC}$ .

When inserted at maximum current ( $q=0$ ), no work is done against the electric field, so energy is conserved, doubling the area due to the increased capacitance.

When inserted at maximum charge, the charge is conserved but the electric field does work, reducing the energy to $E_0/4$ , which halves the area.

The misconception lies in failing to account for the phase-dependent non-conservative work done during the 'impulsive' insertion.

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