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Question #80902

Difficulty: 5/10 | 🟢 Free

The Question

An AC voltage source $V(t) = V_0 \sin(\omega t)$ is connected in series with a resistor of resistance $R$ , an inductor of inductance $L$ , and a parallel plate capacitor. The capacitor is initially filled with a solid dielectric. The dielectric is being withdrawn at a constant speed such that the capacitance decreases linearly with time as $C(t) = C_0 - \beta t$ , where $\beta > 0$ is a constant rate in $\text{F/s}$ . Assume that the observation time is sufficiently small such that $\beta t \ll C_0$ , allowing $C(t)$ to be approximated as $C_0$ in its magnitude, but its rate of change $\frac{dC}{dt} = -\beta$ remains significant. Determine the phase angle $\phi$ by which the source voltage leads the current in the pseudo-steady state.

Options

A.
$\tan^{-1} \left[ \frac{\omega L (\beta^2 + \omega^2 C_0^2) - \omega C_0}{R (\beta^2 + \omega^2 C_0^2) - \beta} \right]$
B.
$\tan^{-1} \left[ \frac{\omega L (\beta^2 + \omega^2 C_0^2) - \omega C_0}{R (\beta^2 + \omega^2 C_0^2) + \beta} \right]$
C.
$\tan^{-1} \left[ \frac{\omega L - (\omega C_0)^{-1}}{R} \right]$
D.
$\tan^{-1} \left[ \frac{\omega L (\beta^2 + \omega^2 C_0^2) - \beta}{R (\beta^2 + \omega^2 C_0^2) - \omega C_0} \right]$

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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

Phase Constant Determination|Phasor Rotation Convention|Bandwidth in RLC|Non-Uniform Dielectric Filling

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the fundamental definition of current in a capacitor, $I = dq/dt = d(CV)/dt$ .

Students commonly misapply the steady-state AC reactance formula $X_C = 1/(\omega C)$ , ignoring the product rule expansion which introduces a $V(dC/dt)$ term.

When the dielectric is withdrawn, the time-varying capacitance acts mathematically as a complex impedance with a negative real part (a negative resistance).

This physically reflects the mechanical work done by the external agent being converted into electrical energy, pumping power back into the circuit.

Failing to account for this non-autonomous nature of the differential equation leads to incorrect phase constant determinations, trapping students who rely solely on standard RLC formulas.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Phase Constant Determination
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free