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Question #43856

Difficulty: 8/10 | 🟢 Free

The Question

An ideal capacitor of capacitance $C$ (in Farads) is initially charged to a charge $Q_0$ (in Coulombs). At $t=0$ , it is connected in series with an inductor of inductance $L$ (in Henrys) and a non-linear resistor. The non-linear resistor has a voltage-current relationship given by $V_R = \alpha I^2$ , where $I$ is the magnitude of the current (in Amperes) and $\alpha$ is a positive constant (in $\text{V}\cdot\text{A}^{-2}$ ). Assuming the resistance opposes the flow of current, find the maximum current $I_{max}$ in the circuit during the first discharge phase.

Options

A.
$\sqrt{\frac{1}{\alpha C} \left[ Q_0 - \frac{L}{2\alpha} \ln\left(1 + \frac{2\alpha Q_0}{L}\right) \right]}$
B.
$\sqrt{\frac{1}{2\alpha C} \left[ Q_0 - \frac{L}{\alpha} \ln\left(1 + \frac{\alpha Q_0}{L}\right) \right]}$
C.
$\sqrt{\frac{1}{\alpha C} \left[ Q_0 - \frac{L}{2\alpha} \left(1 - e^{-\frac{2\alpha Q_0}{L}}\right) \right]}$
D.
$\sqrt{\frac{Q_0^2}{LC} \left[ 1 - \frac{4\alpha Q_0}{3L} \right]}$

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

L-C Circuit Analogy|Transient Current in RL Circuit|SHM Definition via Differential Equation

Why This Question Is Tricky

Common Mistakes

Students often attempt to apply standard energy conservation ( $E_{initial} = E_{final} + E_{dissipated}$ ) to find the maximum current, but fail because the energy dissipated $\int \alpha I^3 dt$ cannot be evaluated without knowing the explicit time dependence $I(t)$ .

Another common trap is assuming the maximum current occurs when the capacitor is fully discharged ( $q=0$ ), which is only true for undamped LC circuits.

This question forces the student to map the circuit to a non-linear damped mechanical oscillator (quadratic drag) and use the phase-space differential equation ( $v dv/dx$ ) to find the exact state where the derivative of current is zero.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	L-C Circuit Analogy
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free