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Question #43071

Difficulty: 8/10 | 🟢 Free

The Question

An ideal capacitor of capacitance $C$ is initially given a charge $Q_0$ . At time $t=0$ , it is connected in a closed series circuit with a real coil having inductance $L$ and internal resistance $r$ . The resistance is small enough ( $r \ll 2\sqrt{\frac{L}{C}}$ ) that the circuit undergoes weakly damped oscillations. The total electromagnetic energy of the circuit decays exponentially over time. What is the total electrical energy dissipated as heat in the coil's resistance from $t=0$ to $t = \frac{L}{r} \ln 4$ ?

Options

A.
$\frac{3Q_0^2}{8C}$
B.
$\frac{Q_0^2}{4C}$
C.
$\frac{Q_0^2}{8C}$
D.
$\frac{15Q_0^2}{32C}$

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

LC Oscillations Energy Cycle|Damped Oscillation Equation|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to mathematically model energy dissipation in a non-ideal LC circuit (a damped harmonic oscillator).

A common trap is confusing the decay rate of the amplitude (charge or current) with the decay rate of the total energy.

The amplitude decays as $e^{-\gamma t}$ where the damping coefficient is $\gamma = \frac{r}{2L}$ .

However, because energy is proportional to the square of the amplitude ( $E \propto q_{max}^2$ ), the energy decays twice as fast, as $e^{-2\gamma t} = e^{-\frac{rt}{L}}$ .

Students who incorrectly apply the amplitude decay factor directly to the energy will calculate a remaining energy of $E_0/2$ and an incorrect dissipated heat of $Q_0^2/4C$ .

Additionally, students might misread the question and select the remaining energy ( $Q_0^2/8C$ ) instead of the dissipated heat.

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