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Question #42782

Difficulty: 8/10 | 🟢 Free

The Question

Two identical capacitors, each of capacitance $C$ , are provided. One is initially charged to a potential difference $V_0$ , while the other is uncharged. At $t=0$ , they are connected in a closed series loop through a resistor. The resistor is thermally insulated from its surroundings. Its resistance $R$ increases with the total heat energy $H$ dissipated in it according to the relation $R(H) = R_0 \left(1 + \frac{H}{U_0}\right)$ , where $R_0$ is the initial resistance and $U_0 = \frac{1}{2} C V_0^2$ is the initial energy stored in the charged capacitor. Assuming the connecting wires have zero resistance and zero heat capacity, find the time $t$ at which the potential difference across the initially charged capacitor drops to $\frac{3V_0}{4}$ .

Options

A.
$\frac{3}{32} R_0 C (8 \ln 2 - 1)$
B.
$\frac{1}{2} R_0 C \ln 2$
C.
$\frac{3}{16} R_0 C (4 \ln 2 - 1)$
D.
$\frac{1}{8} R_0 C (6 \ln 2 + 1)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Common Potential Redistribution|Ohm's Law Validity|Approximations

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to integrate Thermodynamics with Electrodynamics, specifically targeting the misapplication of standard RC circuit formulas.

Students commonly fall into the trap of assuming a standard exponential decay $V(t) = V_0 e^{-t/\tau}$ (Option B), ignoring the fact that Joule heating dynamically alters the resistance.

The 'Lethal Collision' occurs because the differential equation $dV/dt = -V/(R(T)C_{eq})$ appears intractable if time is kept as the independent variable.

The conceptual breakthrough requires recognizing that heat dissipation $H = \int V dq$ is path-independent with respect to time, allowing resistance to be expressed purely as a function of the transferred charge $q$ .

This transforms a complex time-coupled thermal runaway problem into a solvable state-dependent integral.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Common Potential Redistribution
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free