Before You Begin

Question #42847

Difficulty: 8/10 | 🟢 Free

The Question

An isolated, perfectly conducting sphere of radius $a$ in vacuum is initially charged to a potential $V_0$ . At $t=0$ , it is connected to ground through a series combination of a time-varying internal resistor and a variable external load resistor $R_L$ . The internal resistor is a cylinder of length $L$ and cross-sectional area $A$ , filled with a material of constant permittivity $\epsilon$ . Due to heating, the dielectric relaxation time of this material decreases as $\tau(t) = \frac{\tau_0}{1 + \alpha t}$ . A control system continuously adjusts $R_L$ such that the instantaneous power dissipated in $R_L$ is always maximized. Assuming the capacitance of the cylindrical resistor is negligible, which of the following gives the potential of the sphere $V(t)$ ?

Options

A.
$V_0 \exp\left( -\frac{\epsilon A (2t + \alpha t^2)}{16 \pi \epsilon_0 a L \tau_0} \right)$
B.
$V_0 \exp\left( -\frac{\epsilon A (2t + \alpha t^2)}{8 \pi \epsilon_0 a L \tau_0} \right)$
C.
$V_0 \exp\left( -\frac{\epsilon A t}{8 \pi \epsilon_0 a L \tau_0} \right)$
D.
$V_0 \exp\left( -\frac{\epsilon A (t + \alpha t^2)}{8 \pi \epsilon_0 a L \tau_0} \right)$

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

Charged Conducting Sphere|Capacitance Independence|Maximum Power Transfer|Relaxation Time Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Maximum Power Transfer theorem in a dynamic, time-varying system coupled with an RC discharge circuit.

The primary trap is that students often memorize $R_L = R_{int}$ for static DC circuits but fail to realize that when applying it instantaneously to a time-varying source, the total equivalent resistance of the circuit becomes $R_{eq} = 2R_{int}(t)$ .

Many students will incorrectly use $R_{eq} = R_{int}(t)$ (forgetting the load itself in the discharge path) leading to Option B.

Additionally, it tests the fundamental relationship between dielectric relaxation time, permittivity, and conductivity, requiring students to dynamically integrate the time-varying RC constant rather than treating it algebraically.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Charged Conducting Sphere
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free