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Question #42708

Difficulty: 8/10 | 🟢 Free

The Question

A solid conducting sphere of radius $R$ is given an initial charge $Q$ and is immersed in an infinite, homogeneous liquid dielectric medium of relative permittivity $\epsilon_r = 3$ . The sphere is then connected via a long, thin conducting wire to an initially uncharged parallel-plate capacitor of capacitance $C = 8\pi\epsilon_0 R$ located in a vacuum far away. After the switch is closed, charge redistributes until electrostatic equilibrium is reached. What is the total heat dissipated in the wire during this process? (Assume the wire has negligible capacitance and the dielectric medium remains unperturbed).

Options

A.
$\frac{Q^2}{60\pi\epsilon_0 R}$
B.
$\frac{Q^2}{12\pi\epsilon_0 R}$
C.
$\frac{Q^2}{10\pi\epsilon_0 R}$
D.
$\frac{Q^2}{24\pi\epsilon_0 R}$

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

Self-Energy of Charged Sphere|Common Potential Redistribution|Medium Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the understanding of electrostatic self-energy and capacitance in the presence of a dielectric medium.

A common trap is to assume the sphere's capacitance remains $C = 4\pi\epsilon_0 R$ (vacuum capacitance) while calculating either the initial energy or the equivalent capacitance during redistribution.

Students often forget that the dielectric medium undergoes polarization, and the bound charge effectively shields the free charge, altering the energy density of the field ( $u = \frac{1}{2}\epsilon E^2$ ) and increasing the sphere's capacitance to $C = 4\pi\epsilon_0\epsilon_r R$ .

Failing to account for $\epsilon_r$ in both the initial state and the final parallel combination leads to incorrect energy balance equations, yielding the trap answers.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Self-Energy of Charged Sphere
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free