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Question #42918

Difficulty: 8/10 | 🟢 Free

The Question

Three concentric thin conducting spherical shells of radii $R$ , $2R$ , and $3R$ are arranged in free space. The innermost shell carries an initial charge $+Q$ , the middle shell carries $-2Q$ , and the outermost shell carries $+3Q$ . A long, thin conducting wire with an open switch connects the innermost shell to the outermost shell, while the middle shell remains electrically isolated. If the switch is closed and the system is allowed to reach electrostatic equilibrium, what is the total heat dissipated in the connecting wire? (Here $\epsilon_0$ is the permittivity of free space)

Options

A.
$\frac{Q^2}{48\pi\epsilon_0 R}$
B.
$\frac{Q^2}{24\pi\epsilon_0 R}$
C.
$\frac{Q^2}{12\pi\epsilon_0 R}$
D.
$\frac{Q^2}{36\pi\epsilon_0 R}$

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

Gauss Law Applicability|Potential of Concentric Shells|Common Potential Redistribution

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of Gauss's Law and the principle of superposition for electrostatic potential and energy.

A common lethal trap for students is calculating the electrostatic energy by simply summing the self-energies of individual shells ( $U = \sum Q^2/2C$ ), completely ignoring the interaction energies (or equivalently, the energy density of the superimposed electric fields in the regions between the shells).

Another trap is incorrectly calculating an equivalent capacitance by ignoring the isolated middle shell's contribution to the potential difference.

By forcing the calculation of initial and final energies using $U = \frac{1}{2}\sum q_i V_i$ , the problem ensures that only students who account for the full spatial distribution of the electric field will find the correct heat dissipation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Gauss Law Applicability
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free