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Question #42676

Difficulty: 8/10 | 🟢 Free

The Question

Three concentric thin conducting spherical shells A, B, and C have radii $R$ , $2R$ , and $3R$ respectively. Shells A and C are connected by a thin conducting wire and together carry a net charge of $-Q$ . Shell B is given a charge of $+Q$ . The entire system is placed in a region where the reference point for zero electric potential is chosen to be at a radial distance $r = 2.5R$ from the common center. The apparent self-capacitance of shell B in this configuration is defined as $C_B = \frac{Q}{V_B}$ , where $V_B$ is the absolute electric potential of shell B. What is the value of $C_B$ ?

Options

A.
$\frac{160\pi\epsilon_0 R}{3}$
B.
$32\pi\epsilon_0 R$
C.
$24\pi\epsilon_0 R$
D.
$8\pi\epsilon_0 R$

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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

Capacitance Independence|Principle of Superposition|Reference Potential Convention

Why This Question Is Tricky

Common Mistakes

The question tests the student's understanding of the Principle of Superposition and the Reference Potential Convention in a multi-conductor system.

Students often memorize the equivalent mutual capacitance of connected concentric shells ( $C_{eq} = 32\pi\epsilon_0 R$ ) and might blindly select this value, assuming capacitance is always an invariant geometric property.

However, the question explicitly defines an 'apparent self-capacitance' $C_B = Q/V_B$ based on the absolute potential.

To find $V_B$ , students must first determine the charge distribution ( $Q_A = -Q/4$ , $Q_C = -3Q/4$ ) by equating potentials $V_A = V_C$ , and then integrate the electric field from the non-standard reference point ( $r = 2.5R$ ) to shell B.

The trap lies in confusing the invariant mutual capacitance with the reference-dependent absolute potential, and failing to correctly apply Gauss's law to find the potential difference from the new boundary condition.

Quick Facts About This Question

Question Information

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