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Question #42762

Difficulty: 8/10 | 🟢 Free

The Question

A solid conducting sphere of mass $M$ and initial radius $R_0$ (measured in a zero-gravity, uncharged state at temperature $T_0$ ) is placed in deep space. It is given a total charge $Q$ and its temperature is uniformly increased by $\Delta T$ . The material of the sphere has a uniform bulk modulus $K$ and a coefficient of linear thermal expansion $\alpha$ . Assuming all deformations are small ( $\Delta R \ll R_0$ ), which of the following expressions gives the new interior electrostatic potential $V$ of the sphere? \n\n(Given: $V_0 = \frac{Q}{4\pi\epsilon_0 R_0}$ , $G$ is the universal gravitational constant, and $\epsilon_0$ is the vacuum permittivity. Assume the sphere remains perfectly spherical.)

Options

A.
$V_0 \left[ 1 - \alpha \Delta T - \frac{Q^2}{96\pi^2\epsilon_0 R_0^4 K} + \frac{GM^2}{20\pi R_0^4 K} \right]$
B.
$V_0 \left[ 1 - \alpha \Delta T \right]$
C.
$V_0 \left[ 1 - \alpha \Delta T + \frac{Q^2}{96\pi^2\epsilon_0 R_0^4 K} - \frac{GM^2}{20\pi R_0^4 K} \right]$
D.
$V_0 \left[ 1 - 3\alpha \Delta T - \frac{Q^2}{32\pi^2\epsilon_0 R_0^4 K} + \frac{3GM^2}{20\pi R_0^4 K} \right]$

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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

Conductor Interior Potential|Potential vs Field Relationship|Thermal Stress Calculation

Why This Question Is Tricky

Common Mistakes

This question tests the hidden coupling between continuum mechanics and electrostatics.

Students typically fall into the trap of treating conductors as rigid, immutable geometries, calculating the interior potential solely based on the initial radius or only accounting for thermal expansion (Option B).

They fail to account for the mechanical deformation caused by the superposition of Maxwell stress (electrostatic pressure) and self-weight stress (gravitational pressure).

By ignoring the Virial theorem and Hooke's law, they miss that the sphere's capacitance changes due to volumetric strain, which fundamentally shifts the interior potential field.

Option C traps students who invert the signs of the pressures, and Option D traps those who confuse volumetric strain with linear strain.

Quick Facts About This Question

Question Information

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