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Question #43862

Difficulty: 8/10 | 🟢 Free

The Question

A thin conducting spherical shell of initial radius $R$ carries a total charge $Q$ . A small physical patch of initial area $A$ ( $A \ll R^2$ ) is marked on the surface of the shell. The shell is placed in a vacuum and its temperature is uniformly increased by $\Delta T$ . The coefficient of linear expansion of the shell material is $\alpha$ . Let $\Delta F$ be the change in the outward electrostatic force acting on the marked patch, and $\Delta U$ be the change in the total electrostatic self-energy of the shell. Assuming $\alpha \Delta T \ll 1$ , what is the value of the ratio $\frac{\Delta F / F}{\Delta U / U}$ , where $F$ and $U$ are the initial electrostatic force on the patch and the initial self-energy of the shell, respectively?

Options

A.
$2$
B.
$4$
C.
$3$
D.
$1$

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

Electric Field Symmetry|Thermal Expansion of Area and Volume|Surface Integration

Why This Question Is Tricky

Common Mistakes

The primary trap in this problem is failing to account for the thermal expansion of the marked patch area $A$ while correctly expanding the radius $R$ for the charge density $\sigma$ .

Students often write $F = P \cdot A$ and incorrectly assume $A$ is a constant geometric window, leading to $F' = P' \cdot A \propto R^{-4}$ .

This yields a fractional change of $-4\alpha\Delta T$ and an incorrect final ratio of 4.

The problem rigorously tests the coupling of thermo-mechanical area expansion with electrostatic pressure scaling, requiring students to recognize that the charge on the specific physical patch remains constant ( $q = \sigma A$ ) while the local electric field scales as $R^{-2}$ .

Quick Facts About This Question

Question Information

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