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Question #82115

Difficulty: 5/10 | 🟢 Free

The Question

Two identical large, thin, square elastic insulating sheets, each of initial side length $L$ , are placed parallel to each other at a small separation $d$ ( $d \ll L$ ). They carry uniformly distributed total charges $+Q$ and $-Q$ respectively. The sheets have Young's modulus $Y$ , Poisson's ratio $\nu$ , and coefficient of linear thermal expansion $\alpha$ . Each sheet is confined between two smooth, immovable rigid walls separated by distance $L$ , which completely prevent any expansion in the x-direction but allow free frictionless sliding in the y-direction. Initially, the system is at temperature $T_0$ and the sheets are unstressed. If the temperature of both sheets is increased by $\Delta T$ , what is the new electrostatic force of attraction $F$ between the sheets? (Assume $\alpha \Delta T \ll 1$ and neglect any deformation caused by the electrostatic force itself).

Options

A.
$\frac{Q^2}{2 \epsilon_0 L^2} [1 - \alpha \Delta T (1 + \nu)]$
B.
$\frac{Q^2}{2 \epsilon_0 L^2} [1 - \alpha \Delta T]$
C.
$\frac{Q^2}{2 \epsilon_0 L^2} [1 - 2\alpha \Delta T]$
D.
$\frac{Q^2}{2 \epsilon_0 L^2} [1 - \alpha \Delta T (1 - \nu)]$

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

Thermal Stress Calculation|Field due to Infinite Sheet|Variable Mass Systems

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the 'Piezo-Thermal Coupling' trap by forcing students to recognize that the surface charge density in Gauss's Law is a dynamic variable dependent on the thermal strain tensor.

A common lethal mistake is treating the infinite sheet field as a static constant or incorrectly calculating the new area by ignoring the Poisson effect.

Because the sheet is constrained in one dimension, the compressive thermal stress induces an additional transverse strain via Poisson's ratio.

Students who fail to couple the thermoelastic equations with the electrostatic force formula will erroneously choose options lacking the $(1+\nu)$ factor or misapply the sign of the thermal stress.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Thermal Stress Calculation
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free