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Question #80513

Difficulty: 5/10 | 🟢 Free

The Question

An ideal monoatomic gas is enclosed in a horizontal cylinder of cross-sectional area $A$ by a movable, non-conducting, massless piston having a positive charge $q$ . The cylinder is placed in a region where an external electric field exists, pointing into the cylinder (opposing the expansion of the gas). The magnitude of the electric field varies with the displacement $x$ of the piston from its initial position as $E = kx$ , where $k$ is a positive constant. The external atmospheric pressure is $P_0$ . The gas is heated slowly such that the piston moves quasi-statically by a distance $d$ . If the initial volume of the gas is $V_0$ , what is the total heat supplied to the gas during this process?

Options

A.
$\frac{5}{2} P_0 A d + \frac{3}{2} \frac{q k d V_0}{A} + 2 q k d^2$
B.
$\frac{5}{2} P_0 A d + \frac{1}{2} q k d^2$
C.
$\frac{5}{2} P_0 A d + \frac{3}{2} \frac{q k d V_0}{A} + \frac{3}{2} q k d^2$
D.
$\frac{5}{2} P_0 A d + \frac{3}{2} q k d V_0 + 2 q k d^2$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

First Law Application|Work against Conservative Force|Vector Calculus

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 synthesis of the First Law of Thermodynamics with electrostatic mechanics, specifically targeting the misconception that a piston acts solely as a mechanical boundary against constant atmospheric pressure.

Students often fall into the trap of calculating the work done by the gas as simply $P_0 \Delta V$ , ignoring the variable work done against the electrostatic force $qE(x)$ .

Furthermore, even if they account for the electrostatic work in the $W$ term, they frequently fail to realize that the variable external force fundamentally alters the gas pressure $P(x)$ , which in turn changes the final internal energy $U_f$ of the gas.

The inclusion of a dimensionally inconsistent option (Option D) also penalizes students who do not verify the physical dimensions of their derived terms before selecting an answer.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	First Law Application
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free