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Question #42779

Difficulty: 8/10 | 🟢 Free

The Question

A vertical cylindrical container of cross-sectional area $A$ is fitted with a frictionless, movable piston. The piston and the bottom of the cylinder form a parallel plate capacitor. The space between them is filled with an ideal monoatomic gas. The capacitor is given a charge $Q$ and then electrically isolated. The dielectric constant of the gas depends on its volume $V$ as $K(V) = 1 + \frac{V_0}{V}$ , where $V_0$ is a positive constant. The system is initially in equilibrium with a gas volume $V_0$ . Heat is slowly supplied to the gas, causing it to expand to a volume $2V_0$ . The constant external downward force on the piston (including its weight and atmospheric pressure) is $F_{ext}$ . Let $P_{ext} = \frac{F_{ext}}{A}$ . Find the total heat $Q_{heat}$ supplied to the gas during this process. (Ignore the heat capacity of the cylinder and piston, and assume the electric field is completely confined between the plates).

Options

A.
$\frac{5}{2} P_{ext} V_0 + \frac{19 Q^2 V_0}{16 \epsilon_0 A^2}$
B.
$\frac{5}{2} P_{ext} V_0 + \frac{5 Q^2 V_0}{12 \epsilon_0 A^2}$
C.
$\frac{5}{2} P_{ext} V_0 + \frac{37 Q^2 V_0}{48 \epsilon_0 A^2}$
D.
$\frac{5}{2} P_{ext} V_0 + \frac{7 Q^2 V_0}{16 \epsilon_0 A^2}$

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

Energy Density Locality|Coulombic Central Force|Variation of Density with Temperature

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of thermodynamics with electrostatic energy density in a deformable medium.

The primary trap lies in the misapplication of the Ideal Gas Law and the First Law of Thermodynamics by ignoring the electrostatic pressure.

Students commonly assume that the gas pressure $P$ simply equals the external mechanical pressure $P_{ext}$ .

However, because the dielectric constant depends on the gas volume, the electrostatic energy $U_E$ varies with volume, creating an additional attractive electrostatic pressure $P_{elec} = \partial U_E / \partial V$ that the gas must overcome.

Failing to account for this leads to an incorrect calculation of both the change in the gas's internal energy $\Delta U_{gas}$ (since $T = PV/nR$ ) and the mechanical work done by the gas.

The correct approach requires integrating the total pressure $P = P_{ext} + P_{elec}$ to find the work and state variables.

Quick Facts About This Question

Question Information

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