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Question #42194

Difficulty: 8/10 | 🟢 Free

The Question

$n$ moles of an ideal monoatomic gas are trapped in a rectangular container of base area $A = b \times L$ (with $L$ along the $x$ -axis) by a movable, frictionless, conducting piston. The bottom of the container is also a conducting plate, forming a capacitor with the piston. The gas acts as a dielectric whose dielectric constant varies spatially along the length $L$ as $K(x) = 1 + \frac{x}{L}$ . Initially (State 1), the piston is at height $d$ , the gas is at temperature $T_0$ , and a battery of voltage $V_0$ is connected across the plates. The system undergoes the following cycle:

$1 \to 2$ : Reversible isothermal expansion to height $2d$ with the battery connected. $2 \to 3$ : Battery is disconnected, and the gas is cooled isochorically to $T_1$ . $3 \to 4$ : Reversible isothermal compression back to height $d$ . $4 \to 1$ : Gas is heated isochorically to $T_0$ , and the battery is reconnected.

What is the net work done by the external agent on the piston during this complete cycle? (Ignore gravity)

Options

A.
$\frac{3 \epsilon_0 A V_0^2}{16 d} - n R (T_0 - T_1) \ln 2$
B.
$\frac{\epsilon_0 A V_0^2}{8 d} - n R (T_0 - T_1) \ln 2$
C.
$\frac{3 \epsilon_0 A V_0^2}{16 d} + n R (T_0 - T_1) \ln 2$
D.
$\frac{3 \epsilon_0 A V_0^2}{8 d} - n R (T_0 - T_1) \ln 2$

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

Cyclic Process Geometry|Non-Uniform Dielectric Filling|Potential Energy Reference

Why This Question Is Tricky

Common Mistakes

This question tests the application of the First Law of Thermodynamics to an electro-thermodynamic cycle.

A common trap is assuming that the net electrical work over a cycle is zero; however, because the battery is reconnected at a different potential (State 4), the cycle in the Q-V plane is open, and the external agent must account for the dissipated energy.

Additionally, students must correctly integrate the spatially varying dielectric constant as parallel capacitors ( $dC = \epsilon(x) dA / y$ ) rather than a simple average, and correctly switch between constant-voltage and constant-charge electric force formulas ( $|F_e| = \frac{1}{2}V^2 |\frac{dC}{dy}|$ vs $|F_e| = \frac{Q^2}{2C^2}|\frac{dC}{dy}|$ ) to evaluate the mechanical work done by the field.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Cyclic Process Geometry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free