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Question #42808

Difficulty: 8/10 | 🟢 Free

The Question

A uniformly charged spherical cloud of monatomic ideal plasma expands quasi-statically against a controlled external pressure from an initial volume $V_0 \text{ m}^3$ to a final volume $8V_0 \text{ m}^3$ . The cloud is thermally insulated and remains perfectly spherical with a uniform spatial charge density at all times. The expansion is driven by the plasma's internal thermal pressure $P_T \text{ Pa}$ and its effective electrostatic pressure $P_E = -\frac{\partial U_E}{\partial V} \text{ Pa}$ , where $U_E \text{ J}$ is the electrostatic self-energy. If the initial internal thermal energy of the plasma is $E_0 \text{ J}$ and its initial electrostatic self-energy is also $E_0 \text{ J}$ , what is the total work done by the cloud on its surroundings in Joules?

Options

A.
$\frac{5}{4} E_0 \text{ J}$
B.
$\frac{3}{2} E_0 \text{ J}$
C.
$\frac{9}{4} E_0 \text{ J}$
D.
$\frac{13}{8} E_0 \text{ J}$

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

Work against Conservative Force|Work Done in Non-PV Diagrams|Approximations

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply the Generalized Work-Energy Theorem to a system with both thermodynamic and electrostatic degrees of freedom.

The primary trap lies in misinterpreting the electrostatic work during expansion.

Students often correctly apply the adiabatic condition $E_T \propto V^{-2/3}$ for the monatomic gas but incorrectly assume the electrostatic self-energy follows the same polytropic scaling, leading to Option B.

Alternatively, they may attempt to calculate $\int P_E dV$ by incorrectly assuming $P_E \propto 1/V$ or $P_E = U_E/V$ , failing to recognize that $U_E \propto V^{-1/3}$ dictates a specific polytropic index of $n=4/3$ for the electrostatic pressure.

The problem forces the realization that conservative field work can be perfectly mapped to a thermodynamic $P-V$ expansion.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Work against Conservative Force
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free