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Question #42571

Difficulty: 8/10 | 🟢 Free

The Question

A battery of constant electromotive force $E$ and temperature-dependent internal resistance $r(T) = r_0[1 + \alpha(T - T_0)]$ is short-circuited using a wire of zero resistance. The battery is placed in an environment maintained at a constant temperature $T_0$ . It dissipates heat to the surroundings at a rate $P_{\text{diss}} = K(T - T_0)$ , where $K$ is a positive constant. After the battery reaches a steady-state temperature, what is the rate of entropy increase of the universe? (Assume $\alpha > 0$ )

Options

A.
$\frac{K}{2 \alpha T_0} \left( \sqrt{1 + \frac{4 \alpha E^2}{K r_0}} - 1 \right)$
B.
$\frac{E^2}{r_0 T_0}$
C.
$\frac{K}{2 \alpha T_0} \left( 1 - \sqrt{1 - \frac{4 \alpha E^2}{K r_0}} \right)$
D.
$\frac{K}{2 \alpha} \left( \sqrt{1 + \frac{4 \alpha E^2}{K r_0}} - 1 \right) \ln\left(1 + \frac{1}{\alpha T_0}\right)$

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

Entropy and Reversibility|Work Done in Non-PV Diagrams|Internal Resistance Variation

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of non-linear Joule heating with the thermodynamic definition of entropy.

The primary trap is assuming the internal resistance remains $r_0$ (yielding option B) or incorrectly calculating the entropy change by integrating $dQ/T$ over the battery's temperature.

It forces students to recognize that in steady state, the battery's entropy is constant, and all generated heat is irreversibly dumped into the surroundings at $T_0$ .

Students must solve the non-linear power balance equation before applying the Second Law of Thermodynamics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Entropy and Reversibility
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free