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Question #42144

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The Question

A spherical region of radius $R$ contains a weakly conducting ice-water mixture at $0^\circ\text{C}$ with uniform electrical conductivity $\sigma_c$ and permittivity $\epsilon_0$ . An external mechanism maintains a steady, non-uniform volume charge density $\rho(r) = \rho_0 \frac{r}{R}$ throughout the sphere, where $r$ is the radial distance from the center. The resulting electric field causes Joule heating within the mixture. Assuming no heat is lost to the surroundings and the latent heat of fusion of ice is $L_f$ , what is the mass of ice melted in time $t$ ?

Options

A.
$\frac{\pi \sigma_c \rho_0^2 R^5 t}{28 \epsilon_0^2 L_f}$
B.
$\frac{\pi \sigma_c \rho_0^2 R^5 t}{20 \epsilon_0^2 L_f}$
C.
$\frac{\pi \sigma_c \rho_0^2 R^5 t}{14 \epsilon_0^2 L_f}$
D.
$\frac{4\pi \sigma_c \rho_0^2 R^5 t}{9 \epsilon_0^2 L_f}$

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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

Specific Heat Capacity of Phase Mixtures|Flux and Surface Geometry|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of Gauss's Law with local Joule heating and calorimetry.

The primary trap lies in the non-uniform charge distribution $\rho(r)$ .

Students often incorrectly assume they can use the total charge to find an 'average' uniform charge density (leading to Option B), or they calculate the total electric field energy and differentiate it, misapplying the local power dissipation formula $p = \sigma_c E^2$ .

By forcing a pointwise integration of the squared electric field over spherical shells ( $dV = 4\pi r^2 dr$ ), it filters out those who fail to recognize that volumetric heating is non-linear with respect to the electric field and must be integrated locally.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Specific Heat Capacity of Phase Mixtures
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free