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Question #42155

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The Question

A spherical shell of inner radius $a$ and outer radius $b$ has a radially varying thermal conductivity given by $k(r) = \frac{k_0 a}{r}$ , where $k_0$ is a positive constant. The inner surface at $r=a$ is maintained at temperature $T_1$ and the outer surface at $r=b$ is maintained at temperature $T_2$ ( $T_1 > T_2$ ). In steady state, what is the temperature $T(r)$ at a distance $r$ ( $a < r < b$ ) from the center?

Options

A.
$T_1 - (T_1 - T_2) \frac{\ln(r/a)}{\ln(b/a)}$
B.
$T_1 - (T_1 - T_2) \frac{\frac{1}{a} - \frac{1}{r}}{\frac{1}{a} - \frac{1}{b}}$
C.
$T_1 - (T_1 - T_2) \frac{r - a}{b - a}$
D.
$T_1 - (T_1 - T_2) \frac{r^2 - a^2}{b^2 - a^2}$

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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

Thermal Conductivity in Series and Parallel|Steady State Temperature Gradient|Field due to Infinite Sheet

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental derivation of thermal resistance and temperature gradients in non-Cartesian geometries using the Gauss's Law analogy for steady-state heat flux.

The 'Lethal Collision' occurs when students either blindly apply the standard spherical temperature profile $T(r) \propto 1/r$ (which assumes constant $k$ ) or use the linear slab formula $R = L/kA$ while ignoring the $4\pi r^2$ area expansion.

By introducing a radially varying thermal conductivity $k(r) \propto 1/r$ , the $r^2$ area term is partially canceled, resulting in a logarithmic temperature profile identical to a standard cylinder.

This traps students who rely on rote memorization of geometric formulas rather than integrating Fourier's Law from first principles.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Thermal Conductivity in Series and Parallel
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free