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Question #42809

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The Question

A cylindrical capacitor of length $L$ consists of an inner conducting cylinder of radius $a$ and an outer conducting cylinder of radius $b$ (where $L \gg b$ ). The space between the cylinders is completely filled with a solid dielectric material. At an initial temperature $T$ , the dielectric constant varies radially as $\kappa(r) = \frac{r_0}{r}$ , where $r_0$ is a positive constant with dimensions of length. The conductors and the dielectric are made of materials that all have the same coefficient of linear thermal expansion $\alpha$ . Assume that during thermal expansion, the local dielectric constant of any specific material element of the solid dielectric remains invariant. The capacitor is charged to a total charge $Q$ and then electrically isolated. If the temperature of the entire system is increased by a small amount $\Delta T$ (such that $\alpha \Delta T \ll 1$ ), what is the approximate change in the electrostatic energy of the system, $\Delta U$ ?

Options

A.
$-\frac{\alpha \Delta T Q^2 (b-a)}{4\pi \epsilon_0 L r_0}$
B.
0
C.
$+\frac{\alpha \Delta T Q^2 (b-a)}{4\pi \epsilon_0 L r_0}$
D.
$-\frac{\alpha \Delta T Q^2 (b-a)}{2\pi \epsilon_0 L r_0}$

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

Non-Uniform Dielectric Filling|Thermal Expansion of Area and Volume|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the transformation of a scalar field (dielectric constant) under spatial scaling due to thermal expansion, merging electrodynamics with thermodynamics.

A common trap is assuming the spatial functional form $\kappa(r) = r_0/r$ remains fixed in space (Eulerian perspective), which incorrectly leads to $C' = C$ and $\Delta U = 0$ (Option B).

Students must recognize that the dielectric is a solid material; as it expands, a material point moves to $r' = r(1+\alpha \Delta T)$ .

Because the local dielectric constant is invariant, the new spatial profile becomes $\kappa'(r') = r_0(1+\alpha \Delta T)/r'$ .

Failing to apply this Lagrangian perspective, or forgetting that the cylinder length $L$ also expands, leads to incorrect capacitance scaling and the wrong sign or magnitude for the energy change.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Non-Uniform Dielectric Filling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free