Before You Begin

Question #42894

Difficulty: 8/10 | 🟢 Free

The Question

A spherical capacitor consists of an inner conducting sphere of radius $a$ and an outer conducting spherical shell of inner radius $b$ . The space between the conductors ( $a \le r \le b$ ) is filled with a compressible dielectric fluid. Initially uniform, the system is subjected to a steady-state radial temperature gradient. Due to non-uniform thermal expansion, the local particle density of the fluid varies. According to the material's volume-polarizability model, this density gradient results in a spatially varying dielectric constant given by $K(r) = K_0 \frac{b}{r}$ , where $r$ is the radial distance from the center and $K_0$ is the dielectric constant at the outer shell. What is the effective capacitance of this system?

Options

A.
$\frac{4\pi \epsilon_0 K_0 b}{\ln(b/a)}$
B.
$\frac{4\pi \epsilon_0 K_0 a b}{b - a}$
C.
$\frac{4\pi \epsilon_0 K_0 a b^2}{(b-a)^2} \ln\left(\frac{b}{a}\right)$
D.
$\frac{4\pi \epsilon_0 K_0 b^2}{b - a}$

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

Dielectric Constant (K) Shift|Expansion of Cavities in Solids|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply Gauss's Law in non-uniform dielectric media.

A common trap is assuming that the standard capacitance formula $C = \frac{4\pi \epsilon_0 K a b}{b-a}$ can be used by simply plugging in an average value of $K$ , or that the electric field retains its $1/r^2$ dependence.

Students often fail to realize that the fundamental relation $\nabla \cdot \vec{D} = \rho_f$ dictates that $D(r)$ , not $E(r)$ , is independent of the dielectric profile.

By forcing the dielectric constant to vary as $1/r$ due to thermal expansion, the electric field dependence shifts to $1/r$ , fundamentally altering the potential difference integral to a logarithmic form rather than the standard rational function.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Dielectric Constant (K) Shift
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free