---
title: "A spherical capacitor consists of a solid conducting sphere of radius \\(a\\) surrounded by a concentric conducting spherical shell of inner radius \\(b\\). The space between the conductors (\\(a < r < b\\)) is filled with a non-uniform dielectric material whose dielectric constant varies radially according to \\(\\kappa(r) = \\kappa_0 \\dfrac{a}{r}\\), where \\(\\kappa_0\\) is a dimensionless constant. Which of the following expressions represents the capacitance of this capacitor?"
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date_modified: "2026-08-04T08:08:24+00:00"
---

# A spherical capacitor consists of a solid conducting sphere of radius \(a\) surrounded by a concentric conducting spherical shell of inner radius \(b\). The space between the conductors (\(a < r < b\)) is filled with a non-uniform dielectric material whose dielectric constant varies radially according to \(\kappa(r) = \kappa_0 \dfrac{a}{r}\), where \(\kappa_0\) is a dimensionless constant. Which of the following expressions represents the capacitance of this capacitor?

A spherical capacitor consists of a solid conducting sphere of radius \(a\) surrounded by a concentric conducting spherical shell of inner radius \(b\). The space between the conductors (\(a < r < b\)) is filled with a non-uniform dielectric material whose dielectric constant varies radially according to \(\kappa(r) = \kappa_0 \dfrac{a}{r}\), where \(\kappa_0\) is a dimensionless constant. Which of the following expressions represents the capacitance of this capacitor?

![A cross-sectional diagram of two concentric circular conductors centered at the origin. The inner conducting sphere has radius a, shown as a shaded light gray circle of radius a with a black dashed line from the center to its outer edge labeled a. The outer conducting shell has inner radius b, shown as a thin circular ring of radius b concentric with the inner sphere, with a dashed line from the center to the ring labeled b. The annular region between r = a and r = b has a continuous radial gradient shading, dark gray near r = a and fading lighter toward r = b, labeled with the expression \kappa(r) = \kappa_0 \frac{a}{r}. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830904-xSZf8T.jpg)

- **A.** \(C = \dfrac{4\pi \varepsilon_0 \kappa_0 a b}{b - a}\)
- **B.** \(C = \dfrac{4\pi \varepsilon_0 \kappa_0 a}{\ln\left(\dfrac{b}{a}\right)}\)
- **C.** \(C = \dfrac{4\pi \varepsilon_0 \kappa_0 b}{\ln\left(\dfrac{b}{a}\right)}\)
- **D.** \(C = \dfrac{2\pi \varepsilon_0 \kappa_0 (b^2 - a^2)}{a}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118247/*