---
title: "A spherical capacitor consists of two concentric, thin conducting spherical shells of radii \\(a\\) and \\(b\\), where \\(a < b\\). The space between the shells is completely filled with a homogeneous, weakly conducting material having a dielectric constant \\(\\kappa\\) and a uniform resistivity \\(\\rho\\). The capacitor is initially charged so that the inner shell carries charge \\(+Q_0\\) and the outer shell carries charge \\(-Q_0\\), and it is then isolated. Which of the following expressions correctly represents the characteristic time constant \\(\\tau\\) for the exponential decay of charge on the capacitor?"
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url: "https://nerd-notes.com/ubq/118276/"
date_modified: "2026-08-04T08:08:36+00:00"
---

# A spherical capacitor consists of two concentric, thin conducting spherical shells of radii \(a\) and \(b\), where \(a < b\). The space between the shells is completely filled with a homogeneous, weakly conducting material having a dielectric constant \(\kappa\) and a uniform resistivity \(\rho\). The capacitor is initially charged so that the inner shell carries charge \(+Q_0\) and the outer shell carries charge \(-Q_0\), and it is then isolated. Which of the following expressions correctly represents the characteristic time constant \(\tau\) for the exponential decay of charge on the capacitor?

A spherical capacitor consists of two concentric, thin conducting spherical shells of radii \(a\) and \(b\), where \(a < b\). The space between the shells is completely filled with a homogeneous, weakly conducting material having a dielectric constant \(\kappa\) and a uniform resistivity \(\rho\). The capacitor is initially charged so that the inner shell carries charge \(+Q_0\) and the outer shell carries charge \(-Q_0\), and it is then isolated. Which of the following expressions correctly represents the characteristic time constant \(\tau\) for the exponential decay of charge on the capacitor?

![A 2D cross-sectional diagram of two concentric circular boundaries representing spherical shells. The inner circle has radius labeled \(a\) with an arrow pointing from the center to the inner boundary. The outer circle has radius labeled \(b\) with an arrow pointing from the center to the outer boundary. The region between the inner and outer circles is shaded light gray and labeled with text \(\kappa, \rho\). The inner circle carries a label \(+Q_0\) and the outer circle carries a label \(-Q_0\). A vertical dashed line passes through the center as a reference axis. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830915-dMoOgZ.jpg)

- **A.** \(\kappa \varepsilon_0 \rho \left(\dfrac{b}{a}\right)\)
- **B.** \(\kappa \varepsilon_0 \rho\)
- **C.** \(\kappa \varepsilon_0 \rho \left(\dfrac{b - a}{a}\right)\)
- **D.** \(\kappa \varepsilon_0 \rho \left(\dfrac{a b}{(b - a)^2}\right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118276/*