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title: "A spherical capacitor consists of two thin, concentric conducting spherical shells of radii \\(a\\) and \\(b\\), where \\(a < b\\). The region between the shells is completely filled with an inhomogeneous dielectric material whose permittivity varies with radial distance \\(r\\) according to \\(\\varepsilon(r) = \\varepsilon_0 \\left(\\dfrac{r}{a}\\right)\\), where \\(\\varepsilon_0\\) is the permittivity of free space. A charge \\(+Q\\) resides on the inner shell and a charge \\(-Q\\) resides on the outer shell. In terms of \\(\\varepsilon_0\\), \\(a\\), and \\(b\\), which of the following is the capacitance of this spherical capacitor?"
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url: "https://nerd-notes.com/ubq/121137/"
date_modified: "2026-08-23T04:58:16+00:00"
---

# A spherical capacitor consists of two thin, concentric conducting spherical shells of radii \(a\) and \(b\), where \(a < b\). The region between the shells is completely filled with an inhomogeneous dielectric material whose permittivity varies with radial distance \(r\) according to \(\varepsilon(r) = \varepsilon_0 \left(\dfrac{r}{a}\right)\), where \(\varepsilon_0\) is the permittivity of free space. A charge \(+Q\) resides on the inner shell and a charge \(-Q\) resides on the outer shell. In terms of \(\varepsilon_0\), \(a\), and \(b\), which of the following is the capacitance of this spherical capacitor?

A spherical capacitor consists of two thin, concentric conducting spherical shells of radii \(a\) and \(b\), where \(a < b\). The region between the shells is completely filled with an inhomogeneous dielectric material whose permittivity varies with radial distance \(r\) according to \(\varepsilon(r) = \varepsilon_0 \left(\dfrac{r}{a}\right)\), where \(\varepsilon_0\) is the permittivity of free space. A charge \(+Q\) resides on the inner shell and a charge \(-Q\) resides on the outer shell. In terms of \(\varepsilon_0\), \(a\), and \(b\), which of the following is the capacitance of this spherical capacitor?

![A cross-sectional view of two concentric circles centered at the origin. The inner circle has radius a, drawn with a solid line. A straight dimension arrow points from the center to the inner circle at an angle of 135 degrees, labeled a. The outer circle has radius b, drawn with a solid line. A straight dimension arrow points from the center to the outer circle at an angle of 45 degrees, labeled b. The region between the inner and outer circles is shaded with a smooth radial grayscale gradient, labeled \(\varepsilon(r) = \varepsilon_0(r/a)\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461096-Avht8D.jpg)

- **A.** \(\dfrac{4\pi \varepsilon_0 a}{\ln(b/a)}\)
- **B.** \(\dfrac{4\pi \varepsilon_0 a b^2}{b^2 - a^2}\)
- **C.** \(\dfrac{8\pi \varepsilon_0 a b^2}{b^2 - a^2}\)
- **D.** \(\dfrac{8\pi \varepsilon_0 a^2 b}{b^2 - a^2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121137/*