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title: "A spherical capacitor consists of two concentric conducting spherical shells of radii \\(a\\) and \\(b\\) (where \\(b > a\\)) separated by a vacuum. The capacitance of this configuration is given by \\(C = 4\\pi\\varepsilon_0\\dfrac{ab}{b-a}\\). Let \\(d = b – a\\) represent the radial separation between the shells. In the limit where \\(d \\ll a\\), which of the following expressions and physical justifications correctly describes the behavior of the capacitance in terms of the inner shell surface area \\(A = 4\\pi a^2\\) and separation \\(d\\)?"
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date_modified: "2026-08-23T04:58:19+00:00"
---

# A spherical capacitor consists of two concentric conducting spherical shells of radii \(a\) and \(b\) (where \(b > a\)) separated by a vacuum. The capacitance of this configuration is given by \(C = 4\pi\varepsilon_0\dfrac{ab}{b-a}\). Let \(d = b – a\) represent the radial separation between the shells. In the limit where \(d \ll a\), which of the following expressions and physical justifications correctly describes the behavior of the capacitance in terms of the inner shell surface area \(A = 4\pi a^2\) and separation \(d\)?

A spherical capacitor consists of two concentric conducting spherical shells of radii \(a\) and \(b\) (where \(b > a\)) separated by a vacuum. The capacitance of this configuration is given by \(C = 4\pi\varepsilon_0\dfrac{ab}{b-a}\). Let \(d = b - a\) represent the radial separation between the shells. In the limit where \(d \ll a\), which of the following expressions and physical justifications correctly describes the behavior of the capacitance in terms of the inner shell surface area \(A = 4\pi a^2\) and separation \(d\)?

![A cross-sectional view of two concentric circles centered at a common origin. The inner circle has radius labeled \(a\) indicated by a solid arrow from the center to its circumference at a 45-degree angle. The outer circle has radius labeled \(b\) indicated by a solid arrow from the center to its circumference at a 135-degree angle. The radial gap between the inner and outer circle is labeled \(d\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461099-Y8Egh0.jpg)

- **A.** \(C \approx \dfrac{\varepsilon_0 A}{d}\), because the curvature becomes negligible locally as \(d/a \to 0\), so the system asymptotically approaches a parallel-plate capacitor of area \(A\) and separation \(d\).
- **B.** \(C \approx \dfrac{\varepsilon_0 A}{a}\), because the capacitance is dominated by the self-capacitance of an isolated sphere of radius \(a\) as the separation \(d\) vanishes.
- **C.** \(C \approx \dfrac{\varepsilon_0 A d}{a^2}\), because the potential difference between the shells approaches zero linearly with \(d\), causing the capacitance to scale proportionally with \(d\).
- **D.** \(C \approx \dfrac{2\varepsilon_0 A}{d}\), because the inner and outer spherical shells each contribute an independent area of \(A = 4\pi a^2\) in parallel across the gap \(d\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121157/*