---
title: "Three thin, concentric conducting spherical shells have radii \\(R\\), \\(2R\\), and \\(3R\\). The innermost shell of radius \\(R\\) and the outermost shell of radius \\(3R\\) are connected to each other by a thin conducting wire. A battery maintaining a potential difference \\(V_0\\) is connected between the middle shell of radius \\(2R\\) and the wire connecting the inner and outer shells. If the system is initially uncharged, what is the total work done by the battery to fully charge the system?"
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url: "https://nerd-notes.com/ubq/118314/"
date_modified: "2026-08-04T08:08:58+00:00"
---

# Three thin, concentric conducting spherical shells have radii \(R\), \(2R\), and \(3R\). The innermost shell of radius \(R\) and the outermost shell of radius \(3R\) are connected to each other by a thin conducting wire. A battery maintaining a potential difference \(V_0\) is connected between the middle shell of radius \(2R\) and the wire connecting the inner and outer shells. If the system is initially uncharged, what is the total work done by the battery to fully charge the system?

Three thin, concentric conducting spherical shells have radii \(R\), \(2R\), and \(3R\). The innermost shell of radius \(R\) and the outermost shell of radius \(3R\) are connected to each other by a thin conducting wire. A battery maintaining a potential difference \(V_0\) is connected between the middle shell of radius \(2R\) and the wire connecting the inner and outer shells. If the system is initially uncharged, what is the total work done by the battery to fully charge the system?

![A cross-sectional diagram showing three concentric circular thin conducting shells labeled from inside out with radii R, 2R, and 3R centered at the origin. A thin conducting wire connects the innermost shell of radius R to the outermost shell of radius 3R, passing through a small isolated line gap in the middle shell. A battery labeled V_0 is connected with its positive terminal wired to the middle shell of radius 2R and its negative terminal wired to the connection between the innermost and outermost shells. No other labels or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830938-dvOh2g.jpg)

- **A.** \( 16 ull\pi \varepsilon_0 R V_0^2 \)
- **B.** \( 32\pi \varepsilon_0 R V_0^2 \)
- **C.** \( 48\pi \varepsilon_0 R V_0^2 \)
- **D.** \( 64\pi \varepsilon_0 R V_0^2 \)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118314/*