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title: "Three thin, concentric conducting spherical shells have radii \\(a\\), \\(b\\), and \\(c\\), where \\(a < b < c\\). The innermost shell of radius \\(a\\) and the outermost shell of radius \\(c\\) are each connected to ground such that their electric potentials are zero (\\(V = 0\\)). The middle shell of radius \\(b\\) is connected to a power supply that maintains it at a constant positive electric potential \\(V_0\\). In terms of the given quantities and fundamental constants, what is the total electric charge \\(Q_b\\) residing on the middle shell?"
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date_modified: "2026-08-23T04:58:18+00:00"
---

# Three thin, concentric conducting spherical shells have radii \(a\), \(b\), and \(c\), where \(a < b < c\). The innermost shell of radius \(a\) and the outermost shell of radius \(c\) are each connected to ground such that their electric potentials are zero (\(V = 0\)). The middle shell of radius \(b\) is connected to a power supply that maintains it at a constant positive electric potential \(V_0\). In terms of the given quantities and fundamental constants, what is the total electric charge \(Q_b\) residing on the middle shell?

Three thin, concentric conducting spherical shells have radii \(a\), \(b\), and \(c\), where \(a < b < c\). The innermost shell of radius \(a\) and the outermost shell of radius \(c\) are each connected to ground such that their electric potentials are zero (\(V = 0\)). The middle shell of radius \(b\) is connected to a power supply that maintains it at a constant positive electric potential \(V_0\). In terms of the given quantities and fundamental constants, what is the total electric charge \(Q_b\) residing on the middle shell?

![Cross-sectional schematic showing three concentric thin circular outlines representing spherical shells centered at a common origin. The innermost circle has an arrow from the center to its edge labeled a pointing toward the upper-right. The middle circle has an arrow from the center to its edge labeled b pointing horizontally to the right. The outermost circle has an arrow from the center to its edge labeled c pointing toward the lower-right. A thin line extends from the innermost circle to a standard three-bar ground symbol. A thin line extends from the middle circle to a terminal circle labeled V_0. A thin line extends from the outermost circle to a standard three-bar ground symbol. No other labels, lines, or text appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461098-dIQeTM.jpg)

- **A.** \(4\pi\varepsilon_0 V_0 \left( \dfrac{ab}{b - a} \right)\)
- **B.** \(4\pi\varepsilon_0 V_0 \left( \dfrac{ac}{c - a} \right)\)
- **C.** \(4\pi\varepsilon_0 V_0 \left( \dfrac{ab}{b - a} + \dfrac{bc}{c - b} \right)\)
- **D.** \(4\pi\varepsilon_0 V_0 \left( \dfrac{ab}{b - a} + b \right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121154/*