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title: "A solid, uncharged conducting sphere of outer radius \\(R_2\\) is centered at the origin \\(O\\) and contains an off-center spherical cavity of radius \\(R_1\\) whose center is displaced by a distance \\(d\\) along the \\(x\\)-axis from \\(O\\), where \\(d + R_1 < R_2\\). A point charge \\(+Q\\) is fixed at the center of the cavity. Assuming the electric potential is zero at infinity, which of the following is the correct expression for the electric potential \\(V\\) at a point inside the cavity at a distance \\(r_1\\) from the charge, where \\(r_1 < R_1\\)?"
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url: "https://nerd-notes.com/ubq/121135/"
date_modified: "2026-08-23T04:58:15+00:00"
---

# A solid, uncharged conducting sphere of outer radius \(R_2\) is centered at the origin \(O\) and contains an off-center spherical cavity of radius \(R_1\) whose center is displaced by a distance \(d\) along the \(x\)-axis from \(O\), where \(d + R_1 < R_2\). A point charge \(+Q\) is fixed at the center of the cavity. Assuming the electric potential is zero at infinity, which of the following is the correct expression for the electric potential \(V\) at a point inside the cavity at a distance \(r_1\) from the charge, where \(r_1 < R_1\)?

A solid, uncharged conducting sphere of outer radius \(R_2\) is centered at the origin \(O\) and contains an off-center spherical cavity of radius \(R_1\) whose center is displaced by a distance \(d\) along the \(x\)-axis from \(O\), where \(d + R_1 < R_2\). A point charge \(+Q\) is fixed at the center of the cavity. Assuming the electric potential is zero at infinity, which of the following is the correct expression for the electric potential \(V\) at a point inside the cavity at a distance \(r_1\) from the charge, where \(r_1 < R_1\)?

![A cross-sectional diagram showing a large circle of radius \(R_2\) centered at origin \(O\). The region inside the large circle is shaded with light gray representing conducting material, except for a smaller unshaded circular region of radius \(R_1\) located entirely inside the large circle. The center of the small circle is shifted horizontally to the right of \(O\) by a distance \(d\). At the center of the small circle is a solid black dot labeled \(+Q\). A dashed line extends from \(O\) to the outer perimeter of the large circle labeled \(R_2\). A dashed line extends from the center of the small circle to its boundary labeled \(R_1\). A point labeled \(P\) is marked inside the cavity with an arrow pointing from \(+Q\) to \(P\) labeled \(r_1\). A horizontal dimension line between \(O\) and the center of the cavity is labeled \(d\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461095-LLUiQx.jpg)

- **A.** \(\dfrac{Q}{4\pi\varepsilon_0}\left(\dfrac{1}{r_1} - \dfrac{1}{R_1}\right)\)
- **B.** \(\dfrac{Q}{4\pi\varepsilon_0}\left(\dfrac{1}{r_1} + \dfrac{1}{R_2}\right)\)
- **C.** \(\dfrac{Q}{4\pi\varepsilon_0}\left(\dfrac{1}{r_1} - \dfrac{1}{R_1} + \dfrac{1}{R_2}\right)\)
- **D.** \(\dfrac{Q}{4\pi\varepsilon_0}\left(\dfrac{1}{r_1} - \dfrac{1}{R_1} + \dfrac{1}{R_2 - d}\right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121135/*