---
title: "A solid insulating sphere of radius \\(R\\) has a uniform positive volume charge density \\(+\\rho\\). An off-center spherical cavity of radius \\(b\\) (where \\(b < R/2\\)) is scooped out from the interior of the sphere, such that its center is located at a fixed displacement vector \\(\\vec{d}\\) from the center of the large sphere, with \\(|\\vec{d}| + b < R\\). Which of the following statements correctly describes the electric field \\(\\vec{E}\\) inside the cavity and provides the correct justification?"
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url: "https://nerd-notes.com/ubq/124493/"
date_modified: "2026-09-28T14:08:17+00:00"
---

# A solid insulating sphere of radius \(R\) has a uniform positive volume charge density \(+\rho\). An off-center spherical cavity of radius \(b\) (where \(b < R/2\)) is scooped out from the interior of the sphere, such that its center is located at a fixed displacement vector \(\vec{d}\) from the center of the large sphere, with \(|\vec{d}| + b < R\). Which of the following statements correctly describes the electric field \(\vec{E}\) inside the cavity and provides the correct justification?

A solid insulating sphere of radius \(R\) has a uniform positive volume charge density \(+\rho\). An off-center spherical cavity of radius \(b\) (where \(b < R/2\)) is scooped out from the interior of the sphere, such that its center is located at a fixed displacement vector \(\vec{d}\) from the center of the large sphere, with \(|\vec{d}| + b < R\). Which of the following statements correctly describes the electric field \(\vec{E}\) inside the cavity and provides the correct justification?

![A circular cross section of a large sphere of radius \(R\) with center labeled \(O\). A smaller circular cavity of radius \(b\) lies entirely inside the large circle, offset to the right of center \(O\). A vector arrow labeled \(\vec{d}\) extends from \(O\) to the center of the circular cavity. The region of the large circle outside the cavity has light gray shading. A point \(P\) is marked inside the cavity. A position vector arrow labeled \(\vec{r}\) extends from \(O\) to point \(P\), and a dashed vector arrow labeled \(\vec{r}'\) extends from the center of the cavity to point \(P\). A radius line labeled \(R\) extends from \(O\) to the top edge of the large circle. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604497-RoYbxy.jpg)

- **A.** The electric field inside the cavity is zero everywhere because the hollow region contains no enclosed charge, so any closed Gaussian surface drawn entirely within the cavity encloses zero net charge.
- **B.** The electric field inside the cavity is non-uniform and directed radially outward from the center of the large sphere because the field is governed solely by the enclosed charge of the large sphere at distance \(r\), while the hollow cavity contributes no field.
- **C.** The electric field inside the cavity is non-uniform and varies inversely with the square of the distance from the cavity center because the excavated region acts as an effective negative point charge concentrated at the cavity center.
- **D.** The electric field inside the cavity is uniform in magnitude and direction because the linear radial dependence of the interior field of a uniformly charged sphere results in a vector difference that depends only on the constant displacement vector \(\vec{d}\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124493/*