---
title: "An uncharged, infinitely long hollow conducting cylindrical shell has inner radius \\(a\\) and outer radius \\(b\\). A thin line with uniform linear charge density \\(+\\lambda\\) is placed inside the cavity, parallel to the central cylinder axis but displaced off-center by a distance \\(d\\), where \\(0 < d < a\\). The system is allowed to reach electrostatic equilibrium.  | Choice | Potential difference \\(\\Delta V = V_{\\text{outer}} – V_{\\text{inner}}\\) | Total induced charge per unit length \\(\\int_0^{2\\pi} \\sigma(\\theta) a \\, d\\theta\\) | | :— | :— | :— | | A | \\(0\\) | \\(-\\lambda\\) | | B | \\(0\\) | \\(-\\lambda\\left(1 – \\dfrac{d}{a}\\right)\\) | | C | \\(\\dfrac{\\lambda}{2\\pi\\varepsilon_0}\\ln\\left(\\dfrac{b}{a}\\right)\\) | \\(-\\lambda\\) | | D | \\(\\dfrac{\\lambda}{2\\pi\\varepsilon_0}\\ln\\left(\\dfrac{b}{a}\\right)\\) | \\(-\\lambda\\left(1 – \\dfrac{d}{a}\\right)\\) |  Which of the following correctly pairs the electric potential difference \\(\\Delta V\\) between the outer and inner surfaces of the shell with the total induced surface charge per unit length on the inner cavity wall?"
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url: "https://nerd-notes.com/ubq/118324/"
date_modified: "2026-08-04T08:09:04+00:00"
---

# An uncharged, infinitely long hollow conducting cylindrical shell has inner radius \(a\) and outer radius \(b\). A thin line with uniform linear charge density \(+\lambda\) is placed inside the cavity, parallel to the central cylinder axis but displaced off-center by a distance \(d\), where \(0 < d < a\). The system is allowed to reach electrostatic equilibrium.

| Choice | Potential difference \(\Delta V = V_{\text{outer}} – V_{\text{inner}}\) | Total induced charge per unit length \(\int_0^{2\pi} \sigma(\theta) a \, d\theta\) |
| :— | :— | :— |
| A | \(0\) | \(-\lambda\) |
| B | \(0\) | \(-\lambda\left(1 – \dfrac{d}{a}\right)\) |
| C | \(\dfrac{\lambda}{2\pi\varepsilon_0}\ln\left(\dfrac{b}{a}\right)\) | \(-\lambda\) |
| D | \(\dfrac{\lambda}{2\pi\varepsilon_0}\ln\left(\dfrac{b}{a}\right)\) | \(-\lambda\left(1 – \dfrac{d}{a}\right)\) |

Which of the following correctly pairs the electric potential difference \(\Delta V\) between the outer and inner surfaces of the shell with the total induced surface charge per unit length on the inner cavity wall?

An uncharged, infinitely long hollow conducting cylindrical shell has inner radius \(a\) and outer radius \(b\). A thin line with uniform linear charge density \(+\lambda\) is placed inside the cavity, parallel to the central cylinder axis but displaced off-center by a distance \(d\), where \(0 < d < a\). The system is allowed to reach electrostatic equilibrium.

| Choice | Potential difference \(\Delta V = V_{\text{outer}} - V_{\text{inner}}\) | Total induced charge per unit length \(\int_0^{2\pi} \sigma(\theta) a \, d\theta\) |
| :--- | :--- | :--- |
| A | \(0\) | \(-\lambda\) |
| B | \(0\) | \(-\lambda\left(1 - \dfrac{d}{a}\right)\) |
| C | \(\dfrac{\lambda}{2\pi\varepsilon_0}\ln\left(\dfrac{b}{a}\right)\) | \(-\lambda\) |
| D | \(\dfrac{\lambda}{2\pi\varepsilon_0}\ln\left(\dfrac{b}{a}\right)\) | \(-\lambda\left(1 - \dfrac{d}{a}\right)\) |

Which of the following correctly pairs the electric potential difference \(\Delta V\) between the outer and inner surfaces of the shell with the total induced surface charge per unit length on the inner cavity wall?

![Cross-sectional view in the xy-plane of a long hollow conducting cylinder centered at the origin O. Two concentric circles are drawn: an inner circle of radius a and an outer circle of radius b, with the region between them filled with light gray shading to represent the conducting material. A small filled circle represents a line charge parallel to the z-axis, located on the positive x-axis at a distance d from the origin O (where 0 < d < a), labeled +lambda. A radial dashed line from the origin O to the inner circle is labeled a, and a radial dashed line from the origin O to the outer circle is labeled b. A horizontal line segment from the origin O to the line charge is labeled d. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830944-XoOfnD.jpg)

- **A.** \(\Delta V = 0\) and \(-\lambda\)
- **B.** \(\Delta V = 0\) and \(-\lambda\left(1 - \dfrac{d}{a}\right)\)
- **C.** \(\Delta V = \dfrac{\lambda}{2\pi\varepsilon_0}\ln\left(\dfrac{b}{a}\right)\) and \(-\lambda\)
- **D.** \(\Delta V = \dfrac{\lambda}{2\pi\varepsilon_0}\ln\left(\dfrac{b}{a}\right)\) and \(-\lambda\left(1 - \dfrac{d}{a}\right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118324/*