---
title: "A long, hollow cylindrical conductor has an inner radius \\(a\\) and an outer radius \\(b\\). The conductor carries a total current \\(I_0\\) directed along its length parallel to the central axis. Within the conducting material (\\(a \\le r \\le b\\)), the volume current density is non-uniform and given by \\(J(r) = \\dfrac{C}{r}\\), where \\(C\\) is a constant and \\(r\\) is the radial distance from the central axis. Which of the following expressions represents the magnitude of the magnetic field \\(B(r)\\) as a function of radial distance \\(r\\) in the region \\(a < r < b\\)?"
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url: "https://nerd-notes.com/ubq/121276/"
date_modified: "2026-08-23T04:59:18+00:00"
---

# A long, hollow cylindrical conductor has an inner radius \(a\) and an outer radius \(b\). The conductor carries a total current \(I_0\) directed along its length parallel to the central axis. Within the conducting material (\(a \le r \le b\)), the volume current density is non-uniform and given by \(J(r) = \dfrac{C}{r}\), where \(C\) is a constant and \(r\) is the radial distance from the central axis. Which of the following expressions represents the magnitude of the magnetic field \(B(r)\) as a function of radial distance \(r\) in the region \(a < r < b\)?

A long, hollow cylindrical conductor has an inner radius \(a\) and an outer radius \(b\). The conductor carries a total current \(I_0\) directed along its length parallel to the central axis. Within the conducting material (\(a \le r \le b\)), the volume current density is non-uniform and given by \(J(r) = \dfrac{C}{r}\), where \(C\) is a constant and \(r\) is the radial distance from the central axis. Which of the following expressions represents the magnitude of the magnetic field \(B(r)\) as a function of radial distance \(r\) in the region \(a < r < b\)?

![A cross-sectional view of a thick hollow cylindrical conductor centered on the origin. An inner solid circle of radius \(a\) represents an empty white circular core. An outer solid concentric circle of radius \(b\) encloses the cylinder. The annular region between radii \(a\) and \(b\) is shaded light gray. Exactly four encircled dot symbols \(\odot\) are positioned at 90-degree intervals inside the gray region at the top, bottom, left, and right to indicate current flowing out of the page. A solid arrow labeled \(a\) extends from the center to the inner circular boundary at a 45-degree angle above the horizontal. A second solid arrow labeled \(b\) extends from the center to the outer circular boundary at a 45-degree angle below the horizontal. No other labels, lines, arrows, or markings appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461158-baprWO.jpg)

- **A.** \(\dfrac{\mu_0 I_0 (r^2 - a^2)}{2\pi r (b^2 - a^2)}\)
- **B.** \(\dfrac{\mu_0 I_0 (r - a)}{2\pi r (b - a)}\)
- **C.** \(\dfrac{\mu_0 I_0 \ln(r/a)}{2\pi r \ln(b/a)}\)
- **D.** \(\dfrac{\mu_0 I_0 (r^3 - a^3)}{2\pi r^2 (b^3 - a^3)}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121276/*