---
title: "A long, solid cylindrical wire of radius \\(R\\) carries current directed parallel to its central axis. The current density within the wire is non-uniform and depends on the radial distance \\(r\\) from the central axis according to \\(J(r) = C r^2\\), where \\(C\\) is a positive constant. Which of the following integral expressions represents a correct setup using Ampère’s law to determine the magnitude of the magnetic field \\(B(r)\\) at a distance \\(r < R\\) from the central axis?"
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url: "https://nerd-notes.com/ubq/121228/"
date_modified: "2026-08-23T04:58:57+00:00"
---

# A long, solid cylindrical wire of radius \(R\) carries current directed parallel to its central axis. The current density within the wire is non-uniform and depends on the radial distance \(r\) from the central axis according to \(J(r) = C r^2\), where \(C\) is a positive constant. Which of the following integral expressions represents a correct setup using Ampère’s law to determine the magnitude of the magnetic field \(B(r)\) at a distance \(r < R\) from the central axis?

A long, solid cylindrical wire of radius \(R\) carries current directed parallel to its central axis. The current density within the wire is non-uniform and depends on the radial distance \(r\) from the central axis according to \(J(r) = C r^2\), where \(C\) is a positive constant. Which of the following integral expressions represents a correct setup using Ampère's law to determine the magnitude of the magnetic field \(B(r)\) at a distance \(r < R\) from the central axis?

![A cross-sectional view of a solid cylinder of radius \(R\). A large solid circle represents the boundary of the cylinder, centered at the origin. A solid black dot marks the central axis at the center. A solid arrow points from the center toward the upper-right edge at 45 degrees, labeled \(R\). Concentric with the solid circle is an inner dashed circle of radius \(r\), representing an Amperian loop where \(r < R\). A solid arrow points from the center toward the lower-right at 45 degrees to the dashed circle, labeled \(r\). Four small circles with central dots appear within the conductor cross-section to symbolize current directed out of the page. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461136-6vFWUd.jpg)

- **A.** \(B(r) = \dfrac{\mu_0}{2\pi r} \displaystyle\int_0^r C r'^2 \, dr'\)
- **B.** \(B(r) = \dfrac{\mu_0}{2\pi r} \displaystyle\int_0^R C r'^2 (2\pi r') \, dr'\)
- **C.** \(B(r) = \dfrac{\mu_0}{\pi r^2} \displaystyle\int_0^r C r'^2 (2\pi r') \, dr'\)
- **D.** \(B(r) = \dfrac{\mu_0}{2\pi r} \displaystyle\int_0^r C r'^2 (2\pi r') \, dr'\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121228/*