---
title: "A long, solid cylindrical conductor of radius \\(R\\) carries a non-uniform current directed along its central axis. The current density inside the conductor varies with radial distance \\(r\\) from the axis according to \\(J(r) = J_0 \\left(1 – \\dfrac{r}{R}\\right)\\) for \\(r \\le R\\), where \\(J_0\\) is a positive constant. Which of the following expressions gives the magnitude of the magnetic field \\(B(r)\\) inside the conductor at a distance \\(r < R\\) from the central axis?"
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url: "https://nerd-notes.com/ubq/118552/"
date_modified: "2026-08-04T08:11:20+00:00"
---

# A long, solid cylindrical conductor of radius \(R\) carries a non-uniform current directed along its central axis. The current density inside the conductor varies with radial distance \(r\) from the axis according to \(J(r) = J_0 \left(1 – \dfrac{r}{R}\right)\) for \(r \le R\), where \(J_0\) is a positive constant. Which of the following expressions gives the magnitude of the magnetic field \(B(r)\) inside the conductor at a distance \(r < R\) from the central axis?

A long, solid cylindrical conductor of radius \(R\) carries a non-uniform current directed along its central axis. The current density inside the conductor varies with radial distance \(r\) from the axis according to \(J(r) = J_0 \left(1 - \dfrac{r}{R}\right)\) for \(r \le R\), where \(J_0\) is a positive constant. Which of the following expressions gives the magnitude of the magnetic field \(B(r)\) inside the conductor at a distance \(r < R\) from the central axis?

![A circular cross section of a long cylindrical conductor centered at the origin, bounded by an outer solid circle of radius R. A straight line segment extends from the center to the outer circle, labeled R. Inside the outer circle, a concentric dashed circle of radius r, where r < R, represents a circular path of radius r. A second straight line segment extends from the center to the dashed circle, labeled r. Eight evenly spaced small circles with central dots are placed inside the conductor to represent current directed out of the page, parallel to the central axis of the cylinder. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831079-hikr6S.jpg)

- **A.** \(\dfrac{\mu_0 J_0 r}{2}\left(1 - \dfrac{r}{R}\right)\)
- **B.** \(\mu_0 J_0 \left(1 - \dfrac{r}{2R}\right)\)
- **C.** \(\mu_0 J_0 r \left(1 - \dfrac{r}{3R}\right)\)
- **D.** \(\dfrac{\mu_0 J_0 r}{2}\left(1 - \dfrac{2r}{3R}\right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118552/*