---
title: "A long, thick-walled hollow cylindrical solenoid has inner radius \\(a\\) and outer radius \\(b\\). The solenoid carries a non-uniform azimuthal volume current density \\(\\vec{J} = J_0 \\left(\\dfrac{r}{a}\\right) \\hat{\\theta}\\) within its wall (\\(a \\le r \\le b\\)), where \\(J_0\\) is a positive constant and \\(r\\) is the radial distance from the central axis. Outside the solenoid (\\(r > b\\)), the magnetic field is zero. Which of the following expressions gives the magnitude of the magnetic field \\(B(r)\\) inside the solenoid wall, in the region \\(a \\le r \\le b\\)?"
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url: "https://nerd-notes.com/ubq/118614/"
date_modified: "2026-08-04T08:11:50+00:00"
---

# A long, thick-walled hollow cylindrical solenoid has inner radius \(a\) and outer radius \(b\). The solenoid carries a non-uniform azimuthal volume current density \(\vec{J} = J_0 \left(\dfrac{r}{a}\right) \hat{\theta}\) within its wall (\(a \le r \le b\)), where \(J_0\) is a positive constant and \(r\) is the radial distance from the central axis. Outside the solenoid (\(r > b\)), the magnetic field is zero. Which of the following expressions gives the magnitude of the magnetic field \(B(r)\) inside the solenoid wall, in the region \(a \le r \le b\)?

A long, thick-walled hollow cylindrical solenoid has inner radius \(a\) and outer radius \(b\). The solenoid carries a non-uniform azimuthal volume current density \(\vec{J} = J_0 \left(\dfrac{r}{a}\right) \hat{\theta}\) within its wall (\(a \le r \le b\)), where \(J_0\) is a positive constant and \(r\) is the radial distance from the central axis. Outside the solenoid (\(r > b\)), the magnetic field is zero. Which of the following expressions gives the magnitude of the magnetic field \(B(r)\) inside the solenoid wall, in the region \(a \le r \le b\)?

![A perspective view of a long, hollow thick-walled cylinder centered on a vertical dashed axis labeled z. The cylinder has an inner hollow core of radius a, indicated by a dimension arrow from the central axis to the inner boundary labeled a. The outer surface of the cylinder has radius b, indicated by a dimension arrow from the central axis to the outer boundary labeled b. Curved arrows wrapped circumferentially around the cylindrical wall indicate an azimuthal current density J in the counterclockwise direction when viewed from above. The wall thickness between radius a and radius b is shaded light gray. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831110-igJolP.jpg)

- **A.** \(\dfrac{\mu_0 J_0}{3 a r} (r^3 - a^3)\)
- **B.** \(\dfrac{\mu_0 J_0}{2a} (r^2 - a^2)\)
- **C.** \(\dfrac{\mu_0 J_0}{2a} (b^2 - r^2)\)
- **D.** \(\dfrac{\mu_0 J_0}{a} (b^2 - r^2)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118614/*