---
title: "Two long, straight cylindrical wires, 1 and 2, each have radius \\(R\\) and carry the same total current \\(I\\). Wire 1 has a non-uniform current density given by \\(J_1(r) = C r\\) for \\(r \\le R\\), where \\(C\\) is a constant and \\(r\\) is the radial distance from the central axis. Wire 2 has a uniform current density \\(J_2\\).  What is the ratio \\(B_1 / B_2\\) of the magnetic field magnitude \\(B_1\\) inside Wire 1 at \\(r = R/2\\) to the magnetic field magnitude \\(B_2\\) inside Wire 2 at \\(r = R/2\\)?"
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url: "https://nerd-notes.com/ubq/118600/"
date_modified: "2026-08-04T08:11:44+00:00"
---

# Two long, straight cylindrical wires, 1 and 2, each have radius \(R\) and carry the same total current \(I\). Wire 1 has a non-uniform current density given by \(J_1(r) = C r\) for \(r \le R\), where \(C\) is a constant and \(r\) is the radial distance from the central axis. Wire 2 has a uniform current density \(J_2\).

What is the ratio \(B_1 / B_2\) of the magnetic field magnitude \(B_1\) inside Wire 1 at \(r = R/2\) to the magnetic field magnitude \(B_2\) inside Wire 2 at \(r = R/2\)?

Two long, straight cylindrical wires, 1 and 2, each have radius \(R\) and carry the same total current \(I\). Wire 1 has a non-uniform current density given by \(J_1(r) = C r\) for \(r \le R\), where \(C\) is a constant and \(r\) is the radial distance from the central axis. Wire 2 has a uniform current density \(J_2\).

What is the ratio \(B_1 / B_2\) of the magnetic field magnitude \(B_1\) inside Wire 1 at \(r = R/2\) to the magnetic field magnitude \(B_2\) inside Wire 2 at \(r = R/2\)?

![Two side-by-side circular cross-sections of cylindrical wires, labeled Wire 1 on the left and Wire 2 on the right. Each circle has radius \(R\), indicated by a dashed radius line from the center to the outer boundary labeled \(R\). Inside each circle, a smaller concentric dashed circle represents a path of radius \(r = R/2\), labeled \(r = R/2\). For Wire 1, gray shading increases in darkness from the center to the outer edge to represent increasing current density \(J_1(r) = Cr\). For Wire 2, uniform gray shading fills the circle to represent uniform current density \(J_2\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831103-QW6EQo.jpg)

- **A.** \(\dfrac{1}{4}\)
- **B.** \(\dfrac{1}{3}\)
- **C.** \(\dfrac{1}{2}\)
- **D.** \(1\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118600/*