---
title: "A long cylindrical wire of radius \\(R\\) carries a non-uniform current density parallel to its central axis given by \\(J(r) = C r^2\\), where \\(r\\) is the radial distance from the central axis and \\(C\\) is a constant. Let \\(J_{\\text{max}}\\) be the maximum current density at the outer surface of the wire, and let \\(A = \\pi R^2\\) be the cross-sectional area of the wire. What is the total current \\(I\\) flowing through the wire?"
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url: "https://nerd-notes.com/ubq/118372/"
date_modified: "2026-08-04T08:09:45+00:00"
---

# A long cylindrical wire of radius \(R\) carries a non-uniform current density parallel to its central axis given by \(J(r) = C r^2\), where \(r\) is the radial distance from the central axis and \(C\) is a constant. Let \(J_{\text{max}}\) be the maximum current density at the outer surface of the wire, and let \(A = \pi R^2\) be the cross-sectional area of the wire. What is the total current \(I\) flowing through the wire?

A long cylindrical wire of radius \(R\) carries a non-uniform current density parallel to its central axis given by \(J(r) = C r^2\), where \(r\) is the radial distance from the central axis and \(C\) is a constant. Let \(J_{\text{max}}\) be the maximum current density at the outer surface of the wire, and let \(A = \pi R^2\) be the cross-sectional area of the wire. What is the total current \(I\) flowing through the wire?

![A circular cross section of a cylindrical wire of radius R centered at the origin. A concentric thin shaded ring of radius r and thickness dr is shown within the cross section. A radial arrow points from the center to the outer edge, labeled R. A smaller radial arrow points from the center to the shaded ring, labeled r. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830985-rOJy9B.jpg)

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

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118372/*