---
title: "A long cylindrical wire of radius \\(R\\) carries a total current with a nonuniform current density directed parallel to the axis of the wire. The magnitude of the current density as a function of radial distance \\(r\\) from the central axis is given by \\(J(r) = J_0 \\left(1 – \\dfrac{r}{R}\\right)\\) for \\(0 \\le r \\le R\\), where \\(J_0\\) is a constant. Which of the following expressions represents the total current \\(I\\) passing through a cross section of the wire?"
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url: "https://nerd-notes.com/ubq/118406/"
date_modified: "2026-08-04T08:09:56+00:00"
---

# A long cylindrical wire of radius \(R\) carries a total current with a nonuniform current density directed parallel to the axis of the wire. The magnitude of the current density as a function of radial distance \(r\) from the central axis is given by \(J(r) = J_0 \left(1 – \dfrac{r}{R}\right)\) for \(0 \le r \le R\), where \(J_0\) is a constant. Which of the following expressions represents the total current \(I\) passing through a cross section of the wire?

A long cylindrical wire of radius \(R\) carries a total current with a nonuniform current density directed parallel to the axis of the wire. The magnitude of the current density as a function of radial distance \(r\) from the central axis is given by \(J(r) = J_0 \left(1 - \dfrac{r}{R}\right)\) for \(0 \le r \le R\), where \(J_0\) is a constant. Which of the following expressions represents the total current \(I\) passing through a cross section of the wire?

![A circular cross-section of a cylindrical conductor of radius \(R\). A dashed black crosshair marks the center point. A solid black circle of radius \(R\) bounds the cross-section. A single straight black arrow extends from the center point to the outer circular boundary at an angle of 30 degrees above the horizontal, labeled \(R\). Three small circles containing central black dots are positioned along the horizontal axis: one near the center labeled \(J_0\), one halfway to the edge, and one near the outer edge. A light gray gradient fills the circle, darker near the center and fading to white at the outer boundary. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830996-EVccht.jpg)

- **A.** \(\dfrac{\pi J_0 R^2}{3}\)
- **B.** \(\dfrac{\pi J_0 R^2}{2}\)
- **C.** \(\dfrac{2\pi J_0 R^2}{3}\)
- **D.** \(\dfrac{\pi J_0 R^2}{6}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118406/*