---
title: "A hollow cylindrical conductor of inner radius \\(a\\) and outer radius \\(4a\\) carries an electric current parallel to its central axis. The current density within the conductor is non-uniform and given by \\(J(r) = \\dfrac{C}{r}\\), where \\(C\\) is a positive constant and \\(r\\) is the radial distance from the central axis. Three concentric cylindrical shells within the conductor have radial boundaries from \\(a\\) to \\(2a\\) (Region 1), from \\(2a\\) to \\(3a\\) (Region 2), and from \\(3a\\) to \\(4a\\) (Region 3). Which of the following correctly ranks the total currents \\(I_1\\), \\(I_2\\), and \\(I_3\\) passing through Regions 1, 2, and 3, respectively?"
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url: "https://nerd-notes.com/ubq/124787/"
date_modified: "2026-09-28T14:11:15+00:00"
---

# A hollow cylindrical conductor of inner radius \(a\) and outer radius \(4a\) carries an electric current parallel to its central axis. The current density within the conductor is non-uniform and given by \(J(r) = \dfrac{C}{r}\), where \(C\) is a positive constant and \(r\) is the radial distance from the central axis. Three concentric cylindrical shells within the conductor have radial boundaries from \(a\) to \(2a\) (Region 1), from \(2a\) to \(3a\) (Region 2), and from \(3a\) to \(4a\) (Region 3). Which of the following correctly ranks the total currents \(I_1\), \(I_2\), and \(I_3\) passing through Regions 1, 2, and 3, respectively?

A hollow cylindrical conductor of inner radius \(a\) and outer radius \(4a\) carries an electric current parallel to its central axis. The current density within the conductor is non-uniform and given by \(J(r) = \dfrac{C}{r}\), where \(C\) is a positive constant and \(r\) is the radial distance from the central axis. Three concentric cylindrical shells within the conductor have radial boundaries from \(a\) to \(2a\) (Region 1), from \(2a\) to \(3a\) (Region 2), and from \(3a\) to \(4a\) (Region 3). Which of the following correctly ranks the total currents \(I_1\), \(I_2\), and \(I_3\) passing through Regions 1, 2, and 3, respectively?

![A cross-sectional view of four concentric circles sharing a single center point. The innermost circle has radius \(a\) and an unshaded white interior representing an empty cylindrical core. Surrounding this core are three concentric annular shells of equal radial thickness. The first annular shell between radius \(a\) and radius \(2a\) is shaded light gray and labeled 1. The second annular shell between radius \(2a\) and radius \(3a\) is shaded medium gray and labeled 2. The third annular shell between radius \(3a\) and radius \(4a\) is shaded darker gray and labeled 3. A single horizontal dashed line extends from the center point to the right edge, crossing all four boundaries with tick marks and labels \(a\), \(2a\), \(3a\), and \(4a\) located along the line at the respective perimeters. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604675-JCJe8W.jpg)

- **A.** \(I_1 < I_2 < I_3\)
- **B.** \(I_1 > I_2 > I_3\)
- **C.** \(I_2 > I_1 > I_3\)
- **D.** \(I_1 = I_2 = I_3\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124787/*