---
title: "A long coaxial cable consists of a thin cylindrical inner conductor of radius \\(a\\) and a concentric thin cylindrical outer conductor of radius \\(b\\), where \\(a < b\\). The inner conductor carries a uniformly distributed current \\(I\\) in one direction, and the outer conductor carries an equal current \\(I\\) in the opposite direction. Which of the following integral expressions correctly gives the total magnetic energy \\(U\\) stored in the vacuum region between the conductors (\\(a < r < b\\)) for a section of length \\(\\ell\\)?"
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url: "https://nerd-notes.com/ubq/121370/"
date_modified: "2026-08-23T05:00:17+00:00"
---

# A long coaxial cable consists of a thin cylindrical inner conductor of radius \(a\) and a concentric thin cylindrical outer conductor of radius \(b\), where \(a < b\). The inner conductor carries a uniformly distributed current \(I\) in one direction, and the outer conductor carries an equal current \(I\) in the opposite direction. Which of the following integral expressions correctly gives the total magnetic energy \(U\) stored in the vacuum region between the conductors (\(a < r < b\)) for a section of length \(\ell\)?

A long coaxial cable consists of a thin cylindrical inner conductor of radius \(a\) and a concentric thin cylindrical outer conductor of radius \(b\), where \(a < b\). The inner conductor carries a uniformly distributed current \(I\) in one direction, and the outer conductor carries an equal current \(I\) in the opposite direction. Which of the following integral expressions correctly gives the total magnetic energy \(U\) stored in the vacuum region between the conductors (\(a < r < b\)) for a section of length \(\ell\)?

![A perspective view of a coaxial cable segment of length \(\ell\). An inner hollow cylinder of radius \(a\) is centered along a horizontal central axis. A larger, concentric outer hollow cylinder of radius \(b\) surrounds the inner cylinder. At the left circular face, an arrow labeled \(I\) points to the right along the inner cylinder, and an arrow labeled \(I\) points to the left along the outer cylinder. A radial dashed line from the central axis to the inner cylinder is labeled \(a\), and a radial dashed line from the central axis to the outer cylinder is labeled \(b\). A horizontal dimension line spanning the cylinder length is labeled \(\ell\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461216-2M2h5U.jpg)

- **A.** \(\dfrac{\mu_0 I^2 \ell}{8\pi^2} \int_a^b \dfrac{1}{r}\, dr\)
- **B.** \(\dfrac{\mu_0 I^2 \ell}{4\pi} \int_0^b \dfrac{1}{r}\, dr\)
- **C.** \(\dfrac{\mu_0 I^2 \ell}{4\pi} \int_a^b \dfrac{1}{r}\, dr\)
- **D.** \(\dfrac{\mu_0 I^2 \ell}{2\pi} \int_a^b \dfrac{1}{r}\, dr\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121370/*