---
title: "A long coaxial cable consists of a solid inner cylindrical conductor of radius \\(a\\) and a thin, concentric outer conducting cylindrical shell of radius \\(b\\), where \\(b > a\\).  The inner cylinder carries a total current \\(I_0\\) directed out of the page. The current density \\(J\\) inside the inner cylinder is not uniform; it varies with radial distance \\(r\\) from the center according to the equation \\(J(r) = C r^2\\), where \\(C\\) is a positive constant.  The outer shell carries the same total current \\(I_0\\), but directed into the page and distributed uniformly around the shell."
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url: "https://nerd-notes.com/ubq/117874/"
date_modified: "2026-08-04T08:00:13+00:00"
---

# A long coaxial cable consists of a solid inner cylindrical conductor of radius \(a\) and a thin, concentric outer conducting cylindrical shell of radius \(b\), where \(b > a\).

The inner cylinder carries a total current \(I_0\) directed out of the page. The current density \(J\) inside the inner cylinder is not uniform; it varies with radial distance \(r\) from the center according to the equation \(J(r) = C r^2\), where \(C\) is a positive constant.

The outer shell carries the same total current \(I_0\), but directed into the page and distributed uniformly around the shell.

A long coaxial cable consists of a solid inner cylindrical conductor of radius \(a\) and a thin, concentric outer conducting cylindrical shell of radius \(b\), where \(b > a\).

The inner cylinder carries a total current \(I_0\) directed out of the page. The current density \(J\) inside the inner cylinder is not uniform; it varies with radial distance \(r\) from the center according to the equation \(J(r) = C r^2\), where \(C\) is a positive constant.

The outer shell carries the same total current \(I_0\), but directed into the page and distributed uniformly around the shell.

![A flat circle enclosed by a concentric larger ring. A dashed horizontal line starts at the center and extends to the right, passing through both the circle and the ring. The segment from the center to the edge of the inner circle is labeled a. The segment from the center to the outer ring is labeled b. Inside the inner circle, exactly five solid dots represent current vectors pointing out of the page. Along the outer ring, exactly eight 'x' marks represent current vectors pointing into the page. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830412-OQ1DUb.jpg)

**Part a)** **Derive** an expression for the constant \(C\) in terms of \(I_0\), \(a\), and fundamental constants.

**Part b)** Use Ampère's law to **derive** an expression for the magnitude of the magnetic field \(B\) inside the inner cylinder as a function of \(r\), for the region \(r < a\). Express your answer in terms of \(I_0\), \(a\), \(r\), and fundamental constants.

**Part c)** A student proposes replacing the inner cylinder with a new solid cylinder of the same radius \(a\) that carries the same total current \(I_0\) out of the page, but with the current distributed uniformly across its cross section. **Indicate** whether the magnitude of the magnetic field at \(r = a/2\) for the new uniform cylinder is greater than, less than, or equal to the magnitude of the magnetic field at \(r = a/2\) for the original non-uniform cylinder. - [ ] Greater than - [ ] Less than - [ ] Equal to

**Part d)** On the axes provided, **sketch** a graph of the magnitude of the magnetic field \(B\) as a function of the radial distance \(r\) from \(r = 0\) to \(r = 2b\). Label the maximum value of the magnetic field on the vertical axis in terms of \(I_0\), \(a\), and fundamental constants.

**Part e)** An electron travels with speed \(v_0\) parallel to the central axis of the cable in the region \(r > b\). **Determine** the magnitude of the magnetic force exerted on the electron. **Justify** your answer using Ampère's law and the principles of magnetic force.


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117874/*