---
title: "A cylindrical shell conductor of length \\(L\\) has an inner radius \\(a\\) and an outer radius \\(b\\). Current flows radially outward from the inner surface to the outer surface. The resistivity of the material varies with distance \\(r\\) from the central longitudinal axis according to \\(\\rho(r) = \\rho_0 \\left(\\dfrac{r}{a}\\right)\\), where \\(\\rho_0\\) is a positive constant. What is the total electrical resistance of the cylindrical shell conductor?"
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url: "https://nerd-notes.com/ubq/118432/"
date_modified: "2026-08-04T08:10:03+00:00"
---

# A cylindrical shell conductor of length \(L\) has an inner radius \(a\) and an outer radius \(b\). Current flows radially outward from the inner surface to the outer surface. The resistivity of the material varies with distance \(r\) from the central longitudinal axis according to \(\rho(r) = \rho_0 \left(\dfrac{r}{a}\right)\), where \(\rho_0\) is a positive constant. What is the total electrical resistance of the cylindrical shell conductor?

A cylindrical shell conductor of length \(L\) has an inner radius \(a\) and an outer radius \(b\). Current flows radially outward from the inner surface to the outer surface. The resistivity of the material varies with distance \(r\) from the central longitudinal axis according to \(\rho(r) = \rho_0 \left(\dfrac{r}{a}\right)\), where \(\rho_0\) is a positive constant. What is the total electrical resistance of the cylindrical shell conductor?

![A perspective view of a hollow cylindrical conductor of length L. The cylinder has an inner radius labeled a and an outer radius labeled b, both measured from the central longitudinal axis labeled z. Radial arrows originating at the inner surface extend outward through the cylindrical shell toward the outer surface to indicate the direction of current flow. A small concentric thin cylindrical surface of radius r and thickness dr is shown within the shell. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831002-UEcMhb.jpg)

- **A.** \(\dfrac{\rho_0 (b - a)}{2\pi a L}\)
- **B.** \(\dfrac{\rho_0}{2\pi L} \ln\left(\dfrac{b}{a}\right)\)
- **C.** \(\dfrac{\rho_0 (b^2 - a^2)}{4\pi a^2 L}\)
- **D.** \(\dfrac{\rho_0 (b^2 - a^2)}{2\pi a^2 L}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118432/*