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title: "A cylindrical resistor of length \\(L\\) consists of a weakly conducting material filling the region between an inner cylindrical conductor of radius \\(a\\) and a coaxial outer cylindrical conductor of radius \\(b\\). The material has a non-uniform resistivity that depends on the radial distance \\(r\\) from the central axis according to \\(\\rho(r) = \\rho_0 \\left(\\dfrac{r}{a}\\right)^2\\), where \\(\\rho_0\\) is a positive constant. A potential difference is established between the two conductors such that current flows radially outward. Which of the following expressions represents the correct integral setup to determine the total electrical resistance \\(R\\) between the inner and outer cylindrical surfaces?"
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url: "https://nerd-notes.com/ubq/121568/"
date_modified: "2026-08-23T05:24:03+00:00"
---

# A cylindrical resistor of length \(L\) consists of a weakly conducting material filling the region between an inner cylindrical conductor of radius \(a\) and a coaxial outer cylindrical conductor of radius \(b\). The material has a non-uniform resistivity that depends on the radial distance \(r\) from the central axis according to \(\rho(r) = \rho_0 \left(\dfrac{r}{a}\right)^2\), where \(\rho_0\) is a positive constant. A potential difference is established between the two conductors such that current flows radially outward. Which of the following expressions represents the correct integral setup to determine the total electrical resistance \(R\) between the inner and outer cylindrical surfaces?

A cylindrical resistor of length \(L\) consists of a weakly conducting material filling the region between an inner cylindrical conductor of radius \(a\) and a coaxial outer cylindrical conductor of radius \(b\). The material has a non-uniform resistivity that depends on the radial distance \(r\) from the central axis according to \(\rho(r) = \rho_0 \left(\dfrac{r}{a}\right)^2\), where \(\rho_0\) is a positive constant. A potential difference is established between the two conductors such that current flows radially outward. Which of the following expressions represents the correct integral setup to determine the total electrical resistance \(R\) between the inner and outer cylindrical surfaces?

![A perspective diagram showing a coaxial cylindrical geometry of length \(L\). An inner solid cylinder of radius \(a\) is aligned along a central horizontal dashed axis. An outer concentric cylindrical shell of radius \(b\) surrounds it, with the annular volume between them shaded with light gray cross-hatching. A two-headed dimension arrow indicates the total length \(L\). A straight arrow labeled \(a\) points from the central axis to the inner cylinder surface. A second straight arrow labeled \(b\) points from the central axis to the outer cylinder surface. Exactly four outward radial arrows labeled \(I\) point from the inner surface toward the outer surface. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787462643-h9lLt2.jpg)

- **A.** \(\displaystyle \int_a^b \dfrac{\rho_0 r}{2\pi a^2 L}\,dr\)
- **B.** \(\displaystyle \int_a^b \dfrac{\rho_0 r^2}{2\pi a^3 L}\,dr\)
- **C.** \(\displaystyle \int_a^b \dfrac{\rho_0 a}{2\pi L r^2}\,dr\)
- **D.** \(\displaystyle \int_a^b \dfrac{\rho_0}{2\pi a L}\,dr\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121568/*