Chapter 25 Problem 30

Problem

The resistor

A hollow cylindrical resistor with inner radius $r_{1}$ and outer radius $r_{2}$ , and length $l$ , is made of a material whose resistivity is $\rho$

(a) Show that the resistance is given by

$R={\frac {\rho }{2\pi l}}\ln {\frac {r_{2}}{r_{1}}}$

for current that flows radially outward. [Hint: Divide the resistor into concentric cylindrical shells and integrate.]

(b) Evaluate the resistance $R$ for such a resistor made of carbon whose inner and outer radii are 1.0 mm and 1.8 mm and whose length is 2.4 cm

Solution

(a)

$dR=\rho {\frac {dr}{2\pi rl}}$ Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. TeX parse error: Double subscripts: use braces to clarify"): {\displaystyle R=\int dR=\int _{r}_{1}^{r}_{2}\rho {\frac {dr}{2\pi rl}}={\frac {\rho }{2\pi l}}\ln {\frac {r_{2}}{r_{1}}}}