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

(c) What is the resistance in part (b) for current flowing parallel to the axis?

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}}}}