Chapter 25, Problem 27P

(III) A hollow cylindrical resistor with inner radius r₁ and outer radius r₂, and length ℓ, is made of a material whose resistivity is ρ (Fig. 25–36). (a) Show that the resistance is given by

$R=\frac{\rho }{2\pi \mathcal{l}}\ln \frac{r_2}{r_1}$

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. (Choose ρ = 15 × 10⁻⁵ Ω · m.) (c) What is the resistance in part (b) for current flowing parallel to the axis?

FIGURE 25–36 Problem 30.