Consider a conductor of length l with a circular cross section, as depicted in the figure. The cross-sectional radius changes linearly from a to b . The material has a resistivity of ho . Given that b - a l , determine the resistance of the conductor.

Step-by-step solution

Consider an elemental disc of thickness dx at a distance x from the end with radius a . The radius of this disc is given by r = a + b-a l x . The resistance of this elemental disc is dR = dx r^2 . The total resistance of the conductor is obtained by integrating dR from x = 0 to x = l : R = ₀^ l dx (a + b-a l x )^2 Let y = a + b-a l x . Then dy = b-a l dx . When x = 0 , y = a , and when x = l , y = b . Substituting these into the integral: R = l (b-a) _ a ^ b y⁻² dy R = l (b-a) [ - 1 y ]_ a ^ b R = l (b-a) ( 1 a - 1