Question

Shown a conductor of length `l` having a circular cross section. The radius of cross section varies linearly form `a to b`. The resistivity of the material is`(rho)`.Assuming that `b-altltl`,find the resistance of the conductor.

Answer 1

Let at distacne x, the radius is r. Then,
`r = a +Deltar = a+((b-a)/(l))x`
`R= intdR = int(rhodx)/(pir^2) = int_0^l (rhodx)/(pi[a+((b-a)/(l))x]^2)`
`.`
Let `a + ((b-a)/(l)) x = z or ((b-a)/(l))dx = dz`
when z goes from a to b, we get
`R= intdR = int_a^b (rhol)/(pi(b-a)) (dz)/(z^2) = (rhol)/(pi(b- a)) int_a^bh (dz)/(z^2)`
`=(rhol)/(pi(b-a)) [(1)/(a)-(1)/(b)]=(rhol)/(piab)`