Figure shows a conductor of length l carrying current i and having a circular cross-section. The radius of cross section varies linearly from a to b. assuming that `(b-a)ltltl` calculate current density at distance x from left end.

Explanation

`dR due to the small strip dx at a distance
y, `R = (rhodx/ y_2)` …….(i)
` tan theta = ((y-a)/x) = ((b-a)/L)`
` rArr ((y-a)/x) = ((b-a)/L)`
` rArr L xx ( y-a) = x(b-a)`
` rArr Ly - La = xb - xa `
` rArr L dy/dx -0 = b-a (diff. w.r. t. x) `
` dx = Ldy/ (b-a)`
Putting the value of dx in eqn. (i)`
`dR = (rho Ldy / pi (y^2) (b-a))`
` dR = (rhoL/ pi (b-a)). dy/y^2 `
` rArr (int_(0)^R) dR = (rhoL/ pi (b-a)) (int_(a)^b) dy/y^2 `
` R = (rhoL/pi (b-a)) ([-1/y]_(a)^b)`
` = (rhoL/pi (b-a))(-1/b)(-1/a)`
` (rho L / pi ab)`.