A tapered bar of length L and end diameters `D_(1)` and `D_(2)` is made of a material of electrical resistivity `rho`. The electrical resistance of the  bar is

Options

A. `(4rho L)/(pi(D_(1)+D_(2))^(2))`
B. `(4rho L)/(pi(D_(1)-D_(2))^(2))`
C. `(rho pi sqrt(D_(1)+D_(2)))/(4L^(2))`
D. `(4rhoL)/(pi D_(1)D_(2))`

Answer

D

Explanation

Similarity of triangles,

`(r-a)/(x)=(b-a)/(l)`
(Here, `a=D_(1)//2,b=D_(2)//2`)
`implies" "r=a+((b-a)/(l))x`
Resistance of elemental disc is
`dR=(rho dx)/(pi r^(2))=(rho dx)/(pi[a+((b-a)/(l))x]^(2))`
Total resistance of tapered wire is
`R= int dR=underset(0)overset(l)int(rho dx)/(pi[a+((b-a)/(l))x]^(2))`
`implies" "R=(rho)/(pi)[-(1)/([a+((b-a)/(l))x])(l)/((b-a))]_(0)^(l)`
`=(rho)/(pi)xx(l)/(b-a)[(1)/(a)-(1)/(b)]=(rho l)/(pi ab)`
`=(4 rho l)/(pi D_(1)D_(2))`