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

Since radius of left end is a and that of right end is b, therefore increase in radius over length l is (b-a).
Hence rate of increase of radius per unit length `=((b-a)/(l))`
Increase in radius over length `x=((b-a)/(l))x`
since radius at left end is a, radius at distance `x=r=a+((b-a)/(l))x`
Area at this particualr section `A=pir^(2)=pi[a+((b-a)/(l))xx]^(2)`
Hence curret density `J=(i)/(A)=(i)/(pir^(2))=(i)/(pi[a+(x(b-a))/(l)]^(2))`