Question
A cylindrical conductor of length L and radius R has its
radius increasing linearly from R at one end to 2R at the other. A constant current I flows through it. If the resistivity of the material is ρ, find the total resistance of the conductor.Solution
To find the total resistance of the conductor, we need to consider a small element of the cylindrical conductor of thickness dx at a distance x from the end with radius R. Let the radius of the conductor at a distance x from the end with radius R be r(x). Since the radius increases linearly from R to 2R over a length L, the rate of increase of the radius with respect to the length is (2R-R)/L = R/L. So, the radius at a distance x is given by: r(x) = R + (R/L)x = R(1+x/L) The cross-sectional area of this small element at distance x is: 
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