A conductor of length L has a non-uniform cross-section.The radius of cross-section varies linearly from a to b.The resistivity of the material is .Assuming that (b-a)<<L find the resistance of the conductor across its ends.

PLZ GIVE THE SOLUTION

ANSWER IS

,

simple

radius at a distance x is given by

Take a differential cross section at a distance x

its resistance is

integrating this over the entire length ( 0 to L) we get the desired result .

The cylinder tapers from end to end. Since the radius increases linearly, we have the rate of increase of radius per unit length of the cylinder as dr =

Now, consider a thin strip of thickness dx, at a distance x from the end with radius a. Its radius is a + xdr, or, its radius is

Resistance of this element is: dR = , where A is the elemental area. Hence, on substituting the value for A, we get:

dR =

The overall resistance is got by integrating this under proper limits. For sake of simplicity, set k =

R =

On integrating and putting the limits, we get R = units.