The cross section of a cylindrical conductor is A. The resistivity of the material of the cylinder depends only on distance r from the axis of the conductor as `rho=k/r^2` where k is a constant. Find the resistance per unit length of such a conductor.

Explanation

Let us consider a thin cylindrical pipe of unit length having radius r and thickness dr.
The conductance of thin pipe is , `dY=(1dA)/(rho 1)`.......(1)
when dA= cross area of cylindrical pipe.

Here, `dA=2pir d r`
`therefore` From equation (1), `dY=1/rho.2pirdr`
`becauserho=k/r^2so,dY=1/k.2pir^3dr`
Therefore, electrical conductance of the whole conductor becomes,
`Y=(2pi)/kint_0^ar^3dr` [a= radius of the whole conductor]
`=(2pi)/ktimes[r^4/4]_0^a=(pia^4)/(2k)=A^2/(2pik)[becauseA=pia^2]`
So, resistance `=1/Y=(2pik)/A^2`