A current is flowing through a cylinderical conductor of radius R, such that current density at any cross-section is given by `J = J_(0)(1-(r )/(R ))`, where r is radial distance from axis of the cylinder .Calculate the total current through the cross-section of the conductor.

Explanation

The conductor can be assumed to be made up of a large number of hollow cylinders. Consder one such cylinder of radius 'r' and thickness 'dr' . The current through this cylinder is di, such that current density is given by
`J = (di)/(dA) = (d i)/ (2 pi r dr)`
As` J = sigma E `
`implies J = sigma _(0) (1-(r )/(R ) ) xx E `
or `J = sigma _(0) (1-(r )/(R ) ) ((V)/(L))`
`:. (di)/(2pi rdr)=sigma_(0)(V)/(L) (1-(r )/(R)) `
`implies int di = int ( 2pi sigma_(0)V)/(L) (1-(r )/(R )) rdr`
`implies i = ( 2pi sigma_(0)V)/(L) int_(0)^(R ) ( 1- ( r )/(R )) rdr`
`implies i = ( 2pi sigma_(0)V)/(L) ((r^(2))/(2) - (r^(3))/(3R))_(0)^(R )`
`implies i = ( 2pi sigma_(0)V)/(L) ((R^(2))/(6))= (pi sigma_(0)VR^(2))/(3L)`