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A current is flowing through a cylinderi...

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.

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Consider a small ring of radius r and thickness dr as shown in the figure. The current density at the surface of the ring is
`J = J _(0)(1-(r )/(R ))`
`:. ` The current through the ring is
`dl = JdA`
`implies dl= J_(0)(1-(r )/(R )) 2pi r dr `
As ` i = int di`
`:. i = int _(0)^(R )J_(0)(1-(r )/(R )) 2pi r dr `
`= J_(0) xx 2pi int _(0)^(R ) ( r- ( r^(2))/(R )) dr`
`=J_(0) xx 2 pi [(r^(2))/(2)- (r^(3))/(3R)]_(0)^(R )= (J_(0)xx 2pi R^(2))/(6) = (J_(0)pi R^(2))/(3)`
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