The current density varies with radial distance r as J = a r 2 , in a cylindrical wire of radius R . The current passing through the wire between radial distance R /3 and R /2 is

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Given; J = a r 2 . i = ∫ 1 2 ​ J × 2 π r d r = ∫ R /3 R /2 ​ a r 2 × 2 π r d r = 2 πa ∫ R /3 R /2 ​ r 3 d r = 2 πa ​ 4 r 4 ​ ​ R /3 R /2 ​ = 2 πa ​ [ ( 2 R ​ ) 4 − ( 3 R ​ ) 4 ] = 2 πa R 4 ​ × 81 × 16 65 ​ = 2592 65 πa R 4 ​ ​

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