What is current density ? Derive Ohm's law in form of current density?

Explanation

`rArr` Current density : The electric current density at any point is defined as amount of electric current flowing per unit cross section perpendicular to the current at that point.
Current density is Vector quantity,
`vec(J) = (I)/(vec(A))`
Unit = `(A)/(m^(2)) = Am^(-2)`
Dimensional formula = `M^(0) L^(-2) T^(0) A^(1)`
Ohm.s law V = IR `" " `....(1)
Consider conductor of length l electric field be E then potential difference between ends of conductor be V then,
`E = (V)/(l)`
V = El
resistance R = `(rho l)/(A)`
V = IR
El = `I ((rho l)/(A))`
`(I)/(A) = J`
E = J`rho `
`J = sigma ` E
`vec(J) = sigma vec(E)`
which also represent Ohm.s law (in vector form).