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

(N/A) Current density: The electric current density at any point is defined as the amount of electric current flowing per unit cross-sectional area perpendicular to the current at that point. Current density is a vector quantity.
$\overrightarrow{J} = \frac{I}{A} \hat{n}$
$\text{Unit} = A/m^2 = A \cdot m^{-2}$
$\text{Dimensional formula} = [M^0 L^{-2} T^0 A^1]$
Derivation of Ohm's law in vector form:
Consider a conductor of length $l$ and cross-sectional area $A$. Let $E$ be the electric field applied across it. The potential difference $V$ is given by $V = E \cdot l$.
From Ohm's law,$V = I \cdot R$.
We know that resistance $R = \rho \cdot \frac{l}{A}$,where $\rho$ is the resistivity.
Substituting $V$ and $R$ in Ohm's law:
$E \cdot l = I \cdot \left( \frac{\rho \cdot l}{A} \right)$
$E = \left( \frac{I}{A} \right) \cdot \rho$
Since current density $J = I/A$,we have:
$E = J \cdot \rho$
Using conductivity $\sigma = 1/\rho$,we get:
$E = J / \sigma$
$J = \sigma \cdot E$
In vector form,this is written as $\overrightarrow{J} = \sigma \overrightarrow{E}$,which is the vector form of Ohm's law.