Current density

We define the current density ($\vec{J}$, in units of $A/m^2$), for some charge density, equal to the current per unit area through the surface, at some velocity $\hat{u}$:

$$\vec{J}={\rho }_v\hat{u}$$

We then define the current as the flow of charge through a surface $S$ (often the cross-section of a conductor at a particular point) as:

$$I={\iint }_S\vec{J}\cdot d\vec{S}$$

When a conductor has current through it, electrons tend to drift with a given speed ${\vec{v}}_d$ in the direction opposite that of the applied electric field.

$$\vec{J}=(ne){\vec{v}}_d$$

Ohm’s law can be restated in field-form:

$$\vec{J}=\sigma \vec{E}=\frac{\vec{E}}{\rho }$$

where $\sigma$ is the conductivity and $\rho$ is the resistivity.

Addendums

A special case is if the current is uniform across the surface and parallel to $d\vec{A}$.

$$J=\frac{I}{A}$$

We relate to electric field and resistivity with:

$$\vec{E}=\rho \vec{J}$$