Current Density

Consider a wire carrying current . To define the current, we cut the wire perpendicularly and count the number of charges passing through the wire per unit time. Assume the charges are uniformly spread at velocity .

It given time , the charge will move a distance of , and thus the total charge going through the cross section will be given as:

And we calculate current as:

What if the cross-section wasn't perpendicular?

If we follow the approach of counting the number of charges per unit time, we will count fewer charges. But the current doesn't change based on how the wire is cut, so our definition is incomplete.

Definition

We define current density as a vector where the direction of the vector is along the direction of the motion of positive charge:

of units (obviously)

Now we can re-define current:

Definition

Given a cross-sectional surface, current is defined as:

which is obvious since has units (see integral, Interpretation of Integrals, dot product)

With our new definition, the change in angle between and increases, which will decrease the dot product along with the increase in surface area.

Equation

Since the velocity will depend on the material and is proportional to the E-field, we write:

where the conductivity of a material with units siemens per meter.