A quick post about the direction of current

Without a second thought we might say that the current heads to somewhere, from one component to another. But is it true that any current does have an orientation?



Let's begin at the definition of current: the amount of charge flowing through a given area in a specific time. That is:

Hence, how much the area is given can actually have an affect on the current. And that is why we should not consider current to be a vector: it is a quantity relating to area.

Under some circumstances, not all directions of moving charges are prependicular to the area (or parallel to its normal). That is why we need to define something else to solve this problem: though the electric field inside a conductor doesn't change, when we alter the section area, we can receive different current values.

Since we need to remove the factor of area, we can introduce current density:

Which refers to the current through a point on the given area, regardless of the amount of the area. Therefore, the current density can bear orientation and thus be a vector.

Or in the case where the current density varies elsewhere on the given area Ω:

And that is it: current (I) is actually (integration of) inner product of current density and area. It is not current but current density that bears the direction we talk about most of the time.

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According to Ohm's law:

for a linear resistor. Also recall the definition of R regarding ρ (resistivity), that of I regarding J (current density), and that of V regarding E (electric field):

Where L is the length of the resistor and Ω is a given surface. Suppose that E does not vary inside the resistor and either does J on Ω, we can rewrite the sentence:

And:

In fact, inside the resistor these vectors have the same direction. Thus:

where σ is the conductivity of the resistor, the reciprocal of resistivity.