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Background

I am learning about current density in semiconductors. If you imagine a cube of silicon, drift current is defined as the amount of coulombs per time passing through a slice of area of the silicon.

Question

This got me thinking about electric current in other contexts, like solving for currents through circuits using ohm's law. I haven't learned a very precise definition for what that current actually is. I know it's the amount of charge that enters then exits through something per time. But what is it defined as passing through? Is it some volume (3D), an area (2D), a line (1D), or a point (0D) in the conductor?

One idea I have is that the current is defined as the charge passing through the entire area of a conductor perpendicular to the direction of flow, or the charge passing through a volume of conductor with any length but with area being the entire area perpendicular to the flow. A volume sees the same current as an area because it is like a stretched area, since the charge that goes in one end, comes out the other, just like for an area with zero thickness. The amount of current going through a line and point would be zero, since for something to experience current, it would need to have a plane perpendicular to the direction of flow.

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One idea I have is that the current is defined as the charge passing through the entire area of a conductor perpendicular to the direction of flow,

This one.

A volume sees the same current as an area because it is like a stretched area, since the charge that goes in one end, comes out the other, just like for an area with zero thickness.

If your circuit is operating in the lumped circuit limit (no charge accumulates in the wires, no significant changing magnetic fields pass through the loops formed by the wires) then this is also valid.

But you could also look at it this way: You can divide the surface of your volume into two parts with current flowing inward on one surface, and current flowing outward on the other. These two currents will be equal (but opposite in sign). And now we're talking about current flowing through surfaces again rather than volumes.

For that matter, in the lumped circuit limit, you can (and should) really think of current flowing in complete loops. But at the physical level what it really means is that at any point along the loop, if you cut the loop with a surface, the current through that surface would be the same.

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