Is current truly a scalar quantity or a tensor?

[PhysicsEnjoyer31415]

TL;DR Summary

From what i know all quantities are tensors , divided into rank 0,rank 1 ,rank 2 , rank 3 ..rank( n )according to their components which is 3^n . Current is supposed to be a scalar quantity right? It does not follow vector rules . Our book says it is a tensor quantity(ofcourse yes) but my question is "Is current truly a scalar quantity" ? If a bunch of electron were to drift towards different directions , should that be considered as current in different directions

Electric field is present . The net movement of electrons would flow in direction opposite to the electric field yes.But if we were to take a small cross section of the conductor and say 3 electrons are moving in slightly different directions would that not mean that it has 3 components to the current at this particular point? Or say n number electrons ? Can it not go up to rank (n) tensor ? Please enlighten me on this topic , and please correct any misunderstanding of mine about the topic. Thank you!

 

[jbriggs444]

If we are talking about current (current density) in the middle of a conductive block then treating it as a vector would seem quite applicable. We take the vector dot product of the current density (a vector) with a directed area element (also a vector) to get the rate at which charge passes through the cross-sectional area.

Speaking of individual electrons just muddies the water. We are talking about a continuum model.

I found a posting here on physics.stackexchange.com that makes the point that "current" (the integral of current density over a surface) is a scalar while "current density" is a vector.

 

[PhysicsEnjoyer31415]

If we are talking about current (current density) in the middle of a conductive block then treating it as a vector would seem quite applicable. We take the vector dot product of the current density (a vector) with a directed area element (also a vector) to get the rate at which charge passes through the cross-sectional area.

Speaking of individual electrons just muddies the water. We are talking about a continuum model.

If we were talking about circuit theory and Kirchoff's laws then that would be a different story. The current through a circuit element there would be a signed scalar.

Oh okay ...but why do we not consider individual electrons while saying current is scalar or tensor? Is it based on frame of reference by any chance?
If say i were to not consider current density(vector) and rather only current with 3 electrons flowing what would that be . Do i consider it a collective system ? Or individually approach the 3 electrons ? Or some combination of both..??
A bit of clarification : i did not mean to talk about the vector quantity current density in the post before but rather the current sorry for any misunderstanding.

 

[jbriggs444]

Oh okay ...but why do we not consider individual electrons while saying current is scalar or tensor? Is it based on frame of reference by any chance?

"Current" is a summary of the motion of statistically significant numbers of charge carriers. Enough charge carriers that it would be pointless tracking their individual motions and enough so that their net movement is essentially indistinguishable from a continuous flow.

If you have only three electrons moving then I would not think that "current" is a meaningful concept. You might identify a cross-section over which to measure current. But then the current at any given time would either be $+\infty$, $-\infty$ or (almost always) $0$.

 

[Vanadium 50]

...but why do we not consider individual electrons

For the same reason we don't use General Relativity to solve blocks on inclined plane problems. It makes things more complicated and does not give a better answer.

 

[Mister T]

But if we were to take a small cross section of the conductor and say 3 electrons are moving in slightly different directions would that not mean that it has 3 components to the current at this particular point? Or say n number electrons

You can imagine temperature changing by different amounts in different directions. That doesn't make temperature a vector quantity.

Current doesn't obey the laws of vector addition, for example.

I'm sure there are deeper reasons.

 

[Nugatory]

From what i know all quantities are tensors…

Not quite. A tensor also has to satisfy the tensor transformation rules, and indeed that’s what makes them useful.

… divided into rank 0,rank 1 ,rank 2 , rank 3 ..rank( n )according to their components which is 3^n

When you get to relativity you’ll encounter rank-n tensors with $4^N$ components. It may be better to think of a rank-N tensor as a mathematical object that maps N vectors to a scalar (or equivalently, N-1 vectors to a vector).