- Current is a scalar I with units of [J/s]. It is defined as I=dQdt.
- Current density is a vector →J (with magnitude J) with units of [J/s/m2]. It is current per unit cross-sectional area, and is defined as →J=nq→vd (where n is the number of moving q-charges with drift velocity vd).
Why is I defined to not have a direction? Current density →J is defined as a vector, so why is current I not?
There are many questions about current vs. current density
... like this, this and this, but none answers my question about one being vector and the other being scalar. Is it simply just a definition? It just seems so obvious to define current as a vector too.
Another but equivalent definition of current density is I=∫→J⋅d→A. Mathematically, the dot product gives a scalar. But, for me this doesn't give much explanation still, as we could just as well have mathematically defined current as a vector and then used the area in scalar form in an equivalent formula like: →I=∫→J⋅dA.
Is it just a definition without further reason, or is there a point in keeping I in scalar form?
Answer
According to my understanding, indeed you could define a physical quantity like →I=I→nd
The current is defined according to a surface At (and is a local quantity: the position of the surface). Since the surface may be tilted (not perpendicular to →nd), then in general, we should write I=nq→At.→vd=nqAt→nAt.→vd=nqAtcos(θ).vd
That is, let's talk about the densities. We have dI=nqvddA=nqvdcos(θ)dAt
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