Concept understanding — Electric Current and Conventional Current Direction
Definition. Electric current is the rate of flow of charge across a chosen cross-sectional area of a conductor: for a net charge ΔQ crossing in a time Δt, the average current is Iavg=ΔQ/Δt, and the instantaneous current is the limit I=limΔt→0ΔQ/Δt=dQ/dt. The SI unit is the ampere (A), with 1A=1C/s: a current of one ampere means one coulomb of charge is crossing the chosen cross-section every second.
Why current has no net value with no field applied. A conductor's free electrons are always in random thermal motion, colliding continuously with the fixed lattice of positive ions, even with no battery connected. Because this motion has no preferred direction, exactly as many electrons cross any cross-section moving one way as moving the other way at any given instant, so the net charge transfer -- and hence the current -- is exactly zero, even though the electrons themselves are moving very fast. Only once a battery sets up a potential difference (and hence an electric field) across the conductor does a genuine net drift appear, superimposed on top of the random motion, and only then does a measurable current flow.
Current is a scalar. Although the charges that make up a current, and the vector current density built from them, both have a definite direction, the current I itself is formally defined through the scalar (dot) product I=J⋅A=JAcosθ of the current-density vector with the chosen surface's area vector. Because a dot product of two vectors always yields a single number rather than another vector, I does not obey the vector addition law -- currents at a junction combine by simple algebraic (not vector) addition, exactly as used in Kirchhoff's current rule. I can be assigned a positive or negative sign depending purely on which way the surface's normal vector is chosen, but this sign is a bookkeeping convention, not evidence of I being a vector quantity in the physical sense. …