Why isn't current measured in Coulomb meters per second? I've been trying to relearn electronics and have run into a conceptual block. Everywhere I look defines current as something along the lines of
$$
i = \frac{dq}{dt}
$$
where $i$ is current, $q$ is charge and $t$ is time.
Describing it as the rate of charge passing through a surface. But the definition seems odd to me because both charge and time are directionless quantities, but current has a direction, at least conceptually.
The units too seem directionless, but I would expect to see a unit like Coulomb meter per second. This has a direction and also would have the property that if the charges pass through at a rate of once a second but in one case go twice as fast as in the other then the value of one would be twice as high as the other, whereas they'd be the same for the way current is defined above.
I suspect the answer hinges on the way the surface is defined, but in my searching I haven't found anything.
 A: Current is not a vector; it doesn't have a direction in the same way that a velocity has a direction.
There is a good analogy between electric current in a wire and the flux or rate of flow of water through a pipe. The idea here is that we put an imaginary surface right across the interior of the pipe. It can be on the skew, nor need it be plane. The (volume) rate of flow of water in the pipe is the volume of liquid crossing this surface per unit time. The only direction ('sense' would be a better word) that we assign to the flow rate is X to Y or Y to X, in which X and Y are the two sides of the surface. It is the same with electric current, rate of flow of charge, in a wire.
By contrast, we can describe the flow using vectors. For the water we have the velocity, $\mathbf v$, of the individual chunks of water. For the electric charge we have the velocity of the charge carriers (charge $q$, say), or a special vector called 'current density, $\mathbf J$.
The relationship between the velocity of the water (which can vary from point to point) and the volume flow rate, $\Phi$, is
$$\Phi = \int_S \mathbf v.d\mathbf S$$
in which the integral is evaluated over our imaginary surface, S.
In the same way, electric current, $I$, can be calculated from
$$I=\int_S \nu q \mathbf v.d\mathbf S \ \ \ \text {or}\ \ \ I=\int_S  \mathbf J.d\mathbf S.$$
[$\nu$ is the number of charge carriers per unit volume of the wire.]
A: It is! Sort of.
You are thinking just straight Coulombs which might make sense in a 1D-application or so, you can pretend that charges have no spatial extent and are just individual point particles and travel along well-defined paths... But when you start dealing with three dimensions you start wanting to smear out individual electrons so that they are not point particles but occupy a volume. This leads to a $\text{C}/\text{m}^3$ unit.
Multiplying that by a velocity you get the official unit of current density, $\text{C}/(\text{m}^2 \text{s})$. It is a vector. For a uniformly moving charge distribution $\mathbf J=\rho \mathbf v$.
As the other answer says, the $\text{m}^2$ in the bottom gives us a tempting way to connect this to more conventional currents $\text C/\text s$ by doing an integral over a surface, “how many charges pass through this surface per second?”. And that's very natural for an electronic circuit where the surfaces essentially describe electrical components, “how many charges pass through this resistor/capacitor/inductor/mosfet per second?”. Wiring diagrams exist in an abstract space where there are no physical distances per se. Nobody ever tells you that you drew the circuit diagram wrong because a wire is twice as long as it should be given its propagation time or whatever, signals are held to propagate about the speed of light or some fraction of it, whereas the circuit is significantly smaller than a light microsecond (300 m) and so you need something with the speed of a computer chip or so to really notice these propagation delays and care about them. If your system is not responding at megahertz speeds or higher then signal propagation is usually not your first concern when getting the circuit running, and the unit of meters falls away conveniently.
A: In Maxwell's equation the current vector density appears with dimension $Cm^2s^{-1}$. The circuit current is the component of this perpendicular to the cross section of a wire, integrated over that cross section, giving dimension $C/t$. Therefore current is the component of a vector parallel to the wire.
