How do you rigorously define current?

Question

I'm working through Griffiths, and nowhere in the book is current actually formally defined. That's kind of important, since he then bases the definition of surface and volume current densities off the notion of a line current.

With line currents, current could be defined as the amount of charge that passes through a point per unit time. Okay, that works fine with electrons, but it's not very general and I'd much rather work with line charge densities (in which case the charges passing a point could be viewed as summed delta functions).

With this restriction, I figured it's only really rigorously definable on a curve through space $\gamma (s)$ (where $s$ is some real parameter), carrying some line charge density $\lambda(s, t)$. Often with real currents you can view the line charge as propagating through the curve at some velocity $v$, so that then the current becomes $I=\lambda(s,t)v$, for some point $s$ on the curve. But that's not always the case, even when heeding charge continuity, as in the following example.

Consider the following charge distribution, on a circular curve $\gamma(s) = (R\cos s, R\sin s)$ for $s \in (-\pi,\pi]$: \begin{align} \lambda_1(s,0) = q\delta(s); \\ \lim _{t\rightarrow \infty} \lambda_1(s,t) = \frac{q}{2\pi R}. \end{align}

Physically this represents a point charge $q$ distributing itself over $\gamma$ uniformly as time passes. In this case, how would we define a current? The line charge $\lambda_1$ is not propagating over the curve in any sense, so the first equation for the current doesn't really work.

Moreover, what about a closed loop over which the line charge density is constant? This is most theoretical currents, and yet we cannot define the current without the velocity (which is not determinable by looking at the charge density). That leads me to believe we also need a velocity field... but surely we could do some integral for $\lambda_1$ to get the velocity of the charge anyway?

Is it possible to answer these questions? I'm starting to get very confused, so it would be great if someone could help.

Answer 1

It is actually easier to start with current density, particularly if you want a definition based on charge density. Current density, $\vec J = \rho \vec v$ is the charge density $\rho$ times the velocity $\vec v$. Then current is the integral of the current density over some surface $I = \int \vec J \cdot d\vec A$, typically a cross-section of a wire.

Charge is attached to matter, so both the charge density and the velocity are those of the matter carrying the charge. It can be treated as a continuum or as particles.