What is the formal definition of $dI$? Where $I$ stands for current?

Question

Two days before my magnetostatics test and for some reason the idea of current just stopped making sense. I think I pin pointed the problem to $K$ and $J$, the so called surface and volume current density. Could anyone please write down the formal definition of $dI$? Because $dq/dt$ is not approaching 0 so can't be used here.

G. Smith · Accepted Answer

When a wire carries current $I$, $dvec I$ means
$$dvec I=Idvecell$$
where $dvecell$ is an infinitesimal segment of the (possibly curved) wire. Think of the vector $vecell(lambda)$ as pointing from some origin to various points on the wire, in a parametric description of the wire’s path. (For example, for a circular wire, the parameter $lambda$ along the curve could be the angle $theta$ around the circle.) Then $dvecell$ is an infinitesimal vector along the wire.

Claudio Saspinski · Answer

$Delta q = rho Delta V$ where $rho$ is the charge density and $V$ is the volume. If $rho$ is constant, $$I = frac{dq}{dt} = rho frac{dV}{dt}$$ If the cross sectional area $S$ is constant $$I = rho Sfrac{dx}{dt} = rho Sv$$ where $v$ is the charge average velocity along the conductor. It is another form of defining $I$ for a usual wire.

Answered by Claudio Saspinski on April 29, 2021

What is the formal definition of $dI$? Where $I$ stands for current?

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