I am not sure if this helps, but elaborating on Andrey's answer: now if you recall that occasionally one wants to work with electric currents localised on submanifolds of $\mathbb{R}^3$, what a physicist would call a "wire", or a "conducting plane", de Rham currents provide a natural framework for this. This is much like using distributions to describe point charges.

For example, a current corresponding to a loop wire carrying electric current $I$ represented by an oriented curve $\gamma$ is the 1-current $J$ given by $J(\phi)=I\int_\gamma\phi$. This can be though of as a "generalized" 2-form $J$, which we now can try to integrate over 2-manifold $S$ to obtain the current through this surface, but I am not sure that it makes rigourous sence for arbitrary 1-currents (for this particular one it does, at least for finite number of transversal intersections of $\gamma$ and $S$). We can also take $J(E)=RI^2$, where R is the resistance, and $E$ is the electric field 1-form to express the Ohm's law, et cetera.