Why use current density for surface currents in wires

[iScience]

I can see the purpose of J for currents taking place in a volume like, say.. plasma physics

but for current in a wire, it's all surface current, so why use J?

 

[UltrafastPED]

Why do you think that the current is all at the surface?

For DC, it is spread across the wire.

For AC, it is mostly in the region between the surface and the skin depth. For copper and 60 Hz, this is about 1 cm.

See https://en.wikipedia.org/wiki/Skin_effect
Locate the vertical red line in the figure, or use the formulas.