surface and volume current density; definition and logic seems contradictory to me

Question

Okay, so in Griffiths' Introduction to Electrodynamics, Griffiths clearly defines surface current density as follows:

when charge flows over a surface, we describe it by the surface current density, $K$. Consider a 'ribbon' of infinitesimal width dL running parallel to the current flow. If the current in this ribbon is $dI$, surface current density is $K=dI/dL$.

Now, I searched Google and some websites which are clearly telling me that surface current density is current per unit AREA, not length! Is there something wrong with my understanding of this concept or are both the definitions equivalent?

Griffiths similarly defined volume current density as current per unit area perpendicular to the current flow, while in my opinion, it should be current per unit VOLUME...

This is really very confusing to me, please clarify.

Thanks!

Answer 1

Let's start with charge density $\rho$ which is the charge per unit volume. To get the amount of charge on some object, we'd integrate over the volume. Current is defined as charge per unit time crossing some surface. So to describe a charge density moving, we get a current density $J$ which is amount of charge per area per time... dimensionally it is one less "per length" than $\rho$ and an additional "per time" .

Now look at 2D: $\sigma$ is charge per unit area, the surface current $K$ is dimensionally one less "per length" (the charge is now crossing through a 'line' one the surface instead of an area).

Now look at 1D: $\lambda$ (I think that is what Griffiths used) is the charge per length on a wire, $I$ is dimensionally one less "per length" (the charge is now crossing through a 'point' on the wire).

If that doesn't help and you are still confused about the da in the Biot-Savart like law for surface current, let me know and I'll add more about that.

Answer 2

You are wrong that google results say that surface current density is a current per unit area. I did my own google search, and that was not said in any of the results.

My guess is that you have no problem with an infinitely concentrated charge being a zero-dimensional point, and a current happening along a line. Notice that a current occupies the region that a single charge moves through. So if you think about when happens when zero dimensional point charges move, you get a current that lives on a one-dimensional curve.

Now if the point charge is smeared out onto a one-dimensional line segment, then the corresponding current would be on a 2D ribbon, as described in griffiths. The same way that the charge on a line has dimensions of charge per length, the current density on the ribbon has units of current per length.