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!
Best Answer
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.