Surface current density problem

[aaaa202]

The surface current density, K, is defined as the the current through a unit width perpendicular to the flow. In particular:
K = v\cdotσ
where σ is the surface charge density. Now I have a little trouble understanding this formula intuitively. Can someone describe in pictures how it is interpreted physically?
Also as a side note: Why can you unambigously speak of the current through a ribbon of width dl parallel to the current - what if the current changes as we move parallel to it?

 

[jtbell]

One way to justify that formula is to consider the units: (A/m) = (m/s)(C/m²).

Or: Imagine a long rectangular sheet of charge with width w and uniform surface charge density σ. Draw an imaginary line at right angles across the width of the sheet. Set the sheet in motion with constant speed v while keeping the imaginary line stationary.

In a time interval Δt, a length vΔt of the sheet passes the imaginary line. The total charge contained in that length is ΔQ = (σ)(vΔt)(w). The current passing the imaginary line is I = ΔQ/Δt = σvw. The surface current density along the imaginary line is K = I/w = σv.