# K, the idealized surface current density

Palindrom
K, the "idealized surface current density"

Hey, I don't quite understand that guy, K.

I have an exam on Sunday in E&M, I'm studying from Jackson. I haven't found any definition of 'K'.

If anyone could give me a rigurous definition and an integral form, if there's any, I'd appreciate it.
Oh, and since we're at it, I stumped into that next statement:
"Suppose that the upper half of space is filled with a permeable media, while the other half is empty space. If, in the x-y plane, K is in the x direction, it follows that A (vector potential) is also in that direction in the entire space".
Huh?

Homework Helper
Gold Member
What definition do you have so far with which you are unsatisfied?

Griffiths (pp.211) gives the following definition: "When charge flows over a surface, we describe it by the surface current density K, defined as follows: Consider a "ribbon" of infinitesimal width $dl_\perp$, running parallel to the flow. If the current in this ribbon is $d\vec{I}$, the surface current density is

$$\vec{K}=\frac{d\vec{I}}{dl_\perp}$$

In words, K is the current per unit width-perpendicular -to-flow. In particular, if the (mobile) surface charge density is $\sigma$ and the velocity is $\vec{v}$, then

$$\vec{K}=\sigma \vec{v}$$"

It is not written but I believe we can write the integral form as

$$I_{surface} = \int_{\mathcal{P}}\vec{K}\cdot d\vec{l}$$

where $\mathcal{P}$ is a path across the surface.

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Palindrom
Thanks.
But then, in the statement I gave, why is A in the x direction? I just can't see it.