# 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?

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#### quasar987

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.

#### quasar987

Homework Helper
Gold Member
I don't know what permeable means, I'll have to leave that one to someone else.

#### Palindrom

It may not be the right term. It simply means it's a linear matter for some 'miu'.

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