# Surface current density problem

## Homework Statement

A static surface current density Js(x,y) is confined to a narrow strip in the xy-plane In this
static problem ∇ ⋅Js = 0. Show that the line-integral of Js along any cross-section of the strip will yield the same value for the total current I. (The direction of dl in these 2D line-integrals is perpendicular to the line segment; these are not ordinary line-integrals but rather surface integrals in which the third dimension z has shrunk to zero.) Show that I =∫ Jsxdy =∫ Jsydx, where the integrals are over the width of the strip at any desired cross-section.

## Homework Equations

Gauss’s theorem: ∫∫ (∇ ⋅Js)dxdy =∫ Js ⋅dl

## The Attempt at a Solution

I don not understand this line The direction of dl in these 2D line-integrals is perpendicular to the line segment; these are not ordinary line-integrals but rather surface integrals in which the third dimension z has shrunk to zero.

The RHS of Gauss’s theorem = I
LHS:
Integrating from x1 to x2 where x2-x1 =Δx, dl is perpendicular to the line segment.
∫ Js ⋅dl= JsyΔx
similarly Integrating from y1 to y2 where y2-y1 =Δy, dl is perpendicular to the line segment.
∫ Js ⋅dl= JsxΔy

Is this right? How do I prove the 2nd part of the question?