9/27/2005 Surface Current Density.doc 1/4 Jim Stiles The Univ. of Kansas Dept. of EECS Surface Current Density Consider now the problem where we have moving surface charge ()rsρ. The result is surface current! Say at a given point r located on a surface S, charge is moving in direction maxˆa. Now, consider a small length of contour ∆A that is centered at point r , and oriented such that it is orthogonal to unit vector maxˆa. Since charge is moving across this small length, we can define a current I∆ that represents the current flowing across ∆A . S ∆A rmaxˆIa∆9/27/2005 Surface Current Density.doc 2/4 Jim Stiles The Univ. of Kansas Dept. of EECS Note vector maxˆIa∆ therefore represents both the magnitude ()I∆ and direction maxˆa of the current flowing across contour∆A at point r . From this, we can define a surface current density ()rsJ at every point r on surface S by normalizing maxˆIa∆ by dividing by the length ∆A : ()ˆmax0AmpsrlimmsIa∆→∆⎡⎤=⎢⎥∆⎣⎦JAA The result is a vector field ! NOTE: The unit of surface current density is current/length; for example, A/m. Given that we know surface current density ()rsJ throughout some volume, we can find the total current across any arbitrary contour C as: ()rˆsnCIad=⋅∫J A This looks very much like the contour integral we studied in the previous chapter. However, there is one big difference!9/27/2005 Surface Current Density.doc 3/4 Jim Stiles The Univ. of Kansas Dept. of EECS The differential vector ˆnadA is a vector that tangential to surface S (i.e., it lies on surface S), but is normal to contour C! This of course is the opposite of the differential vector dA in that dA lies tangential to the contour: As a result, we find that 0ˆndad⋅=AA. However, note the magnitude of each vector is identical: ˆndadd==AAA For example, consider the planar surface z =3. On this surface is a contour that is a circle, radius 2, centered around the z-axis. For the contour integrals we studied in Section 2-5, we would use: ˆdadφρφ=A However, to determine the total current flowing across the contour, we use ˆˆnaaρ= and ddρφ=A . Note the directions of these two differential vectors are different, but their magnitudes are the same. ˆnadAdAC9/27/2005 Surface Current Density.doc 4/4 Jim Stiles The Univ. of Kansas Dept. of EECS The integral for determining the total current flowing from inside the circle to outside the circle is therefore: ()()()ˆˆˆCC20rrr2nIadadadρπρρφφ=⋅=⋅=⋅∫∫∫JJJA S (z =3)
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