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Unformatted text preview: 6 Circulation and boundary conditions Since curlfree static electric fields have pathindependent line integrals, it follows that over closed paths C (when points p and o coincide) C E d l = 0 , where the C E d l is called the circulation of field E over closed path C bounding a surface S (see margin). x y z o = p E j l j C S d S Closed loop integral over path C enclosing surface S. Note that the area increment dS of surface S is taken by convention to point in the righthandrule direction with respect to "circulation" direction C. x y z (3,0,0) (3,0,0) (3,4,0) (3,4,0) C Example 1: Consider the static electric field variation E ( x, y, z ) = x x o that will be encountered within a uniformly charged slab of an infinite extent in y and z directions and a finite width in x direction centered about x = 0 . Show that this field E satisfies the condition C E d l = 0 for a rectangular closed path C with vertices at ( x, y, z ) = ( 3 , , 0) , (3 , , 0) , (3 , 4 , 0) , and ( 3 , 4 , 0) traversed in the order of the vertices given. Solution: Integration path C is shown in the figure in the margin. With the help of the figure we expand the circulation C E d l as E = 3 x = 3 x x o xdx + 4 y =0 x 3 o ydy + 3 x =3 x x o xdx + y =4 x ( 3) o ydy = 3 x = 3 x o dx + 0 + 3 x =3 x o dx + 0 = 0 . 1 Note that in expanding C E d l above for the given path C , we took d l as xdx and ydy in turns (along horizontal and vertical edges of C , respectively) and ordered the integration limits in x and y to traverse C in a counterclockwise direction as indicated in the diagram. Vector fields E having zero circulations over all closed paths C are known as conservative fields (for obvious reasons having to do with their use in modeling static fields compatible with conservation theo rems). The concepts of curlfree and conservative fields overlap, that is C E d l = 0 E = 0 over all closed paths C and at each r . x y z E d l C S d S STOKES THM: Circulation of E around close path C equals the flux over enclosed surface S of the curl of E taken in direction of dS....
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 Spring '08
 Kim

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