329lect06 - 6 Circulation and boundary conditions Since...

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6 Circulation and boundary conditions Since curl-free static electric fields have path-independent 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 Δ 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 right-hand-rule 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 ) = ˆ x ρ 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 , 0) , (3 , 0 , 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 ˆ x ρ x o · ˆ xdx + 4 y =0 ˆ x ρ 3 o · ˆ ydy + - 3 x =3 ˆ x ρ x o · ˆ xdx + 0 y =4 ˆ x ρ ( - 3) o · ˆ ydy = 3 x = - 3 ρ x o dx + 0 + - 3 x =3 ρ x o dx + 0 = 0 . 1
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Note that in expanding C E · d l above for the given path C , we took d l as ˆ xdx and ˆ 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 counter-clockwise 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 curl-free 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 STOKE’S 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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