Chapter2.Dipole.surface.charge.pages

Chapter2.Dipole.surface.charge.pages - Fields and potential...

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Fields and potential due to a surface electric dipole layer A surface electric dipole layer is a neutral charge layer with an electric dipole moment per unit area directed perpendicular to the surface. It can be modeled as two surface charge layers, ( r , ) and ( r , ) , lying on each side of the surface defined by F ( r ) = 0. The unit vector n = F ( r )/|∇ F ( r )| is directed from the negative surface charge density to the positive surface charge density (it’s sufficient to replace F ( r ) by F ( r ) in order to adjust the sense of n ). The charge layers lie on the surfaces F ( r ± n / 2 ) and the surface dipole moment density is d ( r ) = 0 lim n ( r , ) = d ( r ) n 0 lim ( r , ) = 0 0 lim ( r , ) = d ( r ) 1.103 Since the surface is neutral (total charge = 0), in the limit that 0 with ( r , ) fixed , n ( r ) ( 1 E 1 ( r ) − 2 E 2 ( r )) = o for electric dipole surface layer 1.104 From Gauss’ law one can determine the electric field contributions, E + and E , and from Eq. (1.99) the potential field contributions, + and, , from each individual layer . The latter contributions are shown in the schematics below. The total E 1 and 1 (above the dipole layer) and total E 2 and 2 (below the dipole layer) are given by E + + E and + + .in each region Schematic of the E field contributions ( r , ) n 1 / 2 ++++++++++ ( r , ) n 1 / 2 ( r , ) n 2 / 2 −−−−−−−−− ( r , ) n 2 / 2 where n 1 = n 2 and the negative sign on the field due to the lower layer comes from the negative charge ”surface layer”.
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Schematic of the field contributions − | z | ( r , )/ 2 ++++++++++ − | z | ( r , )/ 2 | z | ( r , )/ 2 −−−−−−−−− | z | ( r , )/ 2 Thus the total electric fields are: E 1, total ( r , ) = ( r , ) n 1 / 2 ( r , ) n 2 / 2 = 0 above the dipole layer E 2, total ( r , )=− ( r , ) n 1 / 2 + ( r , ) n 2 / 2 = 0 below the dipole layer giving n 1 [ E 1, total E 2, total ]= 0 Between the two layers for finite the electric field is constant, directed downward and equal to, E 3, total ( r , ( r , ) n 1 / 2 ( r , ) n 2 / 2 =− ( r , ) n 1 / . The total potential above the dipole layer (where | z 2 | = + | z 1 | )is 1, total ( r , ) = + ( r , ) + ( r , )=−| z 1 | ( r , )/ 2 + | z 2 | ( r , )/ 2 =  ( r , )/ 2 Below the dipole layer (where | z 1 | = + | z 2 | ) . 2, total ( r , ) = | z 2 | ( r , )/ 2 − | z 1 | ( r , )/ 2  ( r , )/ 2 , and for finite between the two charge layers where | z 2 | = − | z 1 | the total potential is 3, total ( r , z 1 | ( r , )/ 2 + | z 2 | ( r , )/ 2 =
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Chapter2.Dipole.surface.charge.pages - Fields and potential...

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