329sum09hw5sol - ECE-329 Summer 2009 Homework 5 Solution...

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Summer 2009 Homework 5 — Solution July 5, 2009 1. The region between two conducting plates is filled by two dielectrics having permittivities ± 1 = 2 ± o and ± 2 as shown in the next figure. x y z ǫ 1 = 2 ǫ o ǫ 2 z = d z = 1 m z = 0 m In addition, the electric field between the plates is known to be E ( z ) = ( - 3 ± 2 2 ± 1 + ± 2 ˆ z V m , 0 < z < d, - 3 ± 1 2 ± 1 + ± 2 ˆ z V m , d < z < 1 . a) First, let us verify that the component of the displacement field normal to the interface at z = d (which in this case is the z -component) satisfies Maxwell’s boundary condition D z | z = d - = D z | z = d + . Since D z | z = d - = ± 1 · E z | z = d - = - 3 ± 1 ± 2 2 ± 1 + ± 2 and D z | z = d + = ± 2 · E z | z = d + = - 3 ± 1 ± 2 2 ± 1 + ± 2 , then the field given above satisfies Maxwell’s condition. b) The electrostatic potential V ( z ) for 0 < z < 1 is given by V ( z ) = - ´ z 0 E z dz. After performing the integration, we can find that V ( z ) = ( 3 ± 2 2 ± 1 + ± 2 z V , 0 < z < d, 3 ± 1 2 ± 1 + ± 2 z + 3( ± 2 - ± 1 ) 2 ± 1 + ± 2 d V , d < z <
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329sum09hw5sol - ECE-329 Summer 2009 Homework 5 Solution...

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