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

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ECE-329 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 < 1 . 1
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ECE-329 Summer 2009 c) Given that ρ s = 3 o C / m 2 at z = 1 m, and applying Maxwell’s boundary condition ρ s = - D z | z =1 - , we have that 3 o = 3 1 2 2 1 + 2 .
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