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set1solution - ECE 222B, Winter, 2010 Solutions to homework...

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ECE 222B, Winter, 2010 Solutions to homework set 1 1. Assume that you are given the parallel plate capacitor as shown in Figure 1. The region between the plates is filled with two layers of dielectric, each with a finite (and different) conductivity ( σ 1 and σ 2 ). In what follows assume that x and y dependencies can be neglected, and assume that all time variations are slow enough so that magnetic field may be neglected and that the electric field within each of the dielectrics is independent of z . Note, however, that the electric field can be a function of time, and that the fields within the two regions can differ. (a) Assume that for times t < 0 the capacitor is totally discharged and the voltage drop across the plates is 0. At time t = 0 a voltage drop of V 0 is established across the plate and maintained. What are the electric fields E 1 and E 2 in the two dielectric regions just after the voltage drop is established? Justify your solution. You may assume that the electric field points in the z direction. Solution: For very early times no charge can accumulate on the dielectric boundary. Thus, E 1 = E 2 . Since V 0 = ( L/ 2)( E 1 + E 2 ), we obtain E 1 (0) = E 2 (0) = V 0 /L . (b) Assuming that the voltage drop has been established for a time long compared for any transients to die out, what is the final (asymptotic) value of E 1 and E 2 . Note that this can be obtained without solving for the time-dependent evolution of the capacitor. Solution: At late times, steady-state is achieved, and thus J 1 = J 2 at the dielectric boundary (no charge accumulates). Thus, σ 1 E 1 = σ 2 E 2 and V 0 = ( L/ 2)( E 1 + E 2 ), leading to E 1 ( ) = 2 σ 2 σ 1 + σ 2 V 0 L E 2 ( ) = 2 σ 1 σ 1 + σ 2 V 0 L (c) Prove that, for this problem setup σ 2 E 2 ( t ) - σ 1 E 1 ( t ) = - ∂t ( εE 2 ( t ) - εE 1 ( t )) (1) Solution: Starting with the continuity equation ∇ · ~ J = - ∂t ρ = - ∂t ∇ · ~ D and then integrating in z across the dielectric boundary yields J 2 - J 1 = - ∂t ( D 2 - D 1 ). Using J 1 = σ 1 E 1 , J 2 = σ 2 E 2 , D 1 = εE 1 , and D 2 = εE 2 yields the desired result. (d) Calculate the charge per unit area (as a function of time) that ac- cumulates at the boundary between the two dielectrics. What is 1
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the e-folding time associated with the evolution of the charge (and associated electric fields)?
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set1solution - ECE 222B, Winter, 2010 Solutions to homework...

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