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329sp09hw10 - ECE 329 Homework 10 Due Thu April 9 2009 5PM...

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ECE 329 Homework 10 Due: Thu, April 9, 2009, 5PM 1. In a region of space electrostatic potential is known to be Φ ( x, y, z ) = - 4( y 2 + 4) z . a) What is the corresponding electrostatic field E ? b) What is the volumetric charge density ρ in the region if = 2 o ? Hint: you can use the Poisson’s equation in this problem. c) Repeat (b) if = o (1 + e - z 2 ) . Hint: you cannot use the Poisson’s equation in this problem. 2. Consider a simplified model of a vacuum diode consisting of a cathode in the x = 0 plane and an anode in the x = d plane, where the anode is held to a constant potential Φ 0 = 1 V relative to the cathode. If the potential distribution in the region 0 < x < d is given by Φ = Φ 0 ( x/d ) 4 / 3 , find the following: a) Field E at x = d/ 2 , b) Volumetric free-charge density ρ at x = d/ 4 , c) The surface charge density ρ s on the anode. Hint: you make use of relations E = - d Φ dx ˆ x , · o E = ρ , and ˆ n · ( D 1 - D 2 ) = ρ s , treating the interior of vacuum diode as free-space containing a free charge carrier density of ρ = ρ ( x ) .
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