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329fall10hw5sol

# 329fall10hw5sol - ECE 329 Fall 2010 Homework 5 Solution Due...

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Unformatted text preview: ECE 329 Fall 2010 Homework 5 - Solution Due: Oct. 5, 2010 1. a) According to Page 2 in Lecture 11, the mobility is de ned as mobility = qτ m = q mν . Although mν is not explicitly given in the problem, it is related to N e and σ by (see Page 3 in Lecture 11) σ = N e q 2 mν , or mν = N e q 2 σ , therefore, mobility = q mν = q N e q 2 σ = σ N e q = 5 . 8 × 10 7 8 . 45 × 10 28 × 1 . 6 × 10- 19 = 4 . 3 × 10- 3 ( m 2 / ( V · s ) ) , where the unit is derived from v = qτ m E . b) R = l σ · πr 2 = 1000 5 . 8 × 10 7 × π × (1 . × 10- 3 ) 2 = 5 . 5 (Ω) . c) Since the current is uniform across the wire, the current density is J = I πr 2 . The current density is related to the electric eld by J = σE. ∴ E = J σ = I σ · πr 2 = 1 5 . 8 × 10 7 × π × (1 . × 10- 3 ) 2 = 5 . 5 × 10- 3 ( V/m ) . d) The velocity of an electron is v = qτ m E = mobility · E = 4 . 3 × 10- 3 × 5 . 5 × 10- 3 = 2 . 4 × 10- 5 ( m/s ) , so the time to drift from one end to the other is t = l v = 1000 2 . 4 × 10- 5 = 4 . 2 × 10 7 ( s ) . 1 ECE 329 Fall 2010 Figure 1: Circuit model of spherical capacitor 2. a) The capacitance can be calculated using the given expression C = 4 πεa = 4 π (4 ε )(1) = 445 (pF) From Lecture 10 online, we have a general expression for conductance (this can also be calculated using Gauss's Law) G = σ ε C = 4 πσa = 12 . 6 ( μ S) b) The circuit model for the spherical capacitor is a capacitor (capacitance C) and resistor (resis- tance 1/G) in parallel as shown in Figure 1. The two terminals represent the surface of the spherical conducting shell and a point at in nity. Because the voltage across both elements is the same (equal to the voltage between the shell and in nity), they must be in parallel. You may also consider the limiting behavior of the circuit as R → (material outside of r > a becomes a perfect conductor and the capacitor is shorted out) and as R → ∞ (material outside of r > a becomes a perfect dielectric and the resistor is an open circuit) to see why the parallel con g-...
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329fall10hw5sol - ECE 329 Fall 2010 Homework 5 Solution Due...

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