hw3 - ECE 329 Homework 3 Due 5PM 1 In free space nd E if...

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ECE 329 Homework 3 Due: Sept 14, 2010, 5PM 1. In free space, find E if the scalar potential is V ( x, y, z ) = xz 2 V. What is the static charge density ρ ( x, y, z ) ? 2. Given that E = x ˆ x + y ˆ y 4 ˆ z V/m, determine the potential V ( x, y, z ) if V (0 , 0 , 0) = 0 . 3. Coulomb’s field of a charge Q stationed at the origin of a right-handed Cartesian coordinate system can be expressed as E = Q 4 πϵ o r 2 ˆ r = Q 4 πϵ o r 2 ( x, y, z ) r , where r 2 x 2 + y 2 + z 2 and r 0 . a) Verify that ∇ × E = 0 by showing that when ∇ × E is expanded as usual, all of its Cartesian components cancel out exactly. b) Assuming that the electrostatic potential V associated with E is zero at r = , show that V = Q 4 πϵ o r . Hint: In this spherically symmetric situation, E = −∇ V = ∂V ∂r ˆ r . 4. Consider a static charge distribution consisting of two infinitesimally thin, parallel sheets of charge in the z = 0 and z = 2 planes. The bottom sheet (at z = 0 ) has an unknown surface charge density ρ s C/m 2 , while the top sheet has a surface charge density of
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