329fall09hw3 - ECE 329 Homework 3 Due 5 PM 1 Coulomb’s...

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Unformatted text preview: ECE 329 Homework 3 Due: September 15, 2009, 5 PM 1. Coulomb’s field of a charge Q stationed at the origin of a right-handed Cartesian coordinate system can be expressed as Q Q (x, y, z ) E= r= ˆ (1) 2 4π 0 r 4π 0 r2 r where r2 = x2 + y 2 + z 2 and r ≥ 0. a) Verify that × E = 0 by showing that when components cancel out exactly. × E is expanded as usual, all of its Cartesian b) Assuming that the electrostatic potential V associated with field E is zero at r = ∞, verify that V = 4πQ0 r . Hint: In this spherically symmetric situation, V = ∂V r. ∂r ˆ 2. Given the fields E = y x ± xy V/m, determine the circulation C E · dl along a triangular path C in ˆ ˆ the counterclockwise direction having vertices (x, y, x) = (−1, −1, 0), (1, −1, 0), and (1, 1, 0) m. Hint: x = y and dx = dy along the slant edge of contour C . 3. Given that E = 3ˆ + y y + 3z z V/m, determine the electrostatic potential V (1, 2, 3) if V (0, 0, 0) = 0. x ˆ ˆ 4. Consider a static charge distribution consisting of two infinitesimally thin, parallel sheets of charge in the z = 0 and z = 5 planes. The bottom sheet (at z = 0) has a surface charge density of 6 C/m2 , while the surface charge density of the top sheet is unknown and specified as ρ0 C/m2 . The displacement field in the region between the sheets is known to be D = 2ˆ + 10ˆ C/m2 , which is a x z superposition of the field generated by the surface charges together with a constant background field generated by a far away charge source. In the region z > 5 m, it is known that Dz = 4 C/m2 . a) Determine the unknown surface charge density ρ0 b) Write an expression for the volumetric charge density ρ C/m3 in the region in terms of the surface charge densities and appropriately shifted delta functions. c) Determine D in the region z > 5 m d) Determine D in the region z < 0 m 1 ...
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