hw4 (1) - ECE 329 Homework 4 Due 5PM 1 Maxwell’s...

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Unformatted text preview: ECE 329 Homework 4 Due: Sept 20 2010, 5PM 1. Maxwell’s equations for static electric fields- In a region of free space that is current-free and charge-free, is E = 2 x ˆ x + 2 sin( z )ˆ y − 2 z ˆ z a possible electrostatic field? Discuss. 2. Solving Poisson’s/Laplace’s eq. and Boundary Conditions- Depicted in the figure below is the cross sectional geometry of an infinitely-long cylindrical configuration of charge in free space. The circular cylindrical shell of inner radius a and outer radius b carries a uniformly distributed electric charge Qµ C/m. The cylindrical surface at r = b carries a uniformly distributed electric charge Qµ C/m, and the cylindrical surface at r = c carries a uniformly distributed electric charge − 2 Qµ C/m. Given that the electrostatic potential is zero on the surface r = a , use Poisson’s equation to solve for the electrostatic potential V everywhere in space. Your result should be in terms of the geometric parameters of the cross sectional geometry, the permittivity of the free space and the charge density Q. (Hint: In view of the cylindrical symmetry of the charge density the electric field and the electrostatic potential are function of r only; furthermore, the electric field has only the radial component. Using the expressions for the gradient and the divergence in cylindrical coordinates (see Appendix B in Rao’s text), you should be able to show that the Laplacian of the electrostatic potential in cylindrical coordinates is given by: ∇ 2 V = 1 r ∂ ∂r ( r ∂V ∂r ) + 1 r 2 ( ∂ 2 V ∂ϕ 2 ) + ∂ 2 V ∂z 2 ....
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hw4 (1) - ECE 329 Homework 4 Due 5PM 1 Maxwell’s...

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