Week 9 Solns - Electromagnetics Workshop Week 9 Solutions...

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Electromagnetics Workshop – Week 9 Solutions Electrostatic Potential, Dielectrics, & Faraday’s Law 1 Problem 1: A regular hexagon of charge with line charge density ρ A exists in the x-y plane centered around the origin. The hexagon is inscribed within a circle of radius a . a) Find the potential due to this charge distribution along the z-axis assuming the reference is at infinity where the potential is zero. b) Repeat a assuming the reference is at the origin where the potential is 5 [V]. Since the hexagon is regular, it can be split up into six equilateral triangles. Hence, the sides of the triangles are of length a . The height of each triangle is 3 2 a . Since the charge density is uniform we can deduce that the electric field (and hence the potential) will be equally affected by each of the six sides of the hexagon. Hence, using superposition () 2 2 2 2 0 2 6 4 32 a a dy z ay z πε Φ= ++ A [] 22 0 ln V 2 2 aa z z z ⎡⎤ ⎢⎥ −+ + ⎣⎦ A 00 33 0l n 3 5 5 l n 3 CC + =⇒ = AA ( ) 0 ln 5 V 2 z z z + + A x y a A
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Electromagnetics Workshop – Week 9 Solutions Electrostatic Potential, Dielectrics, & Faraday’s Law 2 Problem 2: A hollow cylinder of surface charge 22 sin C/m s ρφ = is centered at the origin. Find the potential due to this cylinder along the z-axis. Assume ( ) 0 Φ∞= . Validate this assumption by evaluating your answer in the limit as z →∞ . () 2 2 2 2 2 0 20 sin 4 h h ad d z z az z π φφ πε ′′ Φ= +− ∫∫ 2 2 2 0 2 4 h h z z z ε [] 2 2 2 2 0 ln V 4 zh a a z a ⎡⎤ ++ + + ⎢⎥ −+ + ⎣⎦ 2 2 2 2 0 lim lim 1l n 1 0 4 a a z zz a + + =⇒ Φ = = + x y z h s ρ a
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Electromagnetics Workshop – Week 9 Solutions Electrostatic Potential, Dielectrics, & Faraday’s Law 3 Problem 3: An infinitely long cylindrical shell of uniform charge density 0 s ρ
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This note was uploaded on 10/25/2011 for the course ECE 2317 taught by Professor Staff during the Spring '08 term at University of Houston.

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Week 9 Solns - Electromagnetics Workshop Week 9 Solutions...

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