hw4_solution - Eigiflf*fnfir‘ ECE604 Homework 4 Out:...

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Unformatted text preview: Eigiflf*fnfir‘ ECE604 Homework 4 Out: Tuesday, February 1, 2005 Due: Tuesday, February 8, 2005 Problems given by number are from the text by Ramo, Whinnery and Van Duzer (3rd edition, 1994). l) 7.12h Also give an expression for the added capacitance per unit length AND evaluate numerically. Hints: a) Use a superposition solution CI) = (111+ CD2 , where CD1 = Voy/ a is the solution for the parallel plate capacitor without the extra conducting strip. b) In solving for CD2 , solve separately for x 2 0 and x S 0. Once you have the Solution for x 2 0 , the solution for x S 0 follows almost by inspection. c) Use your solution for (1)2 in computing the added capacitance. 2) a) Shown below are two infinitely long, very thin insulated conducting plates which are held at potentials V0 and 0 as indicated. (The plates are infinite in the z—direction and semi-infinite in the other dimension.) qéflo YT \ I; ¢ 06‘. ‘. —....__.__l__._..._.___ ___J____> V= 0 X Determine the potential distribution for 0 < ¢ < 0t and for 0t < (1) < 27:. b) Shown below are two infinitely long, very thin conducting plates separated by a dielectric of permittivity 8, which are held at potentials V0 and 0 as indicated. (The plates are infinite in the z-direction and finite in the x-y plane.) (i) Neglecting the effects of fringing, determine the capacitance of the structure per unit length. (ii) Neglecting the effects of fringing and without using the relationship WE = %C V2 , determine the energy stored in the structure per unit length. Show that your result is in fact equal to %C V2. 3) A solid dielectric cylinder of radius a and length 2L is uniformly polarized with polarization P, where P is directed axially (i.e. P = p2 ; where p = const.). Determine the electric field, E, along the cylinder axis inside and outside of the cylinder (Le. E(z) Ix=y=0). Hint: Identify the bound charge at the surfaces. These bound charges act as a source for E, which can be obtained through integration. 4) A sphere of radius a has electric charge distributed on its surface so that the electric field inside is uniform and given by E = E02 (for r < a) The sphere is hollow, i.e. 8 = so everywhere inside the sphere. The sphere is embedded in an insulator, with dielectric constant 8. a) Determine the surface charge density ps(6) on the surface of the sphere. Hint: First solve for the potential using Laplace’s equation and appropriate boundary conditions. 5) 6) 7) 8) 9) b) Make a sketch showing representative E—field lines and the distribution of surface charge. Comment on the behavior of the field lines at the boundary. 2.3b 2.3c 2.4b Also find Hq, (r) outside the beam. 2.4e 2.5 '1 4211' , As Seen in figures at fight '. 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This note was uploaded on 12/12/2010 for the course ECE 604 taught by Professor Staff during the Spring '08 term at Purdue University-West Lafayette.

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hw4_solution - Eigiflf*fnfir‘ ECE604 Homework 4 Out:...

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