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University of Rochester Department of Electrical & Computer Engineering EE234/434 Homework #4 due 13 Feb, 2008 -----------------------------------------------------------------1) Consider the geometry below of a pair of electrodes containing a liquid dielectric of dielectric constant k2 in a surrounding ambient medium k1 (also a liquid). The piston at the right prevents the liquid from moving. Note that the position of the left surface of the liquid is at defined to be x. L1 d1 x V k1 L2 k2 d2 Fext width = w a) Find an expression for the system capacitance as a function of the position of the left surface of the liquid x. b) What is the net force Fext that must be exerted upon the piston on the right side to keep the liquid from moving? 2) Work problem #3.3 from Woodson and Melcher for the non-linear dielectric. NOTE: because the dielectric in this device is non-linear, the usual capacitance cannot be defined. Instead, you must use the integral form of Gauss’s Law to find a terminal relation between the net charge and the voltage and mechanical displacement. This relation will be non-linear but you can integrate to obtain the coenergy. 3) Work problem #3.16 from Woodson & Melcher. NOTE: all electrical and mechanical constraints external to the lossless electromechanical coupling can be imposed only after an expression for the force of electrical origin fe has been obtained in terms of the chosen state variables. 2 of 2 pages 4) The coaxial, rotating capacitive transducer shown below in cross-section, has length L. concentric dielectric arc d = thickness k = dielectric const. k q R cross-sectional view gap g << L,R (a) Find an expression for the device capacitance C(q), assuming that 0 < q < p. Ignore all fringing fields. (b) Use coenergy to determine the torque of electrical origin Te.