HW84_3 - at ) , , ( = = ϕ = z a r . Problem 3 In free...

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Electromagnetics School of Electrical and Computer Engineering University of Tehran Homework 3 Fall Semester 1384 Due 84/8/9 Problem 1 The corners of a rectangular current loop are located at the points ) 0 , , ( 1 1 b a , ) 0 , , ( 1 2 b a , ) 0 , , ( 2 2 b a , and ) 0 , , ( 2 1 b a of a rectangular coordinate system. Assume that 0 , , , 2 1 2 1 > b b a a . The loop is infinitely thin and carries a current of I in counter-clockwise direction. Consider the vector magnetic potential A r generated by this current loop on the plane 0 = ϕ of the corresponding spherical coordinate system. Show that in this plane, for 2 2 1 2 2 1 ) ( ) ( b b a a r + + + >> , a) 3 / 1 ˆ r A x A x = r and b) 2 / 1 ˆ r A y A y = r . Problem 2 In a cylindrical coordinate system the region ) 0 , 2 0 , 0 ( π < ϕ < z a r is occupied by microscopic electric dipoles the polarization vector of which is z P P o ˆ = r with o P as a constant. Assume free space everywhere and determine the electric potential Φ generated by this dipoles
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Unformatted text preview: at ) , , ( = = ϕ = z a r . Problem 3 In free space, the region inside a hemisphere of radius a is uniformly filled by identical, z-directed, microscopic electric dipoles. The center of hemisphere is at the origin and its north pole is on the positive z-axis. Measurement of the electric potential Φ along the z-axis shows that the dipoles produce 2 / 3 z = Φ for a z &gt;&gt; . a) Evaluate the polarization vector P r of the dipoles. b) Determine the electric field generated by the dipoles at the origin. Problem 4 A uniform volume current of density z J J o ˆ = r [A/m 2 ] exists in the region a r &lt; ≤ of a cylindrical coordinate system. Determine the vector magnetic potential produced by this volume current. (Assume = A r for = r .) M. Shahabadi...
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