309-2008-Solutions7

309-2008-Solutions7 - ECE 309 Electromagnetic Fields...

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Unformatted text preview: ECE 309 Electromagnetic Fields University of Virginia Fall 2008 Homework # 7 Solutions 1. B-field at center of regular N-sided polygon. For an N-sided regular polygon, each side subtends an angle of 2 = 2 /N : The B-field at the center will be the superposition of the B-fields for each side of the polygon. Using the expression (derived in class) for the field a distance b cos from the center of a wire of length = 2 b sin , we have ~ B one side = I 4 b cos 2sin n where n is the direction of the field (into or out of the page). Thus for the N-sided polygon, ~ B ( P ) = N I 4 b cos 2sin n or, ~ B ( P ) = n NI 2 b tan N as N , /N becomes small and the small-angle approximation pertains: tan N N Thus, ~ B ( P ) = n I 2 b , for N which is the same as the field at the center of a circular loop. 2. Current flowing down a cylindrical wire. This problem exhibits cylindrical symmetry, so we can apply Amperes Law to find ~ B , I ~ B d ~ = I enc = Z S ~ J d~a where ~ J = kr z . Using a coaxial circle (with radius r ) as the Amperian loop, we note that B is azimuthally- directed and constant along such a loop. Thus 2 rB = k Z 2 d Z r r 2 dr = 2 k 3 r 3 r < R 2 k 3 R 3 r > R Thus, the B-field is, ~ B = k 3 r 2 , for r R ~ B = k 3 R 3 r , for r R 3. The Hall Effect. (a) The force on a moving charge is given by the Lorentz force Law, ~ F = q~v ~ B The current consists of positive charges moving to the right with average velocity ~v , so ~v...
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This note was uploaded on 10/16/2010 for the course ECE 309 taught by Professor Weikle during the Spring '08 term at UVA.

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309-2008-Solutions7 - ECE 309 Electromagnetic Fields...

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