fa01em_sol

fa01em_sol - Proficiency Exam, Fall 2001 Friday, September...

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Unformatted text preview: Proficiency Exam, Fall 2001 Friday, September 14 Part 1 9:00 a.m. to 1:00 p.m. Useful constants: e = 1 . 60 10- 19 C hc = 1240 eV nm c = 3 . 00 10 8 m s m e = 0 . 511 MeV c 2 k = 1 4 = 8 . 99 10 9 Nm 2 C 2 = 8 . 85 10- 12 C 2 Nm 2 4 = 10- 7 N A 2 g = 9 . 81 m s 2 1. Four infinitely long wires parallel to the z-axis are arranged with ( x, y ) = (0 , d ) , (0 ,- d ) , ( d, 0) , and (- d, 0) , respectively. (Here d > .) The wires at (0 , d ) and (0 ,- d ) carry a current of magnitude i in the positive z-direction, and the wires at ( d, 0) and (- d, 0) carry a current of magnitude i in the negative z-direction. (a) What are the magnitude and direction of the magnetic field at any point along the x-axis? For better understanding we try to explicitly calculate almost all part of the problem. First of all consider a point on the x-axis and a point on each one of the wires. We have to determine contribution from each wire to the total magnetic field for a point on the x-axis. 1. First from the wire with coordinate (0 , d, z ). Here we denote the source points with prime and the field points unprimed, see figure. The magnetic field from an element of length dl at point ( x, , 0) is given by Biot-Savart law. d ~ B 1 = I 4 ~ dl ~ R R 3 (1) ~ R = ~ r- ~ r (2) ~ r = ( x, , 0) , ~ r = (0 , d, z ) (3) ~ R = ( x,- d, z ) (4) d ~ l = z dl (5) d ~ l ~ R = ( x d + y x ) (6) R r r d l y z x I Substitute Eqs.2-5 in Eq.1 and integrate ~ B 1 = I 4 Z - dz x d + y x x 2 + d 2 + z 2 3 2 (7) let x 2 + d 2 = a 2 and tan( ) = z a dz a = (1 + tan 2 ( )) d d = dz a (1 + tan 2 ( )) = dz a (1 + ( z a ) 2 ) note also 1 1 + ( z a ) 2 = 1 1 + tan 2 ( ) = cos 2 ( ) and, z ]- [ ]- 2 2 [ (8) Rewrite Eq.7 we have ~ B 1 = I 4 a 2 ( x d + y x ) Z - dz a (1 + ( z a ) 2 ) s (1 + z a ) 2 integrand is even function of z f ( z ) = f (- z ) ~ B 1 = I 4 x d + y x x 2 + d 2 2 Z 2 d cos( ) note that cos( ) 0 for...
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fa01em_sol - Proficiency Exam, Fall 2001 Friday, September...

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