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Unformatted text preview: Assignment #4, PHYSICS 354/401 ELECTRICITY & MAGNETISM II Mar. 2010 Assignment #4, Page 1 of 4 Solution to Assignment # 4 259 . 6.2, p Given B is uniform. The area of an arbitrary current loop (not necessarily flat) can be resolved into components parallel and perpendicular to B . The component parallel to B contributes no torque. The projected areas can further be broken down into tiny loops so that one can make use of B m N = which is valid for a localized magnetic dipole. Choose the coordinate axes so that B is along the z axis without loss of generality. a Id m d = is a differential magnetic moment. = N I da da da B x y z z ( , , ) ( , , ) 0 0 . Since B is constant, [i.e. not a function of the variables of integration], it makes no difference whether we sum the areas first before taking the cross product or viceversa. Cross first: z x z y z x z y z z y x B m y B m x B da y I B da x I B da da da I N ) , , ( ) , , ( = = = Sum first: z x z y z z y x z z y x B m y B m x B m m m B da da da I N ) , , ( ) , , ( ) , , ( ) , , ( = = = Thus N I da B I da B m B = = = iff B is uniform in magnitude and direction. 259 . 6.3, p Rotate Figure 6.7 ( p. 259) 90 , to have both dipoles ( m 1 , m 2 ) pointing in the zdirection. (a) Use Eq. 6.2: , cos 2 1 2 B I R F = since only the horizontal component of B 1 contribute to a force. Here is the angle B 1 makes with x at l d I 2 of m 2 as shown in the diagram. This formula is based on fact that the current loop m 2 has a finite R , and the B field is no longer uniform over the loop, (direction varies although magnitude is reasonably constant). + = sin cos 2 4 ) , ( 3 1 1 1 3 1 1 1 1 r m r r m r B o ( r 1 >> dimensions of m 1 ) ( ) ( ) 1 1 1 3 1 1 3 1 1 3 1 1 3 1 1 1 1 1 1 1 sin , 1 cos with 4 3 sin cos 3 4 cos sin sin cos 2 4 cos sin cos r R r z r R r m r m r m r m B B B B B o o o r x x r = = = + = + = + = , 6 4 4 3 2 4 1 2 1 4 1 1 2 r m m r R m R I F o o = 2 2 2 R I m = ....
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This note was uploaded on 12/15/2010 for the course PHYS 354 taught by Professor Josephshin during the Winter '10 term at Concordia Canada.
 Winter '10
 JosephShin
 Current, Magnetism

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