Homework_8 - Homework 8 solutions 5.20. (a) The force is (...

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Unformatted text preview: Homework 8 solutions 5.20. (a) The force is ( ) ( ) 3 . V S d x da = + F M B M n B Consider the i th component of the first integrand. This is ( ) ( ) ( ) ( ) ( ) . m m i k ijk jlm k kl im km il k k k i l l k i k k i k k k k i i k k i i M M M M B B B B x x x x B M M B B M x x x B M x = =- =- =- + = - + M B B M B M The integrand of the second integral is ( ) ( ) ( ) . = - M n B B M n B n M Let q be a fixed but arbitrary vector. Taking the scalar product of q and F , we get ( ) ( ) ( ) ( ) 3 . k i k i V S B q M d x d x = - + + - q F q B M q B M B M q q M B S Converting the surface integral to a volume integral by using the divergence theorem, we find ( ) ( ) ( ) ( ) ( )( ) 3 3 . k i k i V k i k i V B q M d x x B q M d x x = +- = - - + q F q B M q M B q B M q M B B q M Consider ( ) ( )( ) ( )( ) 0, j i i j j i j i M M q B q B x x - - =- = q B M q M B B q M where we have used 0. = B Hence 3 . k i k i V B q M d x x = q F Using 0, = B this becomes ( ) 3 3 3 . k i i i k k i k i i i k i k i i k k k k V V S V M q B B M M q M d x q B d x q M B dS q B d x x x x x = =- =- q F Since q is arbitrary, we finally get the desired result ( ) ( ) 3 . S V dS d x = - F M n B M B (b) Because the magnetization is uniform inside the sphere, there is only the surface contribution to the force. In terms of spherical polar coordinates and Cartesian unit vectors, ( ) ( ) ( ) sin cos sin sin cos , sin cos sin sin cos , 1 sin sin , 1 sin cos . x y M B B R B B R = + + = + + = + = + M i j k n i j k Hence ( ) ( )( ) ( ) ( )( ) 2 2 0 0 2 2 0 0 sin cos sin cos sin sin sin sin cos cos 1 sin sin sin , sin cos sin cos sin sin sin sin cos cos 1 sin cos sin . x x S y y S F B dS MR B R d d F B dS MR B R d d = = + + + = = + + + M n M n Noting that the integrals of the terms linear in sin , cos , sin 2 are zero, this simplifies to 2 2 3 2 0 0 2 2 3 2 0 0 cos cos sin sin sin sin sin . cos cos sin sin cos sin cos . x y F MR B R d d F MR B R d d = + = + The first terms in the integrands also contribute zero to the integrals. Hence 2 3 3 2 2 3 3 2 sin sin sin sin ....
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Homework_8 - Homework 8 solutions 5.20. (a) The force is (...

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