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Unformatted text preview: Homework 10 Answers, 95.657 Fall 2011, Electromagnetic Theory I Dr. Christopher S. Baird, UMass Lowell Problem 1 Jackson 5.1 Starting with the differential expression d B = I 4 d l ' x x ' x x ' 3 for the magnetic induction at the point P with coordinate x produced by an increment of current I d l ' at x ', show explicitly that for a closed loop carrying a current I the magnetic induction at P is B = I 4 where is the solid angle subtended by the loop at the point P. This corresponds to a magnetic scalar potential, M = - I /4. The sign convention for the solid angle is positive if the point P views the inner side of the surface spanning the loop, that is, if a unit normal n to the surface is defined by the direction of current flow via the right-hand rule, is positive if n points away from the point P , and negative otherwise. This is the same convention as in Section 1.6 for the electric dipole layer. SOLUTION: Start with the differential expression and put all the constants on the other side to get them out of the way: 4 I d B = d l ' x x ' x x ' 3 This is a vector equation that must hold for all components. If we take one component in a general way, then it will apply to all components. Let us take the i th Cartesian component. 4 I d B i = x i [ d l ' x x ' x x ' 3 ] now integrate over a closed loop to get the total field due to the loop: 4 I B i = x i [ d l ' x x ' x x ' 3 ] We want to try to do this integral. Use the identity x x ' x x ' 3 = ' ( 1 x x ' ) 4 I B i = x i [ d l ' ' ( 1 x x ' ) ] Use the vector identity a ( b c )= b ( c a ) 4 I B i = d l ' [ ' ( 1 x x ' ) x i ] Use Stoke's theorem to convert the line integral to an area integral over the surface bounded by the closed line integral. 4 I B i = [ ' [ ' ( 1 x x ' ) x i ] ] n ' da ' Now use the identity: ' ( a b )= a ( ' b ) b ( ' a )+( b ' ) a ( a ' ) b and set a = ' ( 1 x x ' ) and b = x i ' ( ' ( 1 x x ' ) x i )= x i ( ' 2 ( 1 x x ' ) )+( x i ' ) ' ( 1 x x ' ) Use this identity: 4 I B i = [ x i ( ' 2 ( 1 x x ' ) ) ] n ' da ' + [ ( x i ' ) ' ( 1 x x ' ) ] n ' da '...
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