P30_015 - a/ 2 R = a a 2 + 4 x 2 multiplies the expression...

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15. We imagine the square loop in the yz plane (with its center at the origin) and the evaluation point for the Feld being along the x axis (as suggested by the notation in the problem). The origin is a distance a/ 2 from each side of the square loop, so the distance from the evaluation point to each side of the square is, by the Pythagorean theorem, R = p ( a/ 2) 2 + x 2 = 1 2 p a 2 +4 x 2 . Only the x components of the Felds (contributed by each side) will contribute to the Fnal result (other components cancel in pairs), so a trigonometric factor of
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Unformatted text preview: a/ 2 R = a a 2 + 4 x 2 multiplies the expression of the Feld given by the result of problem 11 (for each side of length L = a ). Since there are four sides, we Fnd B ( x ) = 4 i 2 R a a 2 + 4 R 2 a a 2 + 4 x 2 = 4 i a 2 2 ( 1 2 )( a 2 + 4 x 2 ) 2 p a 2 + 4( a/ 2) 2 + 4 x 2 which simpliFes to the desired result. It is straightforward to set x = 0 and see that this reduces to the expression found in problem 12 (noting that 4 2 = 2 2)....
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