P30_016.PDF

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Chapter 30 - 30.16
16. Our y axis is along the wire with the origin at the top endpoint, and the current is in the positive y
direction. All segments of the wire produce magnetic fields at P that are into the page. According to
the Biot-Savart law, the magnitude of the field any (infinitesimal) segment produces at P is given by
dB = µ 0 i
4 π
sin θ
r 2
dy
that line) are functions of y . Replacing r with y 2 + a 2 and sin θ with a/r = a/ y 2 + a 2 ,weintegrate
from y =
a to y = 0. The total field is
B = µ 0 ia
4 π
0
( y 2 + a 2 ) 3 / 2 = µ 0 ia
1
a 2
y
( y 2 + a 2 ) 1 / 2
0
= µ 0 i
4 πa
a
a 2 + a 2
a
4 π
−a
which simplifies to the desired result (noting that
4 2 = 8
).
where θ (the angle between the segment and a li ne draw n from the segment to P ) and r ( the length of
dy
1
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