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23. (a) We refer to the (very large) wire length as
L
and seek to compute the ﬂux per meter: Φ
B
/L
.
Using the righthand rule discussed in Chapter 30, we see that the net Feld in the region between
the axes of antiparallel currents is the addition of the magnitudes of their individual Felds, as
given by Eq. 3019 and Eq. 3022. There is an evident reﬂection symmetry in the problem, where
the plane of symmetry is midway between the two wires (at what we will call
x
=
`/
2, where
`
=20mm=0
.
020m)
;thenetFe
ldatanypo
in
t0
<x<`/
2 is the same at its “mirror image”
point
`
−
x
. The central axis of one of the wires passes through the origin, and that of the other
passes through
x
=
`
. We make use of the symmetry by integrating over 0
<x<`/
2andthen
multiplying by 2:
Φ
B
=2
Z
`/
2
0
BdA
=2
Z
d/
2
0
B
(
Ldx
)+2
Z
`/
2
d/
2
B
(
Ldx
)
where
d
=0
.
0025 m is the diameter of each wire. We will use
R
=
d/
2, and
r
instead of
x
in the
following steps. Thus, using the equations from Ch. 30 referred to above, we Fnd
Φ
B
L
=2
Z
R
0
µ
µ
0
i
2
πR
2
r
+
µ
0
i
2
π
(
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 Fall '08
 SPRUNGER
 Physics, Current

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