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where
µ
is the Earth’s dipole moment and
λ
m
is the magnetic latitude. The ratio of the
field magnitudes for two different distances at the same latitude is
B
B
r
r
2
1
1
3
2
3
=
.
With
B
1
being the value at the surface and
B
2
being half of
B
1
, we set
r
1
equal to the
radius
R
e
of the Earth and
r
2
equal to
R
e
+ h
, where
h
is altitude at which
B
is half its
value at the surface. Thus,
1
2
3
3
=
+
R
Rh
e
e
b
g
.
Taking the cube root of both sides and solving for
h
, we get
()
( )( )
13
3
2
1
2
1 6370km
1.66 10 km.
e
hR
=−=−
=×
(b) We use the expression for
B
obtained in Problem 3255, part (a). For maximum
B
, we
set sin
λ
m
= 1.00. Also,
r
= 6370 km – 2900 km = 3470 km. Thus,
( ) ( )
72
2
2
2
2
0
max
3
3
6
4
4
10
T m A
8.00 10 A m
s
i
n
131
.
0
0
4
4m
3.83 10 T.
m
B
r
µµ
−
−
π×
⋅
×
⋅
=+
λ
=
+
π
π3
.47×10
(c) The angle between the magnetic axis and the rotational axis of the Earth is 11.5°, so
λ
m
= 90.0° – 11.5° = 78.5° at Earth’s geographic north pole. Also
r = R
e
= 6370 km. Thus,
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 Spring '08
 Any
 Physics

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