Geoscience Reference
In-Depth Information
of the field is
B
r
:
B
r
(
r
,
θ
,
φ
)
=−
µ
0
∂
V
∂
r
1
r
2
µ
0
m
cos
θ
4
π
∂
∂
r
=
(3.4)
2
µ
0
m
cos
θ
4
π
r
3
=−
The component of the field in the
direction is
B
θ
:
B
θ
(
r
,
θ
,
φ
)
=−
µ
0
1
θ
r
∂
V
∂θ
4
π
r
3
∂
µ
0
m
=
(cos
θ
)
∂θ
=−
µ
0
m
sin
θ
4
π
r
3
(3.5)
The third component is
B
φ
:
1
r
sin
θ
∂
V
∂φ
B
φ
(
r
,
θ
,
φ
)
=−
µ
0
=
0
(3.6)
Note that, by symmetry, there can obviously be no field in the
φ
(east) direction.
The total field strength at any point is
B
r
+
B
2
θ
+
B
2
B
(
r
,
θ
,
φ
)
=
φ
4
π
r
3
1
+
3 cos
2
µ
0
m
=
θ
(3.7)
Along the north-polar axis (
θ
=
0) the field is
B
r
(
r
,0,
φ
)
=−
µ
0
m
2
π
r
3
(3.8)
B
θ
(
r
,0,
φ
)
=
0
θ
=
90
◦
) the field is
B
r
(
r
, 90,
φ
)
=
0
B
θ
(
r
, 90,
φ
)
=−
µ
0
m
4
π
r
3
On the equator (
(3.9)
180
◦
) the field is
B
r
(
r
, 180,
φ
)
=
µ
0
m
2
π
r
3
B
θ
(
r
, 180,
φ
)
=
0
and along the south-polar axis (
θ
=
(3.10)
If we define a constant
B
0
as
µ
0
m
4
B
0
=
(3.11)
π
R
3
where
R
is the radius of the Earth, then Eqs. (3.4) and (3.5)give the components
of the magnetic field at the Earth's surface:
B
r
(
R
,
θ
,
φ
)
=−
2
B
0
cos
θ
(3.12)
B
θ
(
R
,θ,φ
)
=−
B
0
sin
θ
(3.13)
