Geology Reference
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t
Z
P
D
r
Q
r
E
D
O
r 0
R
Figure 5.2 The integration point Q is a distance D from the field point P . PQ
makes an angle t with the fixed line OP . The length of PR is E .
Then, using OZ as the axis of a system of spherical polar co-ordinates ( D , t , p ) with
P as origin,
π
E
D 2 sin t
D
1
D d
V=
dDdt dp
V o
0
0
0
π
r 2 sin 2 t 2
r 0
= π
r cos t
+
sin tdt
0
r 2 π
0
2 r π
0
cos t r 0
cos 2 t sin tdt
r 2 sin 2 t sin tdt
= π
π
r 2 sin 2 t sin tdt
r 0
+
0
r 0
3 r 2
1
=
.
(5.136)
Collecting the results of the foregoing integral transformations, we can write
(5.3) in the form
G
2
2 r 0
3 r 2 P 2 (cosθ)
D d
1
D d
D dS
1
V+
2
Ω
V+
+
Ω
V−V i
V i −V o
S i
U ( r ), r outside S i ,
=
(5.137)
U i ,
r inside S i ,
after substitution for W ( r ) from equation (5.2). The advantage of this expression
is that none of the integrations include the region r < c inside the sphere with
radius equal to the polar radius of the equipotential S i . Taking r < c , the latter of
 
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