Environmental Engineering Reference
In-Depth Information
Similarly, consider the electrical field due to an electrified surface of electric-
charge surface density
μ
(
P
)
. The electric potential reads
μ
(
P
)
u
(
M
)=
d
S
,
(7.141)
r
PM
S
∈
where
r
PM
is the distance between
M
and
P
S
. The integral in Eq. (7.141) is called
the
single-layer potential
.
Consider a dipole formed by two point electric charges of capacity
q
at
P
1
and
P
2
along the
l
-axis, respectively (Fig. 7.5). The
l
-axis is called the dipole
axis. Let
+
q
and
−
Δ
l
be the distance between
P
1
and
P
2
. The constant
p
(=
q
Δ
l
)
is called the
dipole moment
. The electric potential at
M
due to a dipole reads
1
r
2
−
1
p
Δ
(
r
)
q
r
2
−
q
r
1
=
p
1
Δ
1
r
1
u
(
M
)=
=
,
l
Δ
l
which is called the
dipole potential
.Nowlet
0andvary
q
so that
P
1
and
P
2
tend to
P
and
p
is kept as a constant. The electric potential at
M
becomes
Δ
l
→
q
r
2
−
1
r
)
q
r
1
0
p
Δ
(
u
(
M
)=
lim
q
→
+
∞
=
lim
Δ
Δ
l
l
→
1
r
p
∂
(
)
r
2
∂
p
r
=
l
=
−
∂
∂
l
p
r
2
cos
p
r
2
cos
=
−
(
MP
,
l
)=
(
PM
,
l
)
,
(7.142)
x
,
y
,
z
)
where
r
=
MP
and, for
M
(
x
,
y
,
z
)
and
P
(
x
−
∂
r
x
r
=
r
=
(
x
−
x
)
2
+(
y
−
y
)
2
+(
z
−
z
)
2
,
x
=
cos
(
MP
,
Ox
)
,
∂
Fig. 7.5
A dipole formed by two point electric charges
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