Geoscience Reference
In-Depth Information
The integration limits are introduced by the product of Heaviside's step functions
.
u
.// . .
u
.///: We h ave
Z
e
u
./
2
d
u
".a/
D
.
u
/
d
.
u
.// . .
u
.///d;
(3.146)
1
where
u
./ is given by Eq.
3.142
,
u
is the positive zero of this function, and
"
"
C
1
S./
#
:
4
3
.
2
2
d
u
2
"
C
1
1
"
d
D
1/
3
C
1/
2
C
(3.147)
.
2
3.5.5
Charging of Metallic and Small Dielectric Particles
Less general and more observable results can be obtained for metallic particles
"
D1
and small dielectric particles a
ˇe
2
.
C
3.5.5.1
Neutral Particle
Ion
A neutral particle interacts with ions via image forces whose potential is always
attractive (Q
D
0 and
D
0 in Eqs.
3.139
,
3.140
,
3.141
,
3.142
,
3.143
,
3.144
,
3.145
,
3.146
,
3.147
,
3.148
,
3.149
,
3.150
,
3.151
,
3.152
,
3.153
,
3.154
,
3.155
,
3.156
,
3.157
,
3.158
,
3.159
).
We begin by considering a metallic particle ("
!1
, S./
D
1=
2
2
1
).
Equation
3.115
reduces to
u
D
2
1
2
and can be solved analytically to give
1
p
u
:
2
.
u
/
D
1
C
(3.148)
From Eq.
3.145
we can find :
.
.
u
//
D
u
C
2
p
u
:
(3.149)
Equation
3.115
immediately leads to the familiar result (Natanson
1960
):
r
ˇq
2
e
2
2a
Z
1
C
1=
p
u
d
u
D
1
C
e
u
".a/
D
:
(3.150)
0
For dielectric particles we must use the general result, Eqs.
3.146
and
3.147
with
D
0. However, some analytical results can be found in the limit of small particles
a
ˇe
2
. We will derive a two-term (approximate) formula similar to Eq.
3.148
.
