TIlE
STATIC
ELECTRIC FIELD
101
where the last two terms are the socalled Exponential Integrals for which there is no closed form
solution but which are well tabulated
t
.
(d) The result can be shown by simple substitution into the cylindrical coordinate version
of
Poisson's equation, namely
1 8
2
C1>
~CI>
r
2
8cjJ2
+
8z
2
"""
~
No
variation
in
</J
No
variation
in
z
2
I 8 (
8C1»
V
CI>=
r
+
r8r
8r
p(r)
=
Note that differentiation
of
the Exponential Integral terms simply yield the integrands.
The
electron charge density in a hydrogen atom.
(a) Since there is spherical symmetry, we
consider a spherical Gaussian surface
S
with radius
r
and apply Gauss's law as
where
where we used the integral
which can easily be shown using integration by parts twice. As a result, the electric field
due
to
the
electron cloud only
can
be found as
E
1
4qe
[a
3
2r/a
(ar
2
a
2
r a
3
)]

e
+
+
r
41T1:or2
a
3
4
2
2
4

41T~:r2
{
1
e
2r
/
a
[2(~r
+
2
(~)
+
I]
}
Note
that the constant
a
in the above expression is called the
Bohr radius
given by
a
~
0.529
x
10
10
m.
The
electric potential can be found by integrating the electric field. We find
i
r
qe
2r/a[1
1]
qe
1
CI>(r)
= 
Er(r)dr
=

e

+ 


00
41TfO
a
r
41TfO
r
.
See
Chapter 5 of
M.
Abramowitz
and
I.
A.
Stegun,
Handbook
of
Mathematical Functions With Formulas,
..
aphs,
and
mathematical Tables,
National Bureau of Standards,
Tenth
Printing,
1972.
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102
niB
STATIC
ELECTRIC
FIELD
(b)
Using superposition principle,
the
total electric
field
is
given
by
where
ETc
is
the
field
due to the electron cloud only (found
in
part
(a»
and
E1'p
is the field due to
the proton charge
qe
located at the origin given
by
The electric potential can be found
by
integrating the electric
field.
We
find
4·22. Spherical
sheD
of
charge. (a) The total charge in the spherical shell region specified by
a
S
r
S
b
is given
by
Q
= [
p(
r')dv'
=
[21f
r
[b
p(
r')r,2
sin
fJ'
dr'
dfJ'
drp'
iv'
io
io
ia
b
K
lb
=(21T)(2)
2r,2dr,
=
41TK
dr'
=
41TK(b

a)
l
a
r'
a
(b) Due
to
spherical symmetry, the electric field
has
the form E
=
t
E1'(r).
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 Summer '08
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 Electrostatics, Electric charge, E BR

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