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76. We note that for two points on a circle, separated by angle
θ
(in radians), the direct
line distance between them is
r
= 2
R
sin(
/2). Using this fact, distinguishing between the
cases where
N
= odd and
N
= even, and counting the pairwise interactions very carefully,
we arrive at the following results for the total potential energies. We use
k
=
14
π
0
ε
. For
configuration 1 (where all
N
electrons are on the circle), we have
()
1
1
22
1,
even
1,
odd
11
1
,
2
sin
2
2
2
sin
2
NN
jj
Nke
Nke
UU
Rj
θθ
−
−
==
§·
§
·
¨¸
¨
¸
=+
=
¨
¸
¨
¸
©¹
©
¹
¦¦
where
=
2
π
N
.
For configuration 2, we find
U
Nk
e
U
e
N
j
N
N
j
N
2
2
1
2
1
2
2
1
3
2
1
2
1
2
2
1
2
1
2
5
2
,
,
sin
=
=
−
=
=
−
=
−
′
+
F
H
G
G
I
K
J
J
=
−
′
+
F
H
G
G
I
K
J
J
¦
¦
even
odd
b
g
b
g
b
g
b
g
where
′ =
−
2
1
π
N
.
The results are all of the form
U
ke
R
1
2
2
or2
a pure number.
×
In our table, below, we have the results for those
“pure numbers”
as they depend on
N
and on which configuration we are considering. The values listed in the
U
rows are the
potential energies divided by
ke
2
/2
R
.
N
4
5
6
7
8
9
10
11
12
13
14
15
U
1
3.83
6.88
10.96
16.13
22.44
29.92
38.62
48.58
59.81
72.35
86.22
101.5
U
2
4.73
7.83
11.88
16.96
23.13
30.44
39.92
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 Spring '10
 Rebello,NobelS

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