S. M. AL-Jaber / Natural Science 2 (2010) 760-763

Copyright © 2010 SciRes. OPEN ACCESS

763

763

Table 1. The electrostatic energy of the shell and the sphere as

function of space dimension.

N 2

0

(/8)

shell

WQ

2

0

(/4)

sphere

WQ

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

30

40

50

100

1

0.318

0.159

0.101

0.076

0.065

0.060

0.062

0.067

0.078

0.097

0.125

0.169

0.238

0.349

0.531

0.835

1.353

6.8 × 103

2.3 × 106

3.0 × 1010

5.4 × 1036

0.6

0.2

0.11

0.07

0.06

0.051

0.049

0.051

0.057

0.067

0.08

0.11

0.15

0.21

0.31

0.47

0.75

1.23

640

2.18 × 106

2.9 × 1010

5.3 × 1036

value. This behavior is explained as follows: Each of the

quantities )2( NS and 2

(4)VN has a maximum at

9N and thus the electrostatic energy of each system

has a minimum at this value, as shown in Eqs.6 and 12.

Our results also show that the electrostatic energy, for

both systems, becomes infinite in the infinite dimen-

sional space. Furthermore, we considered classical re-

normalization of electrostatic energy for a simplified

model of a classical atom in higher space dimension. It

was shown that the variation in electrostatic energy (the

final minus the initial energy) is exactly the same as that

of the hyper-shell, and thus the singularity persists in the

infinite dimensional space.

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