Discussion of the Question 10/00

CHARGES ON A CIRCLE

The question was:



Several identical point charges (shown in red) are placed on a circular disk (shown in blue) so as to minimize the electrostatic energy of the system. Obviously, single charge can be placed anywhere as depicted in Fig. (a), while a pair of charges will occupy opposite sides of the diameter of the circle, as in Fig. (b). Similarly, three charges will form an equilateral triangle, as in Fig. (c). What can you say about the geometric arrangement of 4, 5, 6, ... charges?

(6/11/00) Y. Kantor: We are getting many replies claiming that the optimal placement of the charges is the equidistant placement along the boundary of the disk, i.e. on the circle. This seems to be true for number of charges n=2 or 3. But is it true for larger n? It is possible that equally spread charges represent a local energy minimum. But is it a global minimum? As an example, let us compare configuration of equally spaced charges on the boundary with configuration in which one charge is in the center of the circle while the remaining are equally spaced on the boundaries. Two such configurations are depicted for n=7 below.

The following table presents the energies of those two configurations for various values of n. (It is assumed that those are unit charges and the radius of the circle is also unity.) 
            n   with charge in center   without charge in center
            3   2.500000000000000       1.732050807568877    
            4   4.732050807568877       3.828427124746190    
            5   7.828427124746190       6.881909602355869    
            6   11.88190960235587       10.96410161513776    
            7   16.96410161513776       16.13335409673741    
            8   23.13335409673741       22.43892676967297    
            9   30.43892676967297       29.92344919779823    
           10   38.92344919779824       38.62449897970962    
           11   48.62449897970962       48.57567511970017    
           12   59.57567511970017       59.80736151791218    
           13   71.80736151791218       72.34728957471518    
           14   85.34728957471515       86.22096479601028    
           15   100.2209647960103       101.4519980160739    
           16   116.4519980160739       118.0623677300199    
           17   134.0623677300200       136.0726314193666    
           18   153.0726314193665       155.5020983001627    
           19   173.5020983001626       176.3689723517624
           20   195.3689723517624       198.6904720782230
You immediately see that for n=12 it is already worthwhile to put one of the charges in the center, and therefore the configuration where all the charges are on the boundaries is not the global minimum!

Obviously, we did not intend to find the global minimum. We just tried to demonstrate that equally spaced charges on the boundary, are not necessarily the optimal placement of the charges...
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