P25_086 - 86. We note that for two points on a circle,...

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Unformatted text preview: 86. 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( θ/ 2). Using this fact, distinguishing between the cases where N = odd and N = even, and counting the pair-wise interactions very carefully, we arrive at the following results for the total potential energies. We use k = 1 / 4 πε . For configuration 1 (where all N electrons are on the circle), we have U 1 ,N =even = Nke 2 2 R N 2 − 1 X j =1 1 sin( jθ/ 2) + 1 2 U 1 ,N =odd = Nke 2 2 R N − 1 2 X j =1 1 sin( jθ/ 2) where θ = 2 π N . For configuration 2, we find U 2 ,N =even = ( N − 1) ke 2 2 R N 2 − 1 X j =1 1 sin( jθ / 2) + 2 U 2 ,N =odd = ( N − 1) ke 2 2 R N − 3 2 X j =1 1 sin( jθ / 2) + 5 2 where θ = 2 π N − 1 . The results are all of the form U 1 or 2 = ke 2 2 R × a pure number ....
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This note was uploaded on 11/12/2011 for the course PHYS 2001 taught by Professor Sprunger during the Fall '08 term at LSU.

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