ch24-p061 - 61. We note that for two points on a circle,...

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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 48.62 59.58 71.81 85.35 100.2 We see that the potential energy for configuration 2 is greater than that for configuration 1 for N < 12, but for N 12 it is configuration 1 that has the greatest potential energy. (a) N = 12 is the smallest value such that U 2 < U 1 . 61. 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 pair-wise 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 , 2s i n 2 2 i n 2 NN jj Nke Nke UU Rj θθ == ⎛⎞ ⎜⎟ =+ = ⎝⎠ ∑∑ where = 2 π N . For configuration 2, we find 3 1 2, even 2, odd 5 2, i n 2 i n 2 2 Nk e e −− ′′ where ′ = 2 1 π N . The results are all of the form U ke R 1 2 2
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ch24-p061 - 61. We note that for two points on a circle,...

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