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yucheng
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- Homework Statement
- Purcell Electricity and Magnetism Problem 1.7
Potential energy in a two-dimensional crystal Use a computer to calculate numerically the potential energy, per ion, for an infinite two-dimensional square ionic crystal with separation ##a##; that is, a plane of equally spaced charges of magnitude ##e## and alternating sign (as with a checkerboard).
- Relevant Equations
- N/A
Potential energy in a two-dimensional crystal
Consider the potential energy of a given ion due to the full infinite plane. Call it##U_{0}##. If we sum over all ions (or a very large number##N##) to find the total##U##of these ions, we obtain##N U_{0}##. However, we have counted each pair twice, so we must divide by 2 to obtain the actual total energy. Dividing by##N##then gives the energy per ion as##\left(N U_{0} / 2\right) / N=U_{0} / 2##.
(part of the answer given)
My question is, why is the potential energy of a given ion due to to the full infinite plane is ##U_0##, but then the author then moves on to show that the energy per ion is ##U_0/2##. Somehow, this seems scandalous to me!
Consider the potential energy of a given ion due to the full infinite plane. Call it##U_{0}##. If we sum over all ions (or a very large number##N##) to find the total##U##of these ions, we obtain##N U_{0}##. However, we have counted each pair twice, so we must divide by 2 to obtain the actual total energy. Dividing by##N##then gives the energy per ion as##\left(N U_{0} / 2\right) / N=U_{0} / 2##.
(part of the answer given)
My question is, why is the potential energy of a given ion due to to the full infinite plane is ##U_0##, but then the author then moves on to show that the energy per ion is ##U_0/2##. Somehow, this seems scandalous to me!
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