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Imagine an infinitely long one-dimensional ionic erystal-a chain of alternating positive and negative ions with a distance a between them (Fig.). Find the force with which one half of the chain acts on an arbitrary ion and compare the result with the force to acting between the two adjacent ions. Calculation accuracy should be better than 0.001. |
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Answer» `F = -(e^2)/(4 pi epsilon_0 a^2) (1 - 1/(2^2) + 1/(3^2) - 1/(4^2) + …) =` `= -(e^2)/(4 pi epsilon_0 a^2) (3/(1^2.2^2) + 7/(3^2.4^2) + 11/(5^2.6^2) + ..)` Compute the value of the series in the brackets to THREE significant digits. To obtain the REQUIRED accuracy, we may discard all terms below 0.001, i.e. we may take the sum of the first ten terms in the series. We obtain `3/(1^2.2^2) + 7/(3^2.4^2) + ....+ 39/(19^2.20^2)` = `= 0.82128 ~~ 0.82` Hence `F = -0.82 (e^2)/(4pi epsilon_0 a^2)` This means that neglecting the interactions with all the ions except the nearest neighbours results in an ERROR of no greater than 20% . |
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