The potential energy per ion for positive ion is :
A
−2e24πε0dloge2
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B
−e24πε0dloge2
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C
2e24πε0d
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D
−2e24πε0d
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Solution
The correct option is B−e24πε0dloge2
Here the potential energy of interaction between the positive ion at x=0 and all the other ions represents the potential energy (U) per ion in this one dimensional crystal.
Thus, U=U12+U13+U14+U15+......
or U=e(−e)4πϵ0d+e(e)4πϵ02d+e(−e)4πϵ03d+e(e)4πϵ04d+.....
=−e24πϵ0d[1−12+13−14+....]
=−e24πϵ0dloge2 as loge(1+x)=x−x22+x33−x44+... or loge(1+1)=loge2=1−12+13−14+.....