Question

The potential energy per ion for positive ion is :

• A. $\frac{-2e^2}{4\pi {\epsilon }_{0}d}lo{g}_{e}2$
• B. $\frac{-e^2}{4\pi {\epsilon }_{0}d}lo{g}_{e}2$
• C. $\frac{2e^2}{4\pi {\epsilon }_{0}d}$
• D. $\frac{-2e^2}{4\pi {\epsilon }_{0}d}$

Answer 1

The correct option is B $\frac{-e^2}{4\pi {\epsilon }_{0}d}lo{g}_{e}2$

Here the potential energy of interaction between the positive ion at $x=0$ and all the other ions represents the potential energy $\left(U\right)$ per ion in this one dimensional crystal.

Thus, $U=U_{12}+U_{13}+U_{14}+U_{15}+......$

or $U=\frac{e(-e)}{4\pi {ϵ}_{0}d}+\frac{e\left(e\right)}{4\pi {ϵ}_{0}2d}+\frac{e(-e)}{4\pi {ϵ}_{0}3d}+\frac{e\left(e\right)}{4\pi {ϵ}_{0}4d}+.....$

$=\frac{-e^2}{4\pi {ϵ}_{0}d}[1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+....]$

$=\frac{-e^2}{4\pi {ϵ}_{0}d}lo{g}_{e}2$ as $lo{g}_e(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+...$ or $lo{g}_e(1+1)=lo{g}_{e}2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+.....$