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I have found that the Coulomb force in two dimension varies with $\frac 1 r$:
\begin{equation}\tag{2}F=\frac{1}{2\pi\epsilon}\cdot\frac{q_1q_2}{r}\end{equation}
But I was not able to prove it. I think it can be proved using the Laplace equation and other further modifications.
One question relate...