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What is the electric dipole moment of the charge distribution with $q$ at $(0,0,1)$, $q$ at $(0,0,-1)$ and $-2q$ at $(0,0,0)$? I would think that it is $\vec{0}$ by the definition $\vec{p}=\sum\limits_i \vec{r_i}q_i$. So would it follow that the potential field due to it be $0$? Since $V=k {\vec{p}\cdot \hat{r}\over r^2}$ where $\vec{r}$ is the position vector of a point at which we wish to evaluate the potential.

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up vote 2 down vote accepted

That formula only works for short dipoles. Here, you need to use the fudamental potential formula, $\frac{kq_1}{r}$. By this formula, we get potential due to $q$ as $\frac{kq}{2}$, and potential due to $-2q$ as $-2kq$. Net potential is $-\frac{3kq}{2}$.

When we say a short dipole, we mean that the distance between the charges should be negligible with respect to the surroundings.

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