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Electrostatic Equilibrium

A state where the electrostatic forces on charged particles are balanced, resulting in no net movement of the particles.
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The statement of the theorem

Consider a system of NN point charges, q1,q2,…,qNq_1, q_2, \dots, q_N, situated at positions r⃗1,r⃗2,…,r⃗N\vec{r}_1, \vec{r}_2, \dots, \vec{r}_N in R3\mathbb{R}^3. The force F⃗i\vec{F}_i acting on the charge qiq_i due to all other charges qjq_j (j≠ij \neq i) is given by the superposition of Coulomb forces. Electrostatic Equilibrium is achieved if and only if the net force on every charge is zero. Mathematically, this condition is stated as:\n∑j=1,j≠iNF⃗ji=0⃗ for all i=1,2,…,N\sum_{j=1, j \neq i}^{N} \vec{F}_{ji} = \vec{0} \text{ for all } i = 1, 2, \dots, N \nwhere F⃗ji\vec{F}_{ji} is the force exerted by qjq_j on qiq_i, defined by:\nF⃗ji=keqiqj∣r⃗i−r⃗j∣3(r⃗i−r⃗j)\vec{F}_{ji} = k_e \frac{q_i q_j}{|\vec{r}_i - \vec{r}_j|^3} (\vec{r}_i - \vec{r}_j) \nHere, kek_e is the Coulomb constant, and the condition implies that the system configuration (r⃗1,…,r⃗N)(\vec{r}_1, \dots, \vec{r}_N) is a critical point of the potential energy function U(r⃗1,…,r⃗N)U(\vec{r}_1, \dots, \vec{r}_N) with respect to the coordinates.