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Electrostatic Force

The force exerted between electrically charged particles at rest. It's a fundamental concept in electrostatics.
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The statement of the theorem

Consider a system of NN point charges, qiq_i, located at positions r⃗i∈R3\vec{r}_i \in \mathbb{R}^3. The electrostatic force F⃗i\vec{F}_i acting on the charge qiq_i due to the presence of all other charges qjq_j (j≠ij \neq i) is defined by the principle of superposition. The force is given by the vector summation:\n\nF⃗i=∑j=1,j≠iNF⃗ji\vec{F}_i = \sum_{j=1, j \neq i}^{N} \vec{F}_{ji} \n\nwhere F⃗ji\vec{F}_{ji} is the force exerted by qjq_j on qiq_i. This pairwise force is rigorously defined using Coulomb's Law in vector form:\n\nF⃗ji=kqjqi∣r⃗i−r⃗j∣2(r⃗i−r⃗j)∣r⃗i−r⃗j∣=kqjqi∣r⃗i−r⃗j∣3(r⃗i−r⃗j)\vec{F}_{ji} = k \frac{q_j q_i}{|\vec{r}_i - \vec{r}_j|^2} \frac{(\vec{r}_i - \vec{r}_j)}{|\vec{r}_i - \vec{r}_j|} = k \frac{q_j q_i}{|\vec{r}_i - \vec{r}_j|^3} (\vec{r}_i - \vec{r}_j) \n\nHere, k=14πϵ0k = \frac{1}{4\pi\epsilon_0} is the Coulomb constant, ϵ0\epsilon_0 is the permittivity of free space, and r⃗i−r⃗j\vec{r}_i - \vec{r}_j is the displacement vector pointing from qjq_j to qiq_i.