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Superposition Principle (Electrostatics)

The total force on a charge in a region with multiple point charges is the vector sum of the forces due to each individual charge.
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The statement of the theorem

Let S={r⃗1,r⃗2,…,r⃗N}S = \{\vec{r}_1, \vec{r}_2, \dots, \vec{r}_N\} be a finite set of positions in R3\mathbb{R}^3, and let qiq_i be the point charge located at r⃗i\vec{r}_i. We consider the electric field E⃗(r⃗)\vec{E}(\vec{r}) at an observation point r⃗∉S\vec{r} \notin S. The electric field E⃗i(r⃗)\vec{E}_i(\vec{r}) generated by the isolated charge qiq_i is given by Coulomb's Law: E⃗i(r⃗)=kqi∣r⃗−r⃗i∣2(r⃗−r⃗i)∣r⃗−r⃗i∣\vec{E}_i(\vec{r}) = k \frac{q_i}{|\vec{r} - \vec{r}_i|^2} \frac{(\vec{r} - \vec{r}_i)}{|\vec{r} - \vec{r}_i|}. The Superposition Principle asserts that the total electric field E⃗(r⃗)\vec{E}(\vec{r}) is the vector sum of the individual fields: \n\nE⃗(r⃗)=∑i=1NE⃗i(r⃗)=∑i=1Nkqi∣r⃗−r⃗i∣3(r⃗−r⃗i)\vec{E}(\vec{r}) = \sum_{i=1}^{N} \vec{E}_i(\vec{r}) = \sum_{i=1}^{N} k \frac{q_i}{|\vec{r} - \vec{r}_i|^3} (\vec{r} - \vec{r}_i) \n\nFurthermore, since the electric field is a conservative vector field, the total potential V(r⃗)V(\vec{r}) is also additive: \n\nV(r⃗)=∑i=1NVi(r⃗)=∑i=1Nkqi∣r⃗−r⃗i∣V(\vec{r}) = \sum_{i=1}^{N} V_i(\vec{r}) = \sum_{i=1}^{N} k \frac{q_i}{|\vec{r} - \vec{r}_i|}