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Electrostatic Principle of Superposition

The electric 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, where qiq_i is the point charge located at r⃗i∈S\vec{r}_i \in S. The electric field E⃗i(r⃗)\vec{E}_i(\vec{r}) generated by the isolated charge qiq_i at the observation point r⃗\vec{r} is given by Coulomb's Law: E⃗i(r⃗)=kqi∣r⃗−r⃗i∣3(r⃗−r⃗i)\vec{E}_i(\vec{r}) = k \frac{q_i}{|\vec{r} - \vec{r}_i|^3} (\vec{r} - \vec{r}_i). The Electrostatic Principle of Superposition asserts that the total electric field E⃗total(r⃗)\vec{E}_{total}(\vec{r}) at r⃗\vec{r} is the vector sum of the fields generated by each charge: E⃗total(r⃗)=∑i=1NE⃗i(r⃗)=k∑i=1Nqi∣r⃗−r⃗i∣3(r⃗−r⃗i)\vec{E}_{total}(\vec{r}) = \sum_{i=1}^{N} \vec{E}_i(\vec{r}) = k \sum_{i=1}^{N} \frac{q_i}{|\vec{r} - \vec{r}_i|^3} (\vec{r} - \vec{r}_i). Furthermore, the total electric potential V(r⃗)V(\vec{r}) is the scalar sum: V(r⃗)=∑i=1NVi(r⃗)=∑i=1Nkqi∣r⃗−r⃗i∣V(\vec{r}) = \sum_{i=1}^{N} V_i(\vec{r}) = \sum_{i=1}^{N} \frac{k q_i}{|\vec{r} - \vec{r}_i|}. This linearity property holds due to the underlying structure of the electric field as a conservative vector field derived from a scalar potential.