Electric Field Strength from Assemblage of Point Charges

Theorem
Let $p_1, p_2, \ldots, p_n$ be charged particles.

Let $q_1, q_2, \ldots, q_n$ be the electric charges on $p_1, p_2, \ldots, p_n$ respectively.

Let $\mathbf r_1, \mathbf r_2, \ldots, \mathbf r_n$ be the position vectors of $p_1, p_2, \ldots, p_n$ respectively.

Let $\map {\mathbf E} {\mathbf r}$ be the electric field strength at a point $P$ whose position vector is $\mathbf r$.

Then:
 * $\ds \map {\mathbf E} {\mathbf r} = \dfrac 1 {4 \pi \epsilon_0} \sum_i \dfrac {q_i} {\size {\mathbf r - \mathbf r_i}^3} \paren {\mathbf r - \mathbf r_i}$

Proof
Let $q$ be a test charge in the vicinity of $p_1, p_2, \ldots, p_n$ at $\map P {\mathbf r}$.

For all $i$ in $\set {1, 2, \ldots, n}$ let $\mathbf F_i$ denote the force exerted on $q$ by $q_i$.

Let $\mathbf F$ be the force exerted on $q$ by the combined action of $q_1, q_2, \ldots, q_n$.

We have:

Hence the result.