Electric Flux out of Closed Surface surrounding Assemblage of Point Charges

Theorem
Let $Q = \set {q_1, q_2, \ldots}$ be a set of point charges.

Let $S$ be a closed surface surrounding $Q$.

The total electric flux through $S$ is given by:
 * $\ds F = \dfrac 1 {\varepsilon_0} \sum_Q q_i$

Proof
From Electric Flux out of Closed Surface surrounding Point Charge:
 * $F_i = \dfrac {q_i} {\varepsilon_0}$

where $F_i$ is the part of $F$ brought about by $q_i$.

The result follows from Electric Field satisfies Principle of Superposition.