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.

$\blacksquare$

Sources

• 1990: I.S. Grant and W.R. Phillips: Electromagnetism (2nd ed.) ... (previous) ... (next): Chapter $1$: Force and energy in electrostatics: $1.4$ Gauss's Law: $1.4.2$ The flux of the electric field out of a closed surface: $(1.14 \, \text a)$