24.4.3 Multiple Charges

Finally, consider an arbitrary Gaussian surface and a group of charges $q_{1}, q_{2}, q_{3}, \dots$ such as those shown in FIGURE 24.22. What is the net electric flux through the surface?

The figure illustrates a two-dimensional cross section of an arbitrary Gaussian surface.

FIGURE 24.22 Full Alternative Text

By definition, the net flux is

Φ_{e} = \oint \vec{E} \cdot d \vec{A}

From the principle of superposition, the electric field is $\vec{E} = \vec{E}_{1} + \vec{E}_{2} + \overset{}{} \overset{}{} \overset{}{} \overset{}{} \vec{E}_{3} + \dots$ , where $\vec{E}_{1}, \vec{E}_{2}, \vec{E}_{3}, \dots$ are the fields of the individual charges. Thus the flux is

\begin{array}{l} Φ_{e} & = & \oint^{​} {\vec{E}}_{1} \cdot d \vec{A} + \oint^{​} {\vec{E}}_{2} \cdot d \vec{A} + \oint^{​} {\vec{E}}_{3} \cdot d \vec{A} + \dots \\ = & Φ_{1} + Φ_{2} + Φ_{3} + \dots \end{array}

(24.15)

where $Φ_{1}, Φ_{2}, Φ_{3}, \dots$ are the fluxes through the Gaussian surface due to the individual charges. That is, the net flux is the sum of the fluxes due to individual charges. But we know what those are: $q / ϵ_{0}$ for the charges inside the surface and zero for the charges outside. Thus

\begin{array}{l} Φ_{e} & = & (\frac{q_{1}}{ϵ_{0}} + \frac{q_{2}}{ϵ_{0}} + \dots + \frac{q_{i}}{ϵ_{0}} for all charges inside the surface) \\ + (0 + 0 + \dots + 0 for all charges outside the surface) \end{array}

(24.16)

We define

Q_{in} = q_{1} + q_{2} + \dots + q_{i} for all charges inside the surface

(24.17)

as the total charge enclosed within the surface. With this definition, we can write our result for the net electric flux in a very neat and compact fashion. For any closed surface enclosing total charge $Q_{in}$ , the net electric flux through the surface is

Φ_{e} = \oint \vec{E} \cdot d \vec{A} = \frac{Q_{in}}{ϵ_{0}}

(24.18)

This result for the electric flux is known as Gauss’s law.