Gauss's Law for Electricity

The electric flux is defined in analogy to the gravitational flux as

$$\Phi_E = \mbox{\bf S} \cdot \mbox{\bf E} ~~~ \mbox{(electric flux)}$$ (17.5)

where $\mbox{\bf S}$ is the directed area through which the flux passes. Since the electric field obeys an inverse square law, Gauss's law applies to the electric flux $\Phi_E$ just as it applies to the gravitational flux. In particular, since the magnitude of the outward electric field a distance $r$ from a charge $q$ is $E = q/(4 \pi \epsilon_0 r^2)$, the electric flux through a sphere of radius $r$ (and area $4 \pi r^2$) concentric with the charge is $ES = [q/(4 \pi \epsilon_0 r^2 )] \times (4 \pi r^2 ) = q/ \epsilon_0$. This generalizes to an arbitrary distribution of charge as in the gravitational case:

$$\Phi_E = q_{inside} / \epsilon_0 ~~~ \mbox{(Gauss's law for electricity)} ,$$ (17.6)

where $\Phi_E$ in this equation is the outward electric flux through a closed surface and $q_{inside}$ is the net charge inside this surface. This is an expression of Gauss's law for the electric field. Since Gauss's law for electricity and for gravitation are so similar, we can use all our insights from studying gravity on the electric field case.