Electric Flux out of Closed Surface/Examples/Sphere

Example of Use of Electric Flux out of Closed Surface
Let $B$ be a spherical body in space with radius $R$.

Let $Q$ be the total electric charge within $B$

Then the total electric flux out of $B$ is given by:
 * $F = \dfrac Q {\varepsilon_0}$

where $\varepsilon_0$ denotes the vacuum permittivity.

Proof
Let the center of $B$ be located at the origin of a spherical polar coordinate system.

Let $\mathbf r$ denote the position vector of an arbitrary point $P$ on the surface of $B$.

Let the spherical polar coordinates of $P$ be denoted $\polar {r, \theta, \phi}$.

The electric charge density and electric field caused by $B$ are expressible in terms of $\polar {r, \theta, \phi}$ as:
 * $\map \rho {r, \theta, \phi}$

and:
 * $\map {\mathbf E} {r, \theta, \phi}$

Let us define $\d \mathbf S$ as being the infinitesimal area element of the surface of $B$ demarcated by the arcs on $B$ subtending:
 * the polar angles $\theta$ and $\d \theta$ and

and:
 * the azimuthal angles $\phi$ and $\d \phi$.

where $\d \theta$ and $\d \phi$ are likewise infinitesimal.

From Area Element in Spherical Polar Coordinate System:
 * $\d S = R^2 \sin \theta \rd \theta \rd \phi$

Integrating over all space, we have:

from which the result follows.