# Electric Flux out of Closed Surface surrounding Point Charge/Lemma

## Theorem

Let $q$ be a point charge located at the origin of a spherical polar coordinate system.

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

Let $\delta \mathbf S$ be an area element of $S$.

Let $\delta \Omega$ be the solid angle subtended by the projection of $\delta \mathbf S$ from $q$.

The total electric flux through $\delta \Omega$ is given by:

- $F = \dfrac q {4 \pi \varepsilon_0} \delta \Omega$

## Proof

Consider the area element $\delta \mathbf S$ at the position $P$ with position vector $\mathbf r$ whose spherical coordinates are $\polar {r, \theta, \phi}$.

Let $\delta S_p$ be the projected area of $\delta \mathbf S$ to the plane perpendicular to $\mathbf r$.

Let us define $\delta \mathbf S$ as being the subset of the surface of $S$ demarcated by the arcs on $S$ subtending:

- the polar angles $\theta$ and $\delta \theta$ and

and:

- the azimuthal angles $\phi$ and $\delta \phi$.

From the above diagram:

- $\delta S_p = r^2 \sin \theta \rdelta \theta \rdelta \phi$

Let $\mathbf E$ be the electric field generated by $q$ at $P$.

According to Coulomb's Law of Electrostatics:

- the magnitude of $\mathbf E$ is:
- $E = \dfrac q {4 \pi \varepsilon_0 r^2}$

- the direction of $\mathbf E$ is along $\mathbf r$ away from the origin.

Hence the electric flux in the direction away from the origin through $\delta \mathbf S$ is:

\(\ds \mathbf E \cdot \delta \mathbf S\) | \(=\) | \(\ds E \rdelta S_p\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac q {4 \pi \varepsilon_0 r^2} r^2 \sin \theta \rdelta \theta \rdelta \phi\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac q {4 \pi \varepsilon_0} \sin \theta \rdelta \theta \rdelta \phi\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \dfrac q {4 \pi \varepsilon_0} \rdelta \Omega\) |

where $\Omega$ is that solid angle so subtended.

$\blacksquare$

## 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: $(1.11)$