Electric Flux out of Closed Surface
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Theorem
Electric Flux out of Closed Surface surrounding Point Charge
Let $q$ be a point charge.
Let $S$ be a closed surface surrounding $q$.
The total electric flux through $S$ is given by:
- $F = \dfrac q {\varepsilon_0}$
Electric Flux out of Closed Surface surrounding Assemblage of Point Charges
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:
- $\ds F = \dfrac 1 {\varepsilon_0} \sum_Q q_i$
Electric Flux out of Closed Surface surrounding Body with Continuous Charge Distribution
Let $B$ be a body of matter which has a continuous macroscopic charge density $\map \rho {\mathbf r}$.
Let $S$ be a closed surface surrounding $Q$.
The total electric flux through $S$ generated by the electric charge on $B$ is given by:
- $\ds F = \dfrac 1 {\varepsilon_0} \int_V \map \rho {\mathbf r} \rd \tau$
where:
- $V$ is the total volume enclosed by $S$
- $\d \tau$ is an infinitesimal volume element
- $\mathbf r$ is the position vector of $\d \tau$
- $\map \rho {\mathbf r}$ is the macroscopic charge density of the macroscopic electric field at $\mathbf r$
- $\varepsilon_0$ denotes the vacuum permittivity.
Electric Flux out of Closed Surface/Examples
Sphere
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.