Consider a horizontal surface such as the one shown below with an upward normal $d\overrightarrow{A}$. Let $\overrightarrow{E}$ denote an electric field that makes an angle $\theta $ with the horizontal surface and let $E_{\parallel }$ and $E_{\bot }$ denotes its parallel and perpendicular components respectively.

The component $E_{\parallel }$ of the electric field $\overrightarrow{E}$ that is parallel to the surface – and is equal to $E_{\parallel }=E{\mathrm{sin} \left(\theta \right)\ }$ – never crosses the surface.

The component $E_{\bot }$ of the electric field $\overrightarrow{E}$ that is perpendicular to the surface – and is equal to $E_{\perp} = E{\mathrm{cos} \left(\theta \right)\ }$ – goes straight through the surface.

The electric flux, denoted by ${\mathrm{\Phi }}_E$, is a measure of how many electric field lines traverse a given surface. More specifically, it is a measure of the fraction of the electric field that crosses a surface: the greater the fraction, the great the value of the electric flux ${\mathrm{\Phi }}_E$.

Mathematically, the fraction of the electric field that crosses the infinitesimal surface $dA$ is equal to $E_{\bot }=E{\mathrm{cos} \left(\theta \right)\ }$ and the corresponding infinitesimal electric flux it creates is equal to

d{\mathrm{\Phi }}_E=\overrightarrow{E}\cdot d\overrightarrow{A}=E{\mathrm{cos} \left(\theta \right)\ }dA

Thus, to account for the entire surface area, we integrate $d{\mathrm{\Phi }}_E$ over the entire area $A$ which yields

\boxed{{\mathrm{\Phi }}_E=\iint_A{\overrightarrow{E}\cdot d\overrightarrow{A}}=\iint_A{Ecos\left(\theta \right)dA}\ \ \ \ \ \ \left(N\cdot m^2/C\right)}

Special case: if the electric field is constant in magnitude and direction over the entire surface, then the quantity $Ecos\left(\theta \right)$ is constant over the entire surface and can be pulled out of the integral to yield

\boxed{{\mathrm{\Phi }}_E=\iint_A{Ecos\left(\theta \right)dA}=Ecos\left(\theta \right)\iint_A{dA}=EAcos\left(\theta \right)}

where $\displaystyle{\iint_A{dA}=A}$ by definition. As a manner of making the notation compact, the electric flux is then often written as the following dot product:

\boxed{{\mathrm{\Phi }}_E=\overrightarrow{E}\cdot \overrightarrow{A}}

Note: if the electric field $\overrightarrow{E}$ is spatially uniform (the same everywhere), then the above condition is automatically satisfied.

Sign of the electric flux:

If the electric flux through a closed surface is negative, then the net flux is inward.

If the electric flux through a closed surface is positive, then the net flux is outward.

If the electric flux through a closed surface is zero, two possibilities arise

The electric field is everywhere perpendicular to the surface: no field lines traverse the surface and the net flux is zero.

The number of field lines that enter the closed surface is equal to the number of field lines that exit the surface. The net flux is then zero even though the gravitational field does traverse the closed surface.

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