** Flux**, also known as

In the case of electric flux, the flux $\psi$ depends on the number and the magnitude of electric field lines $\vec{E}$ going perpendicularly through a normally perpendicular surface and the surface area $\vec{S}$. In other words, electric flux is 'tubes' of electric field. $\psi = \vec{E}.\vec{S}=E.S.cos(\theta)$. To generalize for any surface, $\psi = \iint \vec{E}.d\vec{S}$ (Note that it is a double integral because it is an area, not a line).

For a closed system with a symmetrical ** Gaussian surface**, $\psi = \oint_S \vec{E}.d\vec{S} = \cfrac{q}{4.\pi.r^2.\varepsilon}{4.\pi.r^2}=\cfrac{q}{\varepsilon}$. This can be interpreted as: The total flux over a given Gaussian surface is equal to the total charge inside divided by the medium permittivity.

A Gaussian surface is a symmetrical closed surface such as cyllinder or sphere.

**Electric Flux Density** is a measure of the electric field independent of the material. It is defined as $D=\varepsilon.E=\cfrac{q}{4.\pi.r^2}=\cfrac{q}{A}$

Using this relationship, a further relation can be made - $\int_S \vec{D}.{d\vec{S}}=q_{enc}$. The law is summarized with the expression - **The electric flux passing through any closed surface is equal to the total charge enclosed**

$C=q/V = (\int D dS)/V = \cfrac{\int D dS}{\int E dl} = \varepsilon . \cfrac{\int dS}{\int dl}$

For a Gaussian surface, the capacitance can easily be calculated. For example, for a parallel plate capacitor with plate distance $d$ and plate area $A$, $C=\varepsilon. A/d$