Formula: 1. Maxwell Equation in Integral Form

$$\oint_A \class{purple}{\boldsymbol{E}} ~\cdot~ \text{d}\boldsymbol{a} ~=~ \frac{\class{red}{Q}}{\varepsilon_0}$$

Electric field

$$ \class{purple}{\boldsymbol{E}} $$ Unit $$ \frac{\mathrm{V}}{\mathrm{m}} = \frac{\mathrm{N}}{\mathrm{C}} = \frac{\mathrm{kg} \, \mathrm{m}}{\mathrm{A} \, \mathrm{s}^3} $$

This quantity is a vector field (it assigns a field vector to each point in space) and tells how large the electric force on a test charge would be if it were placed in a particular location.

Surface

$$ A $$

The (imaginary) surface over which the electric field $ \class{blue}{\boldsymbol{E}} $ is integrated. This can be, for example, a spherical surface or a cylindrical surface. For example, to calculate the $ \class{blue}{\boldsymbol{E}} $ field inside a charged sphere, this imaginary surface is placed inside the charged sphere.

Here $ \text{d}\boldsymbol{a} $ is a infinitesimal piece of the surface. By definition the direction of $\text{d}\boldsymbol{a}$ is perpendicular on the surface.

Electric charge

$$ \class{red}{Q} $$ Unit $$ \mathrm{C} = \mathrm{As} $$

This is the charge that is enclosed by the selected surface $ A $.

Vacuum Permittivity

$$ \varepsilon_0 $$ Unit $$ \frac{\mathrm{As}}{\mathrm{Vm}} $$

The vacuum permittivity is a physical constant that appears in equations involving electromagnetic fields. It has the following experimentally determined value: $$ \varepsilon_0 ~\approx~ 8.854 \, 187 \, 8128 ~\cdot~ 10^{-12} \, \frac{\mathrm{As}}{\mathrm{Vm}} $$