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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:

`$$ \varepsilon_0 ~\approx~ 8.854 \, 187 \, 8128 ~\cdot~ 10^{-12} \, \frac{\mathrm{As}}{\mathrm{Vm}} $$`