# Formula 1. Maxwell Equation in Differential Form Electric field Charge density

## Electric field

`$$ \class{blue}{\boldsymbol{E}} $$`Unit

`$$ \frac{\mathrm V}{\mathrm m} $$`

The divergence field is the scalar product between the Nabla operator \(\nabla\) and the electric field \( \class{blue}{\boldsymbol{E}} \):`\[ \nabla ~\cdot~ \class{blue}{\boldsymbol{E}} ~=~ \begin{bmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{bmatrix} ~\cdot~ \begin{bmatrix} \class{blue}{E_{\text x}} \\ \class{blue}{E_{\text y}} \\ \class{blue}{E_{\text z}} \end{bmatrix} \]`

The divergence field is no longer a vector field but a scalar function. The divergence field \( \nabla ~\cdot~ \class{blue}{\boldsymbol{E}}(x,y,z) \) at the location \((x,y,z)\) can be positive, negative or zero. With positive divergence, there is a positive electric charge (a source) at the location \((x,y,z)\). When the divergence is negative, there is a negative electric charge (a sink) at the \((x,y,z)\) location. If, on the other hand, the divergence at the location \((x,y,z)\) is zero, then there is no electric charge there.

## Charge density

`$$ \class{red}{\rho} $$`Unit

`$$ \frac{\mathrm{C}}{\mathrm{m}^3} $$`

## Vacuum Permittivity

`$$ \varepsilon_0 $$`Unit

`$$ \frac{\mathrm{As}}{\mathrm{Vm}} $$`

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