Alexander Fufaev

Formula: 1. Maxwell Equation in Differential Form

Positive charge - electric field lines

Electric field

Electric field indicates how large and in what direction the electric force on a charge would be if that charge were placed at location \((x,y,z)\).

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

Space charge density indicates how many electric charges per volume there are in a space region.

Vacuum Permittivity

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