Formula 4. Maxwell Equation of Magnetostatics (Differential Form) Magnetic field Current density
$$\nabla \times \class{violet}{\boldsymbol{B}} ~=~ \mu_0 \, \class{red}{\boldsymbol{j}}$$
Magnetic field
$$ \class{violet}{\boldsymbol{B}} $$ Unit $$ $$ Magnetic flux density determines the magnitude and direction of the magnetic force on a moving electric charge.
Curl field
$$ \nabla \times \class{violet}{\boldsymbol{B}} $$ Unit $$ $$ Vector magnetic curl field as cross product between the nabla operator \(\nabla\) and the magnetic field \( \boldsymbol{B} \):\[ \nabla \times \boldsymbol{B} ~=~ \begin{bmatrix} \frac{\partial B_z}{\partial y} - \frac{\partial B_y}{\partial z} \\ \frac{\partial B_x}{\partial z} - \frac{\partial B_z}{\partial x} \\ \frac{\partial B_y}{\partial x} - \frac{\partial B_x}{\partial y} \end{bmatrix} \]
Das Rotationsfeld \(\nabla \times \boldsymbol{B}(x,y,z)\) ist ein Vektorfeld, das angibt, wie stark das Magnetfeld \(\boldsymbol{B}\) am Ort \((x,y,z)\) rotiert.
Current density
$$ \class{red}{\boldsymbol{j}} $$ Unit $$ $$ Electric current density indicates the electric current per cross-sectional area. According to Maxwell's equation, an electric current generates a magnetic curl field around the current (magnetostatics). In the case of a time-varying B-field, a time-varying E-field is also generated.
Vacuum permeability
$$ \mu_0 $$ Unit $$ \frac{\mathrm{Vs}}{\mathrm{Am}} = \frac{ \mathrm{kg} \, \mathrm{m} }{ \mathrm{A}^2 \, \mathrm{s}^2 } $$ The vacuum permeability is a physical constant and has the following experimentally determined value:$$ \mu_0 ~=~ 1.256 \, 637 \, 062 \, 12 ~\cdot~ 10^{-6} \, \frac{\mathrm{Vs}}{\mathrm{Am}} $$