5.3. Steady regime

Steady regime - in a Eulerian description - allows for steady currents, but non-varying fields in an Eulerian description \(\partial_t \equiv 0\).

Continuity equation of electric charge.

\[\nabla \cdot \vec{j} = 0\]

Maxwell’s equations.

\[\begin{split}\begin{cases} \nabla \cdot \vec{e} = \dfrac{\rho}{\varepsilon_0} \\ \nabla \times \vec{e} = \vec{0} \\ \nabla \cdot \vec{b} = 0 \\ \nabla \times \vec{b} = \mu_0 \vec{j} \end{cases}\end{split}\]

Potentials.

\[\begin{split}\begin{aligned} \vec{b} & = \nabla \times \vec{a} \\ \vec{e} & = - \nabla \phi \\ \end{aligned}\end{split}\]