5.4. Slow regime

Slow regime leads to circuit approximations of electromagnetic systems with moderate dimensions at low frequency. For these systems and regimes, the ratio appearing into non-dimensional equations of electromagnetism reads,

\[\dfrac{f L }{c_0} \ll 1 \ .\]

Under this assumption, the equations of electromagnetism can be approximated as

Continuity equation of electric charge.

\[\partial_t \rho + \nabla \cdot \vec{j} = 0\]

Maxwell’s equations.

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

Potentials.

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