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, approximate governing equations of electromagnetism in slow opearint regime is obtained setting \(\frac{fL}{c} = 0\) in the non-dimensional equations of the electromagnetism (5.2),

(5.3)\[\begin{split} \begin{aligned} & \text{continuity equation for charge} && \partial_t \rho_f + \nabla \cdot \vec{j}_f = 0 \\ \\ & \text{Maxwell's equations} && \nabla \cdot \vec{d} - \rho_f = 0 \\ & && \nabla \times \vec{e} + \partial_t \vec{b} = \vec{0} \\ & && \nabla \cdot \vec{b} = 0 \\ & && \nabla \times \vec{h} = \, \vec{j}_f \\ \\ & \text{constitutive equations} && \vec{d} = \vec{e} \\ & && \vec{b} = \vec{h} \\ \\ & \text{EM potentials} && \vec{b} = \nabla \times \vec{a} \\ & && \vec{e} = - \nabla \phi - \partial_t \vec{a} \\ \\ & \text{Lorentz's gauge} && \nabla \cdot \vec{a} + \partial_t \phi = 0 \end{aligned} \end{split}\]

todo check consistency of \(\vec{e} = - \nabla \phi\) with Faraday’s induction law \(\nabla \cdot \vec{e} + \partial_t \vec{b} = \vec{0}\). Non-dimensionalization process could be not 100% ok yet

From the approximate governing equations some consequences follow:

• in slow operating regime wave equations can be approximated with Poisson equations: perturbations can’t travel anymore in form of waves, while their transmission is governed by a diffusive equation. Wave equations becomes Poisson equations, as the wave operator \(\square \sim \nabla^2\) if \(\left( \frac{L F}{c} \right)^2 \ll 1\), as the non-dimensional form of a wave equation follows from

  \[\begin{split}\begin{aligned} \frac{F^2}{c^2} \partial_{tt} f - \dfrac{1}{L^2} \nabla^2 f & = \dots \\ \dfrac{1}{L^2} \left[ \left( \frac{F L}{c}\right)^2 \partial_{tt} f - \nabla^2 f \right] & = \dots \\ - \dfrac{1}{L^2} \nabla^2 f & \sim \dots \end{aligned}\end{split}\]

• radiation is negligible in energy balance, as it’s shown for energy balance in circuit approximation