5.1. Non-dimensional equations of electromagnetism

Non-dimensional form of the equations of electromagnetism are derived here, defining every dimensional physical quantity as the product of a dimensional reference value and the non-dimensional quantity, \(g = G \widetilde{g}\), or with a small abuse of notation to avoid writing lots of tildes and to keep lighter notation,

\[g = G g \ ,\]

using the same symbol to indicate the dimensional and the non-dimensional quantity. Here, capital letters are used for reference dimensional quantities.

This procedure allows to estimate the order of magnitude of the physical quantities involved in the problem: without any independent physical quantities, non-dimensional numbers are set equal to \(1\), whenever it’s possible. Only two equations, namely Ampére-Maxwell equation and the definition of the electric field with the EM potentials, contain 3 terms

Continuity equation of electric charge.

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

Maxwell’s equations.

\[\begin{split}\begin{cases} \nabla \cdot \vec{e} = \dfrac{\rho}{\varepsilon_0} \\ \nabla \times \vec{e} + \partial_t \vec{b} = \vec{0} \\ \nabla \cdot \vec{b} = 0 \\ \nabla \times \vec{b} - \frac{1}{c_0^2} \partial_t \vec{e} = \mu_0 \vec{j} \end{cases}\end{split}\]

Constitutive laws, here for linear isotropic medium

\[\begin{split}\begin{aligned} \vec{d} & = \varepsilon \vec{e} \\ \vec{b} & = \mu \vec{h} \\ \end{aligned}\end{split}\]

Potentials.

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

Gauge. Lorentz’s gauge

\[0 = \dfrac{1}{c^2} \partial_t \phi + \nabla \cdot \vec{a}\]

Wave equations.

…

Assuming characteristic dimensions of the physical quantities involved in the problem exist, and allow to write the governing equations in non-dimensional form with contributions with (approximately at least) the same order of magnitude,

\[\begin{split} \begin{aligned} & F R \, \partial_t \rho_f + \dfrac{J}{L} \nabla \cdot \vec{j}_f = 0 \\ \\ & \dfrac{D}{L} \nabla \cdot \vec{d} - R \rho_f = 0 \\ & \dfrac{E}{L} \nabla \times \vec{e} + {B F} \, \partial_t \vec{b} = \vec{0} \\ & \dfrac{B}{L} \nabla \cdot \vec{b} = 0 \\ & \dfrac{H}{L} \nabla \times \vec{h} - F D \partial_t \vec{d} = J \, \vec{j}_f \\ \\ & D \vec{d} = \varepsilon E \vec{e} \\ & B \vec{b} = \mu H \vec{h} \\ \\ & B \vec{b} = \dfrac{A}{L} \nabla \times \vec{a} \\ & E \vec{e} = - A F \, \partial_t \vec{a} - \frac{\Phi}{L} \nabla \phi \\ \\ & \dfrac{A}{L} \nabla \cdot \vec{a} + \dfrac{F \Phi}{c_0^2} \partial_t \phi = 0 \end{aligned} \qquad , \qquad \begin{aligned} & \partial_t \rho_f + \dfrac{J}{F L R} \nabla \cdot \vec{j}_f = 0 \\ \\ & \nabla \cdot \vec{d} - \dfrac{R L}{D} \rho_f = 0 \\ & \nabla \times \vec{e} + \frac{B F L}{E} \, \partial_t \vec{b} = \vec{0} \\ & \nabla \cdot \vec{b} = 0 \\ & \nabla \times \vec{h} - \dfrac{D L F}{H} \partial_t \vec{d} = \frac{J L}{H} \, \vec{j}_f \\ \\ & \vec{d} = \frac{\varepsilon E}{D} \vec{e} \\ & \vec{b} = \frac{\mu H}{B} \vec{h} \\ \\ & \vec{b} = \dfrac{A}{B L} \nabla \times \vec{a} \\ & \vec{e} = \frac{\Phi}{E L} \left[ - \dfrac{A FL}{\Phi} \, \partial_t \vec{a} - \nabla \phi \right] \\ \\ & \nabla \cdot \vec{a} + \dfrac{F L \Phi}{A c_0^2} \partial_t \phi = 0 \end{aligned} \end{split}\]

All these relations but Ampére-Maxwell’s law and the definition of the electric field in terms of the potentials contains at most two terms: these relations can be used to immediately find the relation between the scales of the problem (if they’re not independent), by setting the non-dimensional numbers equal to \(1\),

(5.1)\[\begin{split}\begin{aligned} J & = R F L & \text{from contintuity} \\ R & = \frac{D}{L} & \text{from Gauss' law for $\vec{d}$} \\ E & = B F L & \text{from Faraday's law} \\ D & = \varepsilon E & \text{from $\vec{d} = \varepsilon \vec{e}$} \\ B & = \mu H & \text{from $\vec{b} = \mu \vec{h}$} \\ A & = B L & \text{from the definition $\vec{b} = \nabla \times \vec{a}$} \\ A & = \dfrac{\Phi F L}{c_0^2} & \text{from Lorentz's gauge} \\ \end{aligned}\end{split}\]

Using these definitions of the characteristic dimensions of the problem, the pairs of terms in Ampére-Maxwell law and in the the definition of the electric field are compared

\[\begin{split}\begin{aligned} \frac{D L F}{H} & = \dfrac{\varepsilon \mu E L F}{B} = \left( \dfrac{FL}{c} \right)^2 \\ \frac{A FL}{\Phi} & = \dfrac{\Phi F L}{c^2} \frac{FL}{\Phi} = \left(\dfrac{FL}{c}\right)^2 \\ \end{aligned}\end{split}\]

todo check obseration at the end of thi section

Setting equal to \(1\) al the independent non-dimensional numbers, (todo is it possible? Are they really independent), including

\[H = JL \qquad , \qquad \Phi = E L \ ,\]

the non-dimensional form of the equations of electromagnetism becomes

(5.2)\[\begin{split} \begin{aligned} & \text{continuity equation for charge} && \partial_t \rho_f + \nabla \cdot \vec{j}_f = 0 \\ \\ & \text{Maxwell's equations} && \nabla \cdot \vec{d} - \rho_f = 0 \\ & && \nabla \times \vec{e} + \partial_t \vec{b} = \vec{0} \\ & && \nabla \cdot \vec{b} = 0 \\ & && \nabla \times \vec{h} - \left(\dfrac{FL}{c}\right)^2 \partial_t \vec{d} = \, \vec{j}_f \\ \\ & \text{constitutive equations} && \vec{d} = \vec{e} \\ & && \vec{b} = \vec{h} \\ \\ & \text{EM potentials} && \vec{b} = \nabla \times \vec{a} \\ & && \vec{e} = - \left( \dfrac{FL}{c} \right)^2 \partial_t \vec{a} - \nabla \phi \\ \\ & \text{Lorentz's gauge} && \nabla \cdot \vec{a} + \partial_t \phi = 0 \end{aligned} \end{split}\]

todo check compatibility in the definition of the charactersitic physical quantities; need to distinguish steady and time-dependent terms?