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=Gg~, or with a small abuse of notation to avoid writing lots of tildes and to keep lighter notation,

g=Gg ,

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.

tρ+j=0

Maxwell’s equations.

{e=ρε0×e+tb=0b=0×b1c02te=μ0j

Constitutive laws, here for linear isotropic medium

d=εeb=μh

Potentials.

b=×ae=taϕ

Gauge. Lorentz’s gauge

0=1c2tϕ+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,

FRtρf+JLjf=0DLdRρf=0EL×e+BFtb=0BLb=0HL×hFDtd=JjfDd=εEeBb=μHhBb=AL×aEe=AFtaΦLϕALa+FΦc02tϕ=0,tρf+JFLRjf=0dRLDρf=0×e+BFLEtb=0b=0×hDLFHtd=JLHjfd=εEDeb=μHBhb=ABL×ae=ΦEL[AFLΦtaϕ]a+FLΦAc02tϕ=0

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)#J=RFLfrom contintuityR=DLfrom Gauss' law for dE=BFLfrom Faraday's lawD=εEfrom d=εeB=μHfrom b=μhA=BLfrom the definition b=×aA=ΦFLc02from Lorentz's gauge

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

DLFH=εμELFB=(FLc)2AFLΦ=ΦFLc2FLΦ=(FLc)2

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,Φ=EL ,

the non-dimensional form of the equations of electromagnetism becomes

(5.2)#continuity equation for chargetρf+jf=0Maxwell's equationsdρf=0×e+tb=0b=0×h(FLc)2td=jfconstitutive equationsd=eb=hEM potentialsb=×ae=(FLc)2taϕLorentz's gaugea+tϕ=0

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