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,
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
Continuity equation of electric charge.
Maxwell’s equations.
Constitutive laws, here for linear isotropic medium
Potentials.
Gauge. Lorentz’s gauge
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,
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
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
todo check obseration at the end of thi section
Setting equal to
the non-dimensional form of the equations of electromagnetism becomes
todo check compatibility in the definition of the charactersitic physical quantities; need to distinguish steady and time-dependent terms?