Comments on the Galilean limits of Maxwell s equations

Transcription

1 Comments on the Galilean limits of Maxwell s equations Francesca Rapetti & Germain Rousseaux Laboratoire J.A. Dieudonné, Université de Nice Sophia-Antipolis & CNRS frapetti@unice.fr,rousseax@unice.fr Journée Modélisation et Calcul, 13 décembre 2011 à Reims Thanks to organizers for the invitation.

2 Introduction Maxwell s eqs. describe electromagnetic phenomena over a wide range of spatial/temporal scales. In case of static situations, use the laws of Coulomb and Biot-Savart! In case of time dependent situations, use the full set of Maxwell s eqs.! Not always. To achieve a preliminary accurate electromagnetic analysis of technical devices, suitable numerical approximations of Maxwell equations are carried out. Some particular models in the low frequency range emerge from neglecting particular couplings of electric and magnetic field related quantities. Applications (of non-relativistic electrodynamics) : motors, sensors, power generators, transformers, micromechanical systems,... F. Rapetti 1

3 We consider slowly time-varying fields, that can be approximated by electro-quasistatic (EQS) fields in high-voltage technology or microelectronics (with capacitive effects) magneto-quasistatic (MQS) fields in electrical machines or transformers (with inductive effects) We discuss on the domains of validity for the quasi-static (Galilean) limits of Maxwell equations, in terms of fields, potentials and gauge conditions, relying on dimensional analysis and comparison among characteristic time scalings. References : Le Bellac and Levi-Leblond, Nuovo Cimento, 1973 Haus and Melcher, Prentice Hall 1989 Ammari et al., SIAM J. Appl. Math. 2000, Schmidt et al., IEEE Trans. Magn de Montigny and Rousseaux, Am. J. Phys., 2007, Rousseaux, Europhys. Lett. 2005, Steinmetz et al., Proc. ISTET 2009,... Larsson, Am. J. Phys., 2007 F. Rapetti 2

4 The Maxwell puzzle Michael Faraday ( ) André-Marie Ampère ( ) E = B t H = J James Clerk Maxwell ( ) Generalized Ampère s law with the introduction of the displacement current to explain dielectric materials Karl Friedrich Gauss ( ) H = J + D t D = ρ William Thomson (or Lord Kelvin, ) B = 0 F. Rapetti 3

5 We distinguish between field intensities (E,H) and flux densities (D,B,J) A field intensity occurs in a path integral A flux density occurs in a surface integral The constitutive relations D = ɛe B = µh J = σe (Ohm s law) Field intensities and flux densities do not depend on the metric Field intensities are related to the flux densities via the constitutive relations which depend on the metric properties of the space, on the chosen coordinate system, and on the macroscopic material properties. The constitutive relations are necessary to close the Maxwell equation system F. Rapetti 4

6 Dimensional analysis A physical law must be independent of the units used to measure the physical variables Straightforward applications : find and check relations among physical quantities by using their dimensions simplify a problem by reducing the number of physical parameters check the plausibility and coherence of derived models The term dimension is more abstract than scale unit : mass is a dimension, Kg is a scale unit in the mass dimension The dimensions of a physical quantity are associated with combinations of mass, length, time, electric intensity. Ex. speed dimension [v] = [M] 0 [L] 1 [T ] 1 [I] 0 Vaschy-Buckingham π theorem (1914) : Given an eq. involving n variables and chosen m variables as fundamental, the eq. may be equivalently rewritten in terms of n m dimensionless parameters. F. Rapetti 5

7 Dimensional proof of Pythagorean theorem : [area] = [L] 2 The area of any triangle depends on its size and shape Size and shape are defined by giving one edge, the largest c, and two angles θ 1, θ 2 Angles are dimensionless quantities. Thus area = c 2 f(θ 1, θ 2 ) with f a nondimensional function of the angles Divide into 2 smaller ones 1, 2, thus area( ) = area( 1 ) + area( 2 ) c 2 f(α, π/2) = a 2 f(α, π/2) + b 2 f(α, π/2). Eliminating f, one gets c 2 = a 2 + b 2, without never specifying the form of f. F. Rapetti 6

8 Dimensional analysis in electromagnetism One considers field problems where E = ee, with e a reference quantity and E a non-dimensional vector (of order 1) l, a characteristic spatial dimension τ, a characteristic time constant the time interval in which significant changes of the field quantities arise v = l τ velocity of the phenomenon in the medium or of the medium c m = 1 µɛ velocity of light in the medium (c in the empty space) spatial differentiation x E is approximated by time differentiation t E is approximated by e τ e l F. Rapetti 7

9 Maxwell s equations when v c : fields When v c two scalings appear. In the empty space : E = t B e l b τ b H = t D µ 0 l ɛ 0e τ 8 8 < e = cb < v c yields : b = e, and v c yields : c e vb (v = l τ ) e = vb b = v c 2 e b v c 2 e (c = 1 µ0 ɛ 0 ). which results in v = c, wrong! This means that for v c, the two scalings (e vb and b v c 2 e) are not simultaneously valid, thus Faraday and Ampère laws cannot be coupled To choose the scaling, we give a look to the field sources (charges or currents)! F. Rapetti 8

10 Maxwell s equations when v c : sources D = ρ ɛ 0e l ρ and H = J b µ 0 l J b µ 0 l cɛ 0 e l J ρc J ρc b µ 0 ɛ 0 ce J ρc cb e if J ρc then e cb (thus e vb ) : MQS (the dielectric effect of charges is negligible) if J ρc, then e cb (thus b v c 2 e ) : EQS (the conducting effect is negligible) F. Rapetti 9

11 Limits of Maxwell equations when v c (E = ee and B = bb) Galilean electric limit, known as ElectroQuasiStatics (EQS), where v c m and b v c 2 m e : 8 >< >: E 0, H = J + t D, D = ρ, t ρ + J = 0 Galilean magnetic limit, known as MagnetoQuasiStatics (MQS), 8 E = t B, >< H J, where v c m and e vb : B = 0, >: J = 0 Galilean stationary limit, known as QuasiStationaryConduction (QSC), 8 E 0, >< D = ρ, where no time derivative appears, at very low frequencies : H J, >: B = 0 F. Rapetti 10

12 Potentials E = t A V, B = A E and B remain unchanged if we take a function ϕ(x, t) and transform both A and V as A A + ϕ, V V t ϕ ϕ is the gauge function a particular choice of A and V is a gauge a gauge fixing is done by imposing a gauge condition Some examples of gauge condition : A = 0 Coulomb A + 1 c 2 t V = 0 Lorenz (c = 1 µ0 ɛ 0 ) A + µɛ t V = µσv Stratton F. Rapetti 11

13 Maxwell s equations when v c : potentials B = A and E = t A V Faraday and Thomson laws are automatically satisfied Gauss law results in V + t ( A) = ρ ɛ 0 Generalized Ampère theorem yields ( A 1 c 2 tt V ) ( A + 1 c 2 t V ) = µ 0 J Under Lorenz gauge A + 1 c 2 t V = 0, the two relations become uncoupled ( 1 c 2 tt A A µ 0J A V ρ/ɛ 0 a V J ρc 2, thus ca V J ρc In MQS, c a V ( V v a), and in EQS c a V (a v c 2 V). F. Rapetti 12

14 Maxwell s equations when v c : potentials A = aa (magnetic vector potential), V = VV (electric scalar potential) B = A yields b a l (always true) E = t A V yields e a τ + V l thus if c a V (EQS) then (v c) v a V, l τ a V, so e V, thus E = V compatible with E 0. l a τ V l if c a V (MQS) then a V is compensated by v c, a τ V l so e a τ + V l, thus E = ta V compatible with E = t B. F. Rapetti 13

15 Maxwell s equations when v c : gauge condition A = aa (magnetic vector potential), V = VV (electric scalar potential) A + 1 c 2 tv = 0, a l + V c 2 τ 0, c a + v V c 0 c a V (MQS) and v c yields A = 0 Coulomb gauge condition c a V (EQS) and v c yields A + 1 c 2 t V = 0 Lorenz gauge condition Coulomb and Lorenz conditions are a particular case of Stratton condition (1941) valid in a medium with electric permittivity ɛ and magnetic permeability µ. A + µɛ t V = µσv Coulomb gauge in a medium (MQS and QSC) : A = µσv Lorenz gauge in a medium (EQS) : A + µɛ t V = 0 F. Rapetti 14

16 Characteristic times : τ e, τ m, τ em Three characteristic times appear representing τ, l in terms of ɛ, µ, σ µ ɛ σ τ l Consider the L numerical part M as a 4 5 matrix R. T Here rank(r) = 3. I Application of the π theorem of dimensional analysis : find α, β, γ, c 1, c 2, c 3 reals s.t. τ/(µ α ɛ β σ γ ) = O(1), l/(µ c 1 ɛ c 2 σ c 3 ) = O(1). by solving the two systems 8 >< α 3β 3γ = 0 α β γ = 0 >: 2α + 4β + 3γ = 1 8 >< >: c 1 3c 2 3c 3 = 1 c 1 c 2 c 3 = 0 2c 1 + 4c 2 + 3c 3 = 0 One gets get (α, β, γ) = (0, 1, 1) and (c 1, c 2, c 3 ) = ( 1 2, 1 2, 1). F. Rapetti 15

17 Characteristic times : τ e, τ m, τ em τ (ɛ/σ) = O(1), thus τ e = ɛ σ (charge relaxation time) l ( p ɛ/µ/σ) = O(1), thus l = p ɛ/µ σ (constitutive length) Since l/l = µσlc = µσl 2 (c/l), a natural choice is to set τ em = l/c and τ m = µσl 2 The quantity τ m is called magnetic diffusion time as the magnetic diffusion coefficient is D m = 1/(µσ) which has dimension [L] 2 [T ] 1 and τ m = l 2 /D m. τ 2 em = τ e τ m (light transit time in the medium) Full set of Quasi-static Static Maxwell s eq. regime regime 0 τ τ em τ em τ τ m, τ e τ τ m, τ e F. Rapetti 16

18 Stratton gauge condition and the characteristic times A + µɛ t V = µσv, a l + µɛ V τ µσv Dimensionless ratios 1 + l τ c m V c m a l l V c m a, I + II III II I v c m V c m a = τ em τ V c m a III I l l V c m a = τ m τ em V c m a = τ em τ e V c m a, III II τ τ e Full set of Maxwell s eq. MQS Statics 0 τ e τ em τ m τ Full set of Maxwell s eq. EQS Statics F. Rapetti 17 0 τ m τ em τ e τ

19 Domains of validity τ e τ em = l l, τ m τem = l l, x := log( τ τ em ), y := log( l l ) τ = τ em, log(τ/τ em ) = log(1) (x = 0), τ = τ e, log(τ/τ em ) = log(l /l) (y = x), τ = τ m, log(τ/τ em ) = log(l/l ) (y = x). F. Rapetti 18

20 Why Galilean limits? R is a reference system moving at relative speed v w.r.t. a fixed one R Assume that R and R have the same origin at t = t = 0 : the coordinate systems (x, t) and (x, t ) are said to be in standard configuration The Lorentz transformation (LT) between R and R in standard configuration is : x = γ(x vt), t = γ(t v x r ), γ = 1/ 1 v2 (Lorentz factor) c2 c 2 When c then γ 1 and LT becomes Galilean transformation (GL) x = x vt, t = t Yes, but c is finite, thus we should impose v c to have γ 1 Setting γ 1 in LT yields x = x vt, t = t v x c 2 (do not characterize a group!!!) We need an additional condition : fulfill causality or not! F. Rapetti 19

21 Galilean transformation (GT) results from LT if v c AND (causality) δx cδt x = x vt, t = t (characterize a group) Carroll transformation (CT) results from LT if v c AND (no causality) δx cδt x = x, t = t v x c 2 (characterize a group) Remark : CT was discovered by Levy-Leblond in 1965, and named after Lewis Carroll author of Alice in Wonderland. The rabbit runs staying at the same place, x = x, while time passing, t t. CT was later abandoned as we live in a Galilean world ; )) EQS and MQS models are GL because they are co-variant (or form-invariant) under GT Remark : The appearing velocity is c u = 1/ ɛ 0 µ 0 independent of specific units c u is a unit converter in the MQS and EQS models Moreover, GL of Maxwell s eqs. do not depend on the system of units F. Rapetti 20

22 Field transformations, after Einstein and Laub (1908) : MQS : v c AND e cb, E = γ(e + v B) + (1 γ) v(v E) v 2, B = γ(b v E c 2 ) + (1 γ) v(v B) v 2, D = γ(d + v H ) + (1 γ) v(v D), c 2 v 2 H = γ(h v D) + (1 γ) v(v H), v 2 A = γ(a vv c 2 ) + (1 γ) v(v A) v 2, V = γ(v v A). E = E + v B, B = B, A = A, D = D + (v H)/c 2, H = H, V = V v A, (MHD) ρ = ρ v J/c 2, J = J. EQS : v c AND e cb, E = E, B = B (v E)/c 2, A = A vv c 2, D = D, H = H v D, V = V, (EHD) ρ = ρ, J = J ρv. F. Rapetti 21

23 Eulerian versus Lagrangian in MQS R, reference system attached to the conductor, R, reference system attached to the laboratory B = B (J = J) but E = E + v B (ρ = ρ 1 c 2 v J) (MQS) Eulerian description : R is the unique reference system C c J = J = σe = σ(e + v B) in C J s v R Presence of a convective term v B C Lagrangian description : Two reference systems R in C c and, R in C C c J = σe in C J s v R Transmission conditions on the conductor boundary MQS eqs. are form invariant from R to R C R F. Rapetti 22

24 Eulerian Galerkin approach in MQS : for potentials Diffusion-convection problem ( 1 µ 0 = ν) : ( ) σ t A (ν A) + σv A = J ext The importance of convection versus diffusion is characterized by the Péclet number P e = hσ v 2ν Galerkin technique shows numerical instabilities when P e > 1, that is when h is too large to model the steep changes of A in presence of high velocities or small skin effects Instabilities are pretty numerical, do not exist in reality for A. To have P e 1, two possibilities : either (1) h or (2) σ v 2ν. Solution (1) is unrealistic in presence of high velocities yielding to unacceptable CPU times Solution (2) is realized by adding a numerical diffusion term in (*) : ν ν(1 + P e ) P e P e = P e (1 + P e ) < 1 for all h, v Numerically, the artificial diffusion method is equivalent, with FD, to having discretized the convective term by a first-order up-wind scheme. with FEM, to having stabilized the original Galerkin pb. Note that the mesh does not move and is conforming on the conductor boundary F. Rapetti 23

25 Galerkin Lagrangian approach in MQS Diffusion-convection problem ( 1 µ 0 = ν) : ( ) σ t A (ν A) = J ext together with 2 transmissions conditions on Γ FEM discretization techniques with either non-overlapping domains and sliding interfaces or overlapping domains TEAM workshop problem 28 : electromechanical coupling for a levitation device exact computed levitation height z (mm) F. Rapetti 24 time (ms)

26 Before concluding : few words on Darwin model EQS model from Maxwell s eqs. by neglecting t B in Faraday s law : E 0 MQS model from Maxwell s eqs. by neglecting t E in Ampère law : B µ 0 J Darwin model from Maxwell s eqs. by neglecting a part of t E in generalized Ampère law : B µ 0 (J + ɛ 0 t E C ), with E C = 0 and E C = ρ/ɛ 0 E = E C + E F with E C the Coulomb part of E and E F the Faraday part In the MQS model, only stationary currents are allowed and these currents cannot explain time changes in the charge density ρ. In the EQS model, there is no magnetic induction. Darwin model can include these aspects : it allows charge conservation in dynamic situations and t B also acts as a source of electric field. Darwin model in terms of potentials (E C = V and E F = t A) V = ρ/ɛ 0, A = µ 0 J + µ 0 ɛ 0 t V, A = 0 Darwin model uses in the EQS range the Coulomb gauge typical of the MQS range!!!!! : (( Darwin model is a mathematical simplification of the Maxwell-Vlasov physical system F. Rapetti 25

27 To conclude Dimensional analysis has to be driven by researcher experience and physical meaning Dimensional analysis allows to select important terms avoiding the computation of irrelevant ones Analogous conclusions on GL can be obtained by asymptotic analysis The proposed approach is local : in a system composed of different parts, consider the proper formulation according to the subdomain nature (ex. RCL circuit) With potentials, the gauge condition to use has to be compatible with the problem physics Thank you for the kind attention! Movies at http ://math.unice.fr/ frapetti/movies peps.html F. Rapetti 26