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Physics & Astronomy International Journal

Review Article Volume 2 Issue 6

Some comments on electromagnetic oscillations in anisotropic cavities-wave equations and boundary conditions

Luiz CL Botelho

Department of Applied Mathematics, Mathematics Institute, Fluminense Federal University, Brazil

Correspondence: Luiz CL Botelho, Department of Applied Mathematics, Mathematics Institute, Fluminense Federal University, Rua Mario Santos Braga, CEP 24220-140 Niterói, Rio de Janeiro, Brazil

Received: October 23, 2018 | Published: November 29, 2018

Citation: Botelho LCL. Some comments on electromagnetic oscillations in anisotropic cavities-wave equations and boundary conditions. Phys Astron Int J. 2018;2(6):562-565. DOI: 10.15406/paij.2018.02.00142

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Abstract

We present several new studies on the mathematical methods formulation of the important problem of electromagnetic oscillations in cavities on anisotropic and axial- anisotropic media, especially on the writing of the relevant dynamical wave equations and their correct boundary conditions. This study is the physical model for applications of a path integral method presented earlier.1

Keywords: anisotropic cavities, electromagnetic oscillations, Maxwell equations, Hodge-Helmotz theorem

Electromagnetic oscillations dynamical wave equations

We start this not by writing the initial value problem for Maxwell equations in the presence of sources and in a compact domain ΩΩ with boundary ΩΩ with spatially variable constitutive parameter (ε(r),μ(r),σ(r))(ε(r),μ(r),σ(r)) 14

×E+μtH=0×E+μtH=0  (1)

×HμεEt=μj+σμE×HμεEt=μj+σμE  (2)

B=0B=0  (3)

(εE)=ρ(εE)=ρ  (4)

E(r,0)=E0(r)E(r,0)=E0(r)  (5)

H(r,0)=H0(r)H(r,0)=H0(r)  (6)

Let us firstly search appropriated boundary conditions to be imposed on the electric and magnetic fields E(r,t)E(r,t) and H(r,t)H(r,t) respectively in order to lead to the problem unicity of the above written set of PDE’s, eq(l)-eq(6).

On basis of eq(l) and eq(2), we have the obvious energy balance equation for the PDE’s system above written

[H(×E+μtH)][(×H)EμεEtEjμEσμEE]=0[H(×E+μtH)][(×H)EμεEtEjμEσμEE]=0  (7)

 In other words, we have the following electromagnetic energy balance equation in ΩΩ 

(12)Ωt(εμE2+μH2d3r)=+ΩμE.J˜)d3r+Ωμσd3x+ΩFlux Poynting vector((E×H).n)dA(12)Ωt(εμE2+μH2d3r)=+ΩμE.J˜)d3r+Ωμσd3x+ΩFlux Poynting vector((E×H).n)dA  (8)

It one supposes that has two different set of solutions for Maxwell equations eq(l) and eq(2), namely (E1,H1) and (E2,H2) , then theirs difference satisfy the obvious inequality

+12t[ε(t)Ω(μ(ΔE)2+μ(ΔH)2d3r)]ΩΔE×ΔHndA+12t⎢ ⎢ ⎢ ⎢ ⎢ε(t)Ω(μ(ΔE)2+μ(ΔH)2d3r)⎥ ⎥ ⎥ ⎥ ⎥ΩΔE×ΔHndA  (9)

since the loss of ohmic energy is positive-definite

ΩμσΔE2d3x0ΩμσΔE2d3x0  (10)

 Now if one chooses boundary conditions that lead to the vanishing of the Poyinting flux on eq(8), one gets the problem unicity since

12tEt0EtE12tEt0EtE  (11)

 Let us analyze examples of boundary conditions that lead to vanishing of the Poynting flux on eq(9)

((E2E1)×(H2H1)Ω=0((E2E1)×(H2H1)Ω=0  (12)

Namely:

  1. E|Ω=0EΩ=0   (13)
  2. H|Ω=0HΩ=0  (14)
  3. ET1,2|Ω=0×HT1,2|ΩET1,2Ω=0×HT1,2Ω           (15)
  4. HT1,2|Ω=0HT1,2Ω=0         (16 a)
  5. E×H|Ω=0E×HΩ=0              (16 b)

where ˜T1,2˜T1,2 denote the tangent vectors on the domain boundary Ω.Ω.

Let us consider now a medium with constant electromagnetic parameters

ε=ε0μ=μ0σ=σ0.ε=ε0μ=μ0σ=σ0.

In this simply case we have the decoupled dynamical equations for Electric and Magnetic fields with the non-absorbing boundary conditions

ΔE=εμ2Et2+σμEt+(μjt+ρ)ΔE=εμ2Et2+σμEt+(μjt+ρ)  (17)

E(r,0)=Ε0(r)E(r,0)=E0(r)  (18)

Et(r,0)=J(r,0)+σE0(r)(×H0(r))Et(r,0)=J(r,0)+σE0(r)(×H0(r))  (19)

ET1,2|Ω=0ET1,2Ω=0  (20)

ΔH=εμ2Ht2+μσHt×J)ΔH=εμ2Ht2+μσHt×J)  (21)

H(r,0)=H0(r)H(r,0)=H0(r)  (22)

Ht(r,0)=1μ(×E0(r))Ht(r,0)=1μ(×E0(r))  (23)

(×H)T1,2|Ω=(JT1,2)|Ω=0(×H)T1,2Ω=(JT1,2)Ω=0  (24)

 Note that the divergence-less of the magnetic field eq (3) in this case can be insured straightforwardly by the equation below:

HTr(r,t)=H(r,t)14πr{Ωdiv(Hr',t)|rr'|)d3r')}HTrr,t=Hr,t14πrΩdiv(Hr,t)rrd3r  (25)

 Let us re-write the problem in terms of vector and scalar potentials with the condition of <t<(Ε(,t)  and  Η(,t)Λ2(Ρ3))<t<(E(,t)andH(,t)Λ2(P3)) ).

Since in this case one has gauge invariance so redundancy on the Maxwell equations solutions one should choose the generalized radiation gauge as a natural gauge fixing to find a unique solution:

1μAεϕtσϕ=01μAεϕtσϕ=0  (26)

 In this case the Maxwell dynamical equations eq(17)-eq(24) take the more invariant form below through the use of the electromagnetic potentials (A,ϕ)(A,ϕ).

{ΔAεμ2t2A=μj+σμtAΔϕμε2t2ϕμσϕt=ρET1,2|Ω=(tA+ϕ).T1,2|Ω=0×H).T1,2|Ω(×(×A)).T1,2|Ω=μ(JT1,2|Ω)⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ΔAεμ2t2A=μj+σμtAΔϕμε2t2ϕμσϕt=ρET1,2Ω=(tA+ϕ).T1,2Ω=0×H).T1,2Ω(×(×A)).T1,2Ω=μ(JT1,2Ω)(27)

At this point we address our readers to implement numerical approximate procedures to solve the above written set of linear boundary value problems in a computer by finite- differences or finite-elements.

Finally we write the associated wave equations associative to the full Maxwell equations in the context of an electromagnetic spatially variable medium namely:

ΔH(εε)(r)2Ht2={×j+σμHt(σ)×E(Hμr)(εε)(r)×(×HσE}ΔH(εε)(r)2Ht2={×j+σμHt(σ)×E(Hμr)(εε)(r)×(×HσE}  (28)

ΔE(εμ)(r)2Et2=μjt+σμEt(ρEEε)(μμ××E)ΔE(εμ)(r)2Et2=μjt+σμEt(ρEEε)(μμ××E)  (29)

plus boundary and initial conditions

At this point we call the reader attention that solving electromagnetic problem in cavities with a non-trivial topological/homological class with potentials, one encounter the severe difficulty of the Helmholtz-Hodge non trivial decomposition of the electromagnetic fields in term of the above mentioned potentials. For instance, B=0B=0 in ΩΩ means that B=×A+BtopB=×A+Btop where BtopBtop is an harmonic vector field configuration associated to the topological-homological characterization of ΩΩ (with ΩΩ being a domain with holes inside for instance, and of difficult determination from the local Maxwell PDE’systems! In this Helmholtz-Hodge context the Maxwell equations written in terms of potential (A,φ)(A,φ) are of form with constant medium electric parameters for instance)

ΔAεμ2t2A=(homological current(×HtopεμtEtop)+j+σEΔAεμ2t2A=⎜ ⎜ ⎜ ⎜homological current(×HtopεμtEtop⎟ ⎟ ⎟ ⎟+j+σE  (30-a)

Δϕεμ2t2ϕ=ρεhomological charge densityμt(×Htop)Δϕεμ2t2ϕ=ρεhomological charge densityμt(×Htop)  (30-b)

radiation gauge fixingAμϕt=0radiation gauge fixingAμϕt=0  (30-c)

++Boundary conditions

Here the homological-topological electromagnetic field configurations are defined by the Hodge theorem.

HodgeHelmholtz theoremH=1μ(×A)+HtopHodgeHelmholtz theoremH=1μ(×A)+Htop  (31-a)

0=×EtopμHtopdt0=×EtopμHtopdt  (31-b)

We note that non trivial topology on the manifold structure of ΩΩ appear dynamically in Maxwell equations as sources of to electric charge and electric currents.

It appears thus, that considering from the beginning the dynamical equations written directly for the strength field(Ε,Η)(E,H), all subtle and very difficulty topological-homological constraints imposed by the Hodge-Helmotz theorem are solved and already built on the boundary conditions imposed directly for the dynamical electromagnetic field equations.

Note that the set of second-order PDE’s eq(17)-eq(18) may enlarge the original set of solutions of the first order eqs(l)-eq(2), opposite to the potential method (A,φ)(A,φ). So, further direct verifications of using unique solutions eq(17)-eq(18) for solving eq(l) and eq(2) should be done at the end of the problem solving.

Let us now consider the anisotropic electromagnetic irradiation case in R3. In this situation we have the tensorial-matricial constituve relationships on the medium electro-magnetic properties between the electric displacement vector D(r,t)D(r,t) and the associated electric field E(r,t)E(r,t) respectively between the magnetic flux vector B(r,t)B(r,t) and the mag- netic vector field H(r,t).H(r,t).

D(r,t)=anisotropic medium permitivity[ε(r).]E(r,t)(Di(r,t)=εC0(R3)εij(r)Ej(r,t)(r))B(r,t)=[μ(r)]H(r,t)εijC0(R3)μij(r)Hj(r,t)=Bi(r,t)D(r,t)=anisotropic medium permitivity[ε(r).]E(r,t)(Di(r,t)=εC0(R3)εij(r)Ej(r,t)(r))B(r,t)=[μ(r)]H(r,t)εijC0(R3)μij(r)Hj(r,t)=Bi(r,t)  (31)

 Since B=0B=0, we have that exists a potential AA, such that

Bi(r,t)=εijkjAk(r,t)Bi(r,t)=εijkjAk(r,t)  (32)

 As a consequence of the Maxwell equations we also have that

Ei(r,t)=tAi(r,t)+xiϕ(r,t)Ei(r,t)=tAi(r,t)+xiϕ(r,t)

 One has thus the following anisotropic wave equation as outcome:

εirsxr(μ1]sq(r)Bq(r,t))t([εijr)Ejr,t))=Ji(r,t)εirsxr(μ1]sq(r)Bq(r,t))t([εijr)Ejr,t))=Ji(r,t)  (34-a)

 which can be re-written as of as

εirsxr([μ1]sgBgεgjkxjAk)t([ε]ij(tAj+ϕj))=Jiεirsxr⎜ ⎜[μ1]sgBgεgjkxjAk⎟ ⎟t([ε]ij(tAj+ϕj))=Ji  (34-b)

 or equivalently

εirsεgjk{([μ1]sg2xrxjAk)+(xr[μ1]sg)xjAk}+[ε]ij2Ajt2=Ji+[ε]ij2txjφεirsεgjk{([μ1]sg2xrxjAk)+(xr[μ1]sg)xjAk}+[ε]ij2Ajt2=Ji+[ε]ij2txjφ  (34-c)

In other words

N(θA,θB)=A|cos(θA)cos(θB)cos(θ)+eiϕsin(θA)sin(θB)sin(θ)|2+CN(θA,θB)=Acos(θA)cos(θB)cos(θ)+eiϕsin(θA)sin(θB)sin(θ)2+C (35)

 After considering the radiation anisotropic gauge

[ε]ij2txjφ(εirsεgjkxj[μ1]sgxjAk)0[ε]ij2txjφ(εirsεgjkxj[μ1]sgxjAk)0  (36)

one has the anisotropic second order Maxwell wave equation for the vector potential A(r,t)A(r,t), decoupled from the scalar potential.

Cmrjk(r)[ε]1mi(εirsεgjk[μ1]sg):ˉCmrkg2xrxjAk+2Amt2=[ε]1mijiCmrjk(r)[ε]1mi(εirsεgjk[μ1]sg):¯¯¯Cmrkg2xrxjAk+2Amt2=[ε]1miji  (37)

where the anisotropic fourth-order electromagnetic medium tensor is explicitly given by

Cmrjk(r)=[ε1r]mi(εirsεgjk[μ1]sg(r))Cmrjk(r)=[ε1r]mi(εirsεgjk[μ1]sg(r))  (38)

The equation for the ϕϕ-electric potential in the choice radiation gauge eq (36) is devoid of dynamical content and given by a sort of Poisson equation through Maxwell equations once known the solution of the vector potential dynamics as given by eq (36)

xi([ε]irxrϕ)=(ρ+(xi[ε]ir)tAr)xi([ε]irxrϕ)=(ρ+(xi[ε]ir)tAr)  (39)

We now show that it is possible to choose the anisotropic radiation gauge eq (36). Let (ˉA,ˉϕ)(¯¯¯A,¯ϕ) be a given fixed electromagnetic potential configuration and (ˉA+Λ,φ+Λt)(¯¯¯A+Λ,φ+Λt) it is gauge transformed.

We now show that it is possible to determine the gauge transformation parameter Λ(r,t)Λ(r,t) with the gauge field transformed electromagnetic potential configuration satisfying the gauge fixing analytical definition eq 34.

So let us suppose that

[εgjkεirs(xr([μ1]sq(r))xjˉAk)]t[[ε]ij(r)xjˉϕ]0[εgjkεirs(xr([μ1]sq(r))xj¯¯¯Ak)]t[[ε]ij(r)xj¯ϕ]0  (40)

So we need to determine ΛΛ such that

[εgjkεirs([μ1]sq)xj(A+Λ)k][t([ε]ij(ϕxj+Λt))][εgjkεirs([μ1]sq)xj(A+Λ)k][t([ε]ij(ϕxj+Λt))]  (41)

We have thus, that the gauge parameter function Λ(r,t)Λ(r,t) satisfies the second order PDE equation below

(εgjkεirsxr([μ1]sg)xj(xkΛ)2t2Λj==source as function of(ˉA,ˉφ)eq(40)(εgjkεirsxr([μ1]sg)xj(xkΛ)2t2Λj==source as function of(¯¯¯A,¯¯¯φ)eq(40)  (42)

Finally, the reader should realize perturbative analytical calculations by considering weakly anisotropy around the isotropic case (g1,21)(g1,21) 

[μ(r)]ij=δij+g1μij(r)[ε(r)]ij=δij+g2εij(r)[μ(r)]ij=δij+g1μij(r)[ε(r)]ij=δij+g2εij(r)  (43)

Maxwell equations in an axial anisotropic conductive medium

Let us start this section 2 by considering the set of Maxwell equations in a medium with constitutive parameters depending solely on the spatial variable z, i.e. our anisotropic medium has a permittivity E=EzE=Ez and a media permeability μ=μ(z)μ=μ(z). Note also the supposed z dependence of the medium conductivityσ(z)σ(z).
The electric flux density BB (see eq(31) and eq(30)) are also supposed to be time variant and solely depending on the spatial variable z

D=D(z,t)=(Dx,Dy,Dz)(z,t)D=D(z,t)=(Dx,Dy,Dz)(z,t)  (44-a)

B=B(z,t)=(Bx,By,Bz)(z,t)B=B(z,t)=(Bx,By,Bz)(z,t)  (44-b)

D=ε(z)E(z,t)D=ε(z)E(z,t)  (44-c)

B=μ(z)H(z,t)B=μ(z)H(z,t)  (44-d)

We have thus the following set of partial differential equations describing the electro- magnetic pulse in a such electromagnetic axial anisotropic dependent medium (ε(z),μ(z))(ε(z),μ(z)) with a source also still depending on the time and the z-variable solely

1ε(z)rt(D(z,t))+(1εz)×D(z,t)=Bt(z,t)1ε(z)rt(D(z,t))+(1εz)×D(z,t)=Bt(z,t)  (45-a)

1μ(z)rt(D(z,t))+(1μz)×B(z,t)tD(z,t)=j(z,t)+σzεzD(z,t)1μ(z)rt(D(z,t))+(1μz)×B(z,t)tD(z,t)=j(z,t)+σzεzD(z,t)  (45-b)

zBz(z,t)=(divB)(z,t)=0zBz(z,t)=(divB)(z,t)=0  (45-c)

zDz(z,t)=(divD)(z,t)=0zDz(z,t)=(divD)(z,t)0=  (45-d)

After re-writing the set of axial anisotropic Maxwell equations eq(45-a)−eq(45-d) in components, it yields

z(1εzDyz)+z(ε'(z)ε(z)Dy)=Bxtzz(1εzDyz)+z(ε(z)ε(z)Dy)=Bxtz  (46-a)

1μ(z)Bxtz(μ'(z)μ2(z))Bxt2t2Dy=tjy+σzεztDy1μ(z)Bxtz(μ(z)μ2(z))Bxt2t2Dy=tjy+σzεztDy  (46-b)

In a decoupled form after some elementary algebra, one gets the fully decoupled component wave equation

(1ε(z2z2Dy+(ε'(z)ε2(zDyz+ε'(z)ε2zDyz+(ε'ε2)(zDy(z)((μ'(z)μ(z)εz))Dyz+(μ'ε'με2(z)Dyμ(z)2t2Dyμ(z)tjy(μσε(z)tDy))(1ε(z2z2Dy+(ε(z)ε2(zDyz+ε(z)ε2zDyz+(εε2)(zDy(z)((μ(z)μ(z)εz))Dyz+(μεμε2(z)Dyμ(z)2t2Dyμ(z)tjy(μσε(z)tDy))   (47)

Similar algebraic procedures give decoupled equations for Dx(z,t)Dx(z,t) and the magnetic flux densityBB, after determining the candidate solutions for the electric flux density D=(Dx,Dy)D=(Dx,Dy).

z(1μ(z)Bxz)z(μ(z)μ2(z)Bx)==ε(z)2Bxt2+Effective source term{(εε)(z)Dyt(z,t)zjy(z,t)+z(σ(z)ε(z)Dy(z,t))}z(1μ(z)Bxz)z(μ(z)μ2(z)Bx)==ε(z)2Bxt2+Effective source term{(εε)(z)Dyt(z,t)zjy(z,t)+z(σ(z)ε(z)Dy(z,t))}  (48)

Formal harmonic solutions for the decoupled eq (47) are easily found for a harmonic source with a definite frequency (<t<)(<t<)  

Dy(z,t)=eiwtΦ(z)Dy(z,t)=eiwtΦ(z)  (49-a)

jy(z,t)=eiwtjy(z)jy(z,t)=eiwtjy(z)  (49-b)

Here

Φ(z)=exp{α(z)12zz0b(ˉz)dˉz}φ(z)Φ(z)=exp⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪α(z)12zz0b(¯z)d¯z⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪φ(z)   (50-a)

b(z)[(2εε2+μμε)1μ]b(z)[(2εε2+μμε)1μ]   (50-b)

d2φdz2+(iwd(z)V(z))φ(z)=w2φjy(z)eα(z)d2φdz2+(iwd(z)V(z))φ(z)=w2φjy(z)eα(z)   (50-c)

d(z)σ(z)ε(z)d(z)σ(z)ε(z)  (50-d)

V(z)=α(z)+(α(z))2+b(z)α(z)+c(z)V(z)=α(z)+(α(z))2+b(z)α(z)+c(z)  (50-e)

a(z)(1εμ)(z)a(z)(1εμ)(z)  (50-f)

c(z)([(εε2)μμεε]1μ)(z)c(z)([(εε2)μμεε]1μ)(z)  (50-g)

We have thus reduced the harmonic electromagnetic wave propagation described in a axial-anisotropic medium to the ordinary differential equation eq(50-c). Let us finally point out that at the limit of higher frequencies and jy(z,t)0jy(z,t)0, one may introduce the effective spatial variable Z.5

Z=ωγzφ(z)=U(Z);d(z)=˜d(Z);etc.Z=ωγzφ(z)=U(Z);d(z)=˜d(Z);etc.  (51)

with γγ being an expansion parameter and get the explicitly solutions at the asymptotic higher frequency limit

U(Z)=A4˜c(ˉZ)exp{iZZ0(-˜c(ˉZ))12dˉZ}           +B4˜c(Z)exp{iZZ0(-˜c(ˉZ))12dˉZ}U(Z)=A4˜c(¯¯¯Z)exp{iZZ0(˜c(¯¯¯Z))12d¯¯¯Z}+B4˜c(Z)exp{iZZ0(˜c(¯¯¯Z))12d¯¯¯Z}  (52)

Useful for scattering problem of an electromagnetic pulse into a slab (or layers) (work in progress). Work on applied settings are in progress.6

Acknowledgments

None

Conflict of interest

Authors declare there is no conflicts of interest.

References

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