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Modeling of electromagnetics, electrostatics and interface problems MATH750, Fall 2009 Department of Mathematics Colorado State University October 2, 2009 MATH750 Governing Equations Maxwell Equations MATH750 Governing Equations Maxwell Equations Helmholtz equations MATH750 Governing Equations Maxwell Equations Helmholtz equations Interface problems Poisson-Boltzmann equation MATH750 Characteristic Quantities of Electromagnetic Fields Physical quantities of interesting: electric ﬁeld E, conductive current J, electric displacement D, magnetic ﬁeld H, magnetic induction B, magnetic polarization M. Conductive current J: ﬂow of electric charge (current), related to E through electric conductivity σ (Ohm’s law): J = σE Electric displacement D: In a dielectric material an applied electric ﬁeld E causes the bound charges in the material (atomic nuclei and their electrons) to slightly separate, inducing a local electric dipole moment: D = ǫ0 E + P = ǫ0 E + ǫ0 χe E = ǫ0 (1 + χe )E = ǫE, where ǫ0 is the dielectric permittivity of the vacuum, χe is the electric susceptibility, and ǫ is the dielectric permittivity of the material. ǫ/ǫ0 = 1 + χe is called the dielectric constant. MATH750 Characteristic Quantities of Electromagnetic Fields The induced dipole after polarization in a dielectric changes the electric ﬁeld in it. D is introduced to avoid the appearance of unknown induced charges in the Gauss law, thus simpliﬁes the model and computation. Relation between H and B follows similarly: B = µ0 H + µ0 M = µ0 (H + χv H) = µ0 (1 + χv )H = µH, where µ0 is the magnetic permeability in vacuum, χv is the magnetic susceptibility, and µ is the magnetic permeability of the material. Over years of shifting, it is now widely accepted that B a fundamental quantity, while H is a derived ﬁeld. MATH750 Derivation of Maxwell equations Maxwell equations, in both integral and differential forms, are derived based on a number fo physical and mathematical laws. Ampere’s law MATH750 Derivation of Maxwell equations Maxwell equations, in both integral and differential forms, are derived based on a number fo physical and mathematical laws. Ampere’s law Faraday’s law MATH750 Derivation of Maxwell equations Maxwell equations, in both integral and differential forms, are derived based on a number fo physical and mathematical laws. Ampere’s law Faraday’s law Gauss’s law MATH750 Ampere’s Law An electric current generate magnetic ﬁeld: in a stable magnetic ﬁeld the integration along a magnetic loop is equal to the electric current the loop enclosed. H · ds = j · da, S A where j is the electric current density over an arbitrary surface A which has boundary S. Applying Stokes theorem F · ds = ∇ × F · da S A we get ∇ × H · da = j · da. A A MATH750 Ampere’s Law An electric current generate magnetic ﬁeld: in a stable magnetic ﬁeld the integration along Electric current includes the conductive current J and the current induced by time-varying electric displacement, i.e., ∂D j=J+ . ∂t Hence ∂D ∇ × H · da = J+ · da. A A ∂t The differential form is therefore ∂D ∇·H=J+ . ∂t MATH750 Faraday’s Law Moving magnet can generate an alternating electric ﬁeld: ∂ E · ds = − B · da, S ∂t A where A is a surface bounded by the closed contour S, both independent of time. Applying Stokes theorem again, we obtain ∂ ∇ × E · da = − B · da, A ∂t A or the differential form ∂B(r , t) ∇ · E(r , t) = − . ∂t MATH750 Negative sign in Faraday’s law To conserve the energy, the direction of the EMF must be such that the induced current would oppose the change in the magnetic ﬂux. This was best expressed by H.F.E. Lenz in Lenz’s Law which can be succintly stated as: The direction of any magnetic induction effect must oppose the cause of the effect. MATH750 Gauss’s Law - Electric Field Electric displacement ﬂux through a close surface Ω is the toal charge enclosed in Ω: D · da = ρe dx. ∂Ω Ω Applying divergence theorem ∇ · Fdx = F · da, Ω ∂Ω we get ∇ · Ddx = ρe dx, Ω Ω or the differential form ∇ · D = ρe . MATH750 Gauss’s Law - Magnetic Field Magnetic induction ﬂux through a close surface Ω is the toal magnet enclosed in Ω: B · da = ρm dx. ∂Ω Ω Applying divergence theorem we get ∇ · Bdx = ρm dx, Ω Ω or the differential form ∇ · B = ρm ≡ 0 since there is no magnetic monopole. MATH750 Maxwell Equations ∂B ∇×E = − ∂t ∂D ∇×H = J+ ∂t ∇·D = ρe ∇·B = 0 Constitutive relations: B = µH, D = ǫE, J = σE, P = χe E, M = µ0 χv H. MATH750 Maxwell Equations Express Maxwell equations in terms of E and B: ∂B ∇×E = − ∂t ∂E ∂P ∇ × B = µ0 ǫ0 + µ0 J + +∇×M ∂t ∂t 1 ∇·E = (ρe − ∇ · P) ǫ0 ∇·B = 0 Current and charge densities due to electric polarization of the material ∂P Jpol = , ρpol = −∇ · P. ∂t Total current and charge density ∂P Jtot = J + Jpol + Jmag = J + + ∇ × M, ρtot = ρe + ρpol = ρ − ∇ · P. ∂t MATH750 Helmholtz Equations It is seen that the electric ﬁeld and the magnetic ﬁeld are coupled together, which makes the computation cumbersome and physical nature implicit. We will de-couple the two equations and identify their diffusion and wave propagation nature more explicitly. Recall that any vector ﬁeld A has a decomposition A = ∇Φ + ∇ × Ψ, i.e., as the sum of the gradient of a scalar ﬁeld Φ and the curl of a vector ﬁeld Ψ. Also recall identities ∇ × ∇Φ ≡ 0, ∇ · ∇ × Ψ ≡ 0, hence ∇ × ∇ × A = ∇(∇ · A) − ∇2 A. MATH750 Helmholtz Equations Apply this to the equation for Ampere’s law, we get ∂D ∇×∇×H = ∇×J+∇× ∂t ∂E = ∇ × (σE) + ∇ × ǫ ∂t ∂ = σ(∇ × E) + ǫ (∇ × E) ∂t ∂B ∂ ∂B = σ − −ǫ ∂t ∂t ∂t ∂H ∂2H = −σµ − ǫµ 2 ∂t ∂t MATH750 Helmholtz Equations Notice 1 ∇ × ∇ × H = ∇(∇ · H) − ∇2 H = ∇ ∇ · B − ∇2 H = −∇2 H, µ thus ∂H ∂2H ∇2 H = σµ + ǫµ 2 . ∂t ∂t Similarly we get the same equation for E if ρe = 0 (source free): ∂E ∂2E ∇2 E = σµ + ǫµ 2 . ∂t ∂t MATH750 Helmholtz Equations Assume that the electric ﬁeld is a plane wave E = E0 e−iωt , and hence ∂E ∂2E = −iωE0 e−iωt = −iωE, 2 = −ω 2 E. ∂t ∂t Having this into the equation, we obtain ∂E ∂2E ∇2 E = σµ + ǫµ 2 ∂t ∂t = −iωσµE − ω 2 ǫµE σ = −ω 2 ǫµ 1 + i E ωǫ = −k 2 E. MATH750 Helmholtz Equations Helmholtz equation can have different nature depending on the material properties and This is an eigenvalue problem: ∇2 E + k 2 E = 0, with squared complex wave number σ k 2 = ω 2 ǫµ 1 + i . ωǫ The solution ω is determined by material properties and the structure. Helmholtz equation can have different nature depending on the material properties and the frequency of the wave. MATH750 Helmholtz Equations If σ/(ωǫ) >> 1, we see natures of diffusion equation: ∂E ∂2E ∂E ∇2 E = σµ + ǫµ 2 = σµ . ∂t ∂t ∂t In this case the electric conductivity is the controlling parameter of the process, while the magnetic susceptibility is weak. This is a parabolic equation. MATH750 Helmholtz Equations If σ/(ωǫ) << 1, we see natures of wave equation. ∂E ∂2E ∂2E ∇2 E = σµ + ǫµ 2 = ǫµ 2 . ∂t ∂t ∂t In this case the dielectric permittivity is the prevailing parameter, while the magnetic permeability is still weak. Dielectric polarization is the controlling process other than conduction. This is a hyperbolic equation. Insulator has σ = 0. The EM wave travels in the absence of source. MATH750 Interface Conditions for Maxwell Equations E1t − E2t = 0 H1t − H2t = Js × n D1n − D2n = ρs B1n − B2n = 0 where Js is surface current density and ρs is the surface charge density. They can be derived by using the integral forms of the equations. MATH750 Boundary and Interface Conditions for Maxwell Equations It was known that the two tangential conditions are necessary while the two normal conditions are redundant. Current studies of the interface methods for Maxwell equations indicate that this is not true. MATH750