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Modeling of electromagnetics_ electrostatics and interface problems

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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

								
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