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.
I
Z
H · ds =
S
j · da,
A
where j is the electric current density over an arbitrary surface A
which has boundary S. Applying Stokes theorem
Z
I
∇ × F · da
F · ds =
A
S
we get
Z
∇ × H · da =
Z
A
A
MATH750
j · da.
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.,
j=J+
Hence
Z
∇ × H · da =
A
∂D
.
∂t
Z A
∂D
J+
∂t
The differential form is therefore
∇·H=J+
MATH750
∂D
.
∂t
· da.
Faraday’s Law
Moving magnet can generate an alternating electric field:
Z
I
∂
B · da,
E · ds = −
∂t A
S
where A is a surface bounded by the closed contour S, both
independent of time.
Applying Stokes theorem again, we obtain
Z
Z
∂
B · da,
∇ × E · da = −
∂t A
A
or the differential form
∇ · E(r , t) = −
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∂B(r , t)
.
∂t
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:
I
Z
D · da =
ρe dx.
Ω
∂Ω
Applying divergence theorem
Z
I
∇ · Fdx =
Ω
F · da,
∂Ω
we get
Z
∇ · Ddx =
Ω
Z
Ω
or the differential form
∇ · D = ρe .
MATH750
ρe dx,
Gauss’s Law - Magnetic Field
Magnetic induction flux through a close surface Ω is the toal magnet
enclosed in Ω:
Z
I
ρm dx.
B · da =
Ω
∂Ω
Applying divergence theorem we get
Z
Z
∇ · Bdx =
ρm dx,
Ω
Ω
or the differential form
∇ · B = ρm ≡ 0
since there is no magnetic monopole.
MATH750
Maxwell Equations
∇·D
∂B
∂t
∂D
= J+
∂t
= ρe
∇·B
= 0
∇×E
∇×H
= −
Constitutive relations:
B = µH,
D = ǫE,
J = σE,
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P = χe E,
M = µ0 χv H.
Maxwell Equations
Express Maxwell equations in terms of E and B:
∂B
∂t
∂P
∂E
+ µ0 J +
+∇×M
∇ × B = µ0 ǫ0
∂t
∂t
1
∇·E =
(ρe − ∇ · P)
ǫ0
∇·B = 0
∇×E = −
Current and charge densities due to electric polarization of the
material
∂P
, ρpol = −∇ · P.
Jpol =
∂t
Total current and charge density
Jtot = J + Jpol + Jmag = J +
∂P
+ ∇ × M,
∂t
MATH750
ρtot = ρe + ρpol = ρ − ∇ · P.
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
∇×∇×H = ∇×J+∇×
∂D
∂t ∂E
= ∇ × (σE) + ∇ × ǫ
∂t
∂
= σ(∇ × E) + ǫ (∇ × E)
∂t
∂ ∂B
∂B
−ǫ
= σ −
∂t
∂t ∂t
= −σµ
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∂2H
∂H
− ǫµ 2
∂t
∂t
Helmholtz Equations
Notice
∇ × ∇ × H = ∇(∇ · H) − ∇2 H = ∇
1
∇ · B − ∇2 H = −∇2 H,
µ
thus
∂H
∂2H
+ ǫµ 2 .
∂t
∂t
Similarly we get the same equation for E if ρe = 0 (source free):
∇2 H = σµ
∇2 E = σµ
MATH750
∂E
∂2E
+ ǫµ 2 .
∂t
∂t
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
+ ǫµ 2
∂t
∂t
= −iωσµE − ω 2 ǫµE
σ
= −ω 2 ǫµ 1 + i
E
ωǫ
= −k 2 E.
∇2 E = σµ
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:
∇2 E = σµ
∂2E
∂E
∂E
+ 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.
∇2 E = σµ
∂E
∂2E
∂2E
+ 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
H1t − H2t
= 0
= Js × n
D1n − D2n
B1n − B2n
= ρs
= 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