I. Basic Electromagnetic Effects in a Medium with Time-Varying Parameters and/or Moving Boundary
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Chapter 1
Initial and Boundary Value
Electromagnetic Problems in a
Time-Varying Medium
An essential point for elaborating a common approach to the investigation of transient electromagnetic phenomena is the evolutionary
character of such phenomena, and an initial moment, when the
non-stationary condition starts, takes an important meaning. The
introduction of the non-stationary initial moment is dictated in
many cases by a necessity to separate the moment of “switching
on” the ﬁeld and the moment of the beginning of non-stationary
behaviour. The non-stationary state, which starts at some deﬁnite
moment of time, is accompanied by the appearance of a transient
(non-harmonic) ﬁeld. These so-called transients can exist for a
long time, being a signiﬁcant part of the total ﬁeld. However, they
fall out of the ﬁeld of vision of a stationary approach when all
periodic processes are assumed to start at the inﬁnite past. It
should be noted that the commonly used approximation of an
adiabatic “switching on” of a process at the inﬁnite past can easily
lead to indeﬁniteness in the problem formulation because of the
irreversibility of the non-stationary phenomenon. Therefore, an
Non-Stationary Electromagnetics
Alexander Nerukh, Nataliya Sakhnenko, Trevor Benson, and Phillip Sewell
Copyright c 2013 Pan Stanford Publishing Pte. Ltd.
ISBN 978-981-4316-44-6 (Hardcover), 978-981-4364-24-9 (eBook)
www.panstanford.com
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10 Initial and Boundary Value Electromagnetic Problems in a Time-Varying Medium
investigation of non-stationary electromagnetic phenomena should
be based on equations which include a general representation
of the medium parameters, where an inhomogeneity has a timedependent shape and time-dependent medium properties inside it.
A mathematical approach to the theory of transient electromagnetic
phenomena should contain a description of both continuous and
abrupt changes of both the ﬁeld functions and the medium
parameters. This technique also has to take into account the
correlation between spatial and temporal changes in the media.
Such a correlation occurs, for example, when a medium boundary
moves in space. In this case a sharp time jump of the medium
parameters occurs at every ﬁxed point passed by the medium
boundary.
The theory of generalised functions [1–6] is an adequate mathematical technique for treating such problems. The generalised functions describe uniformly continuous and discontinuous functions of
the ﬁeld and media parameters. Applying this theory to the classical
electromagnetic equations means a substitution of the generalised
derivatives instead of the conventional (classical) derivatives with a
corresponding modiﬁcation of Maxwell’s equation.
In this chapter a non-stationary electromagnetic problem is
mathematically formulated as a diﬀerential equation in a generalised function space. This allows all conditions for the ﬁelds on the
discontinuity surfaces (boundaries) as well as parameter time jumps
to be included directly into the equations.
1.1 Generalised Wave Equation for an Electromagnetic
Field in a Time-Varying Medium with a Transparent
Object
1.1.1 Generalised Derivatives
To use the space of generalised functions one must consider the
generalised derivatives [1–6], instead of the classic one. Assume
a vector-function a(t, r) has a discontinuity on an arbitrary timevarying surface, S(t), and that its jump value is equal to [a] S =
a+ − a− , where a+ is the magnitude of the vector function on the
positive side of this surface. This side is determined by a normal
vector n, as shown in Fig. 1.1:
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Generalised Wave Equation for an Electromagnetic Field 11
Figure 1.1.
surface.
The orientation of the normal vector on the discontinuity
The generalised derivatives are deﬁned by the following formula:
∂a
∂a
=
− [a]s un δ(S(t)), curla = {curla} + n × [a]s δ(S(t)),
∂t
∂t
(1.1.1)
where the braces mean an ordinary derivative where it exists, δ(S) is
a surface delta-function, and un is a velocity component normal to a
certain surface domain. Here and later, bold characters are used for
vectors.
Time jumps of solid medium parameters can be taken into
account by the formula
∂a
∂a
=
(1.1.2)
− [a]t=0 δ(t),
∂t
∂t
where [a]t=0 = a(t = +0) − a(t = −0).
Transition to the generalised derivatives allows the conditions
for the ﬁelds on the surfaces and at the time points to be included
directly into Maxwell’s equations, where the medium parameters
are discontinuous. These conditions are given by the terms in
the square brackets in Eqs. 1.1.1 and 1.1.2. Compared with the
continuous medium case, the equation form remains unchanged
almost everywhere. To show this, let us consider the classical
Maxwell’s equations in a continuous medium. These equations have
the following form in an SI system:
∂E
∂P
∂B
1
{curlB} = ε0
+
+{curlM}+j {curlE} = −
,
μ0
∂t
∂t
∂t
(1.1.3)
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12 Initial and Boundary Value Electromagnetic Problems in a Time-Varying Medium
where E is the electric ﬁeld strength, B is the magnetic ﬂux density,
P and M are vectors of medium electric and magnetic polarisations,
j is a conductivity current, ε0 = 10−9 36π [F·m−1 ] and μ0 =
4π · 10−7 [H·m−1 ] are the permittivity and permeability of free
space, respectively, and √ε10 μ0 = c = 3 · 108 [m/s] is the velocity
of light in vacuum. The polarisations, P and M, and the electric ﬁeld
ﬂux density D and the magnetic ﬁeld strength H are connected in a
conventional way:
D = ε0 E + P = ε0 (1 + κ ε )E = ε0 εE
1
1
1
H=
B−M=
B=
B
μ0
μ0 (1 + κ μ )
μ0 μ
(1.1.4)
Here, 1 + κ ε = ε, κ ε is an operator of an electrical susceptibility, ε is
an operator of a relative permittivity of the medium, and analogously
1 + κ μ = μ, κ μ is an operator of a magnetic susceptibility and μ is
an operator of a relative permeability of the medium. As κ ε and κ μ
are assumed to be operators, the relations (Eq. 1.1.4) are general
ones and they describe all possible media, including dispersive and
anisotropic ones.
1.1.2 Initial and Boundary Conditions for Electromagnetic
Fields in a Time-Varying Medium
It is convenient to determine the conditions for the ﬁeld on a
discontinuity surface on the basis of Maxwell’s equations in integral
form [7, 8, 9]. They have an invariant form independent of the way
the medium parameters change:
Hdl =
d
dt
L
Dds +
S
S
d
Edl = −
dt
L
S
Bds = 0,
S
Bds
S
Dds = q
jds
(1.1.5)
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Generalised Wave Equation for an Electromagnetic Field 13
where j and q are bulk densities of current and charge in a laboratory
frame of reference. It follows from Eq. 1.1.5 that the boundary
conditions have the following form for an arbitrarily moving surface
[9, 10]:
n × [H]s + un [Dtan ]s = γ −1 (n × (i × n))
n × [E]s − un [Btan ]s = 0
[Dn ]s = σ [Bn ]s = 0, (1.1.6)
where Dtan , Btan are the components of the ﬁelds tangential to the
surface, γ −1 = 1 − βn2 is the relativistic factor, βn = un /c, and i
and σ are the surface current and charge densities in the intrinsic
frame of reference of the moving surface domain, respectively.
The electric ﬁeld ﬂux density, as well as the magnetic induction,
remains continuous at time jumps of the medium features. This
follows from the classical Maxwell’s equations (Eq. 1.1.3) together
with the limiting value of the spatial derivatives of the ﬁeld. Indeed,
yields the following:
integrating the relation curl μ10 B − M = ∂D
∂t
t
D(t = +0) − D(t = −0) = lim
t→0
− t
curl
1
B − M dt = 0
μ0
(1.1.7)
leads to
Analogously, the equation curlE = − ∂B
∂t
α
B(t = +0) − B(t = −0) = − lim
α→0
−α
curlEdt = 0
(1.1.8)
Note that in general case the time derivatives of the electric
ﬂux density and the magnetic induction do not remain continuous.
Indeed, the jump of an electric ﬂux density diﬀers from zero in the
general case:
∂D
∂t
α
= lim
α→0
−α
t=0
= curl
curl
∂
∂t
1
B − M dt
μ0
1
B−M
μ0
=0
t=0
Similarly, for the magnetic induction we have
∂B
∂t
= − [curlE]t=0 = 0.
t=0
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14 Initial and Boundary Value Electromagnetic Problems in a Time-Varying Medium
The electric ﬂux density derivative remains continuous only when
medium’s magnetic features are continuous. Analogously, the
magnetic induction is continuous in the case when the medium
electric features are continuous. If this is not the case, then the initial
condition for the magnetic induction derivative can be derived as
in the case when the permittivity
follows. The equation curlE = − ∂B
∂t
D(t)
changes in time, E(t) = ε(t) , leads to
∂B
∂B
(t = +0) −
(t = −0)
∂t
∂t
α
= − lim
α→0
−α
curl
D(t + α) D(t − α)
∂
Edt = − lim curl
−
α→0
∂t
ε(t + α)
ε(t − α)
1
1
curlD(t + α) −
curlD(t − α)
α→0 ε(t + α)
ε(t − α)
1
1
1
1
= − + curlD(t) + − curlD(t) = −
− − curlD(t)
+
ε
ε
ε
ε
−
1
1
ε
=−
− − curlε− E(−) (t) = −
− 1 curlE(−) (t).
ε+
ε
ε+
(1.1.9)
= − lim
It gives the initial condition for the magnetic induction derivative as
∂B
∂t
=−
t=0
ε−
− 1 curlE(−) (t).
ε+
(1.1.10)
1.1.3 Maxwell’s Equations in Generalised Derivative
Representation
Returning to the diﬀerential form of Maxwell’s equations (Eq. 1.1.3),
we see that the second equation in Eq. 1.1.3 is not changed when the
classical derivatives are replaced by the generalised ones, while the
ﬁrst equation gains an additional term determined by the surface
current i = vn σ + γ −1 (n × (i × n)) in the laboratory frame of
reference:
curlB = μ0 ε0
∂E ∂P
+
∂t
∂t
+ μ0 (curlM + j + iδ(S(t)))
curlE = −
∂B
∂t
(1.1.11)
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Generalised Wave Equation for an Electromagnetic Field 15
In these equations, all the derivatives are generalised; therefore,
they readily contain the boundary conditions for the ﬁelds on
the discontinuity surfaces [11] that distinguishes them from
Eq. 1.1.3. The discontinuity surface S(t) restricts the region V (t)
and is moving in the general case. Equation 1.1.11 forms the
basis for further description and investigation of electromagnetic
phenomena in time-varying inhomogeneous media.
Merging the two equations results in a wave equation in a
generalised derivative representation. This equation describes an
electromagnetic ﬁeld in a medium whose parameters can vary
arbitrarily in time as well as in space (this variation includes the
arbitrarily moving surface as well) [12]:
∂ 2E
∂ 2P
∂M ∂(j + iδ(S))
1
−
(1.1.12)
curlcurlE + ε0 2 = − 2 − curl
μ0
∂t
∂t
∂t
∂t
Using the characteristic function χ , which is equal to unity
inside V (t) and equal to zero outside this region, we can deﬁne
the generalised functions P, M and j that describe the medium
electromagnetic representation in the whole space:
P = χ (P1 − Pex ) + Pex
M = χ (M1 − Mex ) + Mex
(1.1.13)
j = χ j1 + jextr
where the values with the index “1” are deﬁned inside the region
V (t) and those with the index “ex” are deﬁned outside of it. This
external region is further referred to as a “background medium”.
jextr is a current describing extrinsic sources of the ﬁeld (see
Fig. 1.2).
Substituting Eq. 1.1.13 into Eq. 1.1.12 yields
∂2
∂Mex
1
+ 2 (ε0 E + Pex )
curlcurlE + curl
μ0
∂t
∂t
2
∂
∂
= − 2 χ (P1 − Pex ) − curl χ (M1 − Mex )
∂t
∂t
∂(χ j1 + iδ(S)) ∂jextr
−
.
(1.1.14)
−
∂t
∂t
According to the main idea of the approach originated by Khizhnyak
[13], the left-hand side of this equation has the form as in the
background, and the right-hand side is distinct from zero inside
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16 Initial and Boundary Value Electromagnetic Problems in a Time-Varying Medium
Figure 1.2. The medium description and the arrangement of an inhomogeneity and ﬁeld sources in the general problem formulation.
the region V (t) only. This equation allows consideration of the
electromagnetic problem for an arbitrary inhomogeneity placed
in various backgrounds. We will consider two such cases: a nondispersive background, and a plasma as an example of dispersive
one.
1.1.4 Generalised Wave Equation for the Case of a
Non-Dispersive Background
First we consider the generalised wave equation in the nondispersive background that is described by the relative permittivity
and permeability ε and μ, respectively. In this case
Pex = ε0 (ε − 1)E
Mex =
1
(1 − μ−1 )B.
μ0
(1.1.15)
If these operators commute with the operators ∂t∂ and curl, then
we have for the background polarisations in Eq. 1.1.14
∂2
∂Mex
∂ 1
∂ 2 Pex
=
+
curl
ε0 (ε − 1)E + curl
(1 − μ−1 )B
2
2
∂t
∂t
∂t
∂t μ0
∂2
1
= 2 ε0 (ε − 1)E − curl (1 − μ−1 )curlE
∂t
μ0
(1.1.16)
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Generalised Wave Equation for an Electromagnetic Field 17
Now the generalised wave equation takes the form
∂2
∂2
curlcurlE + ε0 μ0 2 εμE = −μ0 μ
χ (P1 − Pex )
∂t
∂t2
∂(χ j1 + iδ(S)) ∂jextr
∂
+
+ curl χ (M1 − Mex ) +
, (1.1.17)
∂t
∂t
∂t
where all parameters of the region V (t) are collected in the righthand side of this equation. Introducing the shorthand notation
1 ∂2
1
∂
Q1 = − 2 2 χ
(P1 − Pex ) − curl χ μ0 μ(M1 − Mex )
v ∂t ε0 ε
∂t
∂
− μ0 μ(χ j1 + iδ(S)),
(1.1.18)
∂t
Eq. 1.1.17 can be rewritten as follows:
1 ∂2
∂jextr
.
(1.1.19)
curlcurlE + 2 2 E = Q1 − μ0 μ
v ∂t
∂t
√
√
The coeﬃcient v = c
εμ = 1
ε0 μ0 εμ in Eq. 1.1.19 is a wavephase velocity in the background.
1.1.5 Generalised Wave Equation for the Case of a
Dispersive Background
Equation 1.1.19 describes the electromagnetic ﬁeld in the case of
the object placed in the uniform non-dispersive medium. Let us now
consider the case of a dispersive background, the simplest and most
applicable of which is a cold isotropic plasma. It is known that such
a plasma is described by the constitutive relations
t
Pex (t, r) = ε0
ωe2 (t − t )E(t , r)dt
Mex = jex = 0,
(1.1.20)
−∞
where ωe =
N e2 /mε0 is a plasma frequency, N is a density
of electrons, and e and m are the electron charge and mass,
respectively.
With these relations, the ﬁrst equation in Eq. 1.1.3 can be
represented in the form valid for the region outside of the object
V (t)
∂Eex
1
curlBex − ε0
− ε0
μ0
∂t
t
−∞
ωe2 Eex dt = 0.
(1.1.21)
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18 Initial and Boundary Value Electromagnetic Problems in a Time-Varying Medium
Inside the object, this equation has the right-hand side
1
curlBin
μ0
=
∂
∂t
− ε0 ∂E∂tin − ε0
Pin − ε0
t
−∞
ωe2 (t
t
−∞
ωe2 Ein dt
− t )Ein dt
+ (curlMin + jin + iδ(S)) .
(1.1.22)
Similarly to the previous case of non-dispersive medium, we can
also introduce the discontinuous functions
t
P = χ Pin + (1 − χ )ε0
ωe2 (t − t )E(t , r)dt
M = χ Min j = χ jin
−∞
(1.1.23)
and expand Eq. 1.1.22 on the whole space. This gives the equation
deﬁned in the whole considered space.
⎛
⎞
t
t
∂E
∂ ⎝
1
2
2
− ε0
P − ε0
curlB − ε0
ωe Edt =
ωe (t − t )Edt ⎠
μ0
∂t
∂t
−∞
−∞
+ (curlM + j + iδ(S)) (1.1.24)
Combining with the Maxwell’s second equation yields inhomogeneous Klein–Gordon’s equation
curlcurlE +
where
∂jextr
ω2
1 ∂2
,
E + 2e E = Q1, p − μ0
2
2
c ∂t
c
∂t
⎡
Q1, p
⎛
2
∂
= −μ0 ⎣ 2 ⎝P − ε0
∂t
(1.1.25)
⎞
t
ωe2 (t − t )Edt ⎠
−∞
∂
∂(j1 + iδ(S))
+ curl M +
.
∂t
∂t
(1.1.26)
Equations 1.1.19 and 1.1.25 are deﬁned on the whole time axis
and in the whole space and describes the electromagnetic ﬁeld
completely, even though they contain only the electric ﬁeld. The
magnetic ﬁeld B can be determined from the second equation in
Eq. 1.1.3 according to which B = −
t
curlEdt. The magnetic ﬁeld
−∞
is deﬁned uniquely if the condition of zero-value limit is fulﬁlled for
the electromagnetic ﬁeld when t → −∞.