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 → −∞.