Lehner G. Electromagnetic Field Theory for Engineers and Physicists

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404 Time Dependent Problems I (Quasi Stationary Approximation)

The time is the relaxation time, discussed in Sect. 4.2. For the case of the skin

effect, a necessary requirement is that the frequency is

These prerequisites and the related approximations can be dropped when

considering the displacement current, as we will do in Chapter 7. Specifically in

Sect. 7.12, we will return to the currently discussed problems and solve some of

which exactly, from the perspective of the wave theory. The limits of the quasi

stationary theory will become more apparent in this context. Then, we will also see,

how the solutions of the quasi stationary approximation transform into the wave

theory and vice versa.

---» t

2π

------=

---«

---=

7 Time Dependent Problems II (Electromagnetic

Waves)

7.1 Wave Equations and their simplest Solutions

7.1.1 The Wave Equations

Now we consider Maxwell’s equations in their complete form. Assume that the

medium in which we will solve them is uniform, i.e., are constant over all

space.

(7.1)

(7.2)

(7.3)

(7.4)

(7.5)

(7.6)

(7.7)

Taking the curl of (7.2) and substitute by use of (7.7), (7.1), (7.6), and (7.5) yields

Making use of the vector identity

gives

(7.8)

On the other hand, taking the curl of (7.1) and substitute by use of (7.6), (7.5),

(7.2), and (7.7) yields

εµκ,,

H∇× g

t∂

∂D

E∇×

t∂

∂B

–=

B∇• 0=

D∇• ρ=

D εE=

g κE=

B µH=

E∇×()∇×

t∂

∂

B∇×()– µ

t∂

∂

H∇×()– µ

t∂

∂

t∂

∂D

+()–== =

t∂

∂

κE

t∂

∂

εE+()– µκ

t∂

∂

E µε

∂

∂t

----------

––==

A∇×()∇× A∇•()∇∆A– A∇•()∇∇

A–==

∇

EE∇•()∇– µκ

t∂

∂E

µε

∂

∂t

----------

G. Lehner, Electromagnetic Field Theory for Engineers and Physicists,

DOI 10.1007/978-3-540-76306-2_7, © Springer-Verlag Berlin Heidelberg 2010

406 Time Dependent Problems II (Electromagnetic Waves)

and with (7.3) obtain

(7.9)

The two equations (7.8) and (7.9) represent the so-called wave equations for E and

B in their most general form for uniform media.

Inside a charge-free and non-conducting dielectric, vacuum (or free space) in

particular, we have

giving rise to the more specific wave equations

(7.10)

. (7.11)

7.1.2 The simplest Case: Plane Wave in an Insulator

Initially, only the simplest solutions of the wave equations (7.10) and (7.11) shall

be examined. It shall be assumed that E and B depend on only one of the three

Cartesian coordinates for example, z, but are time dependent.

(7.12)

(7.13)

Both fields have to be source free since , i.e.,

(7.14)

. (7.15)

It follows that

(7.16)

H∇×()∇× g∇×

t∂

∂

D∇×()+ κ E∇× ε

t∂

∂

E∇×()+==

t∂

∂B

– ε

∂

∂t

----------

–

---

B∇×()∇×==

---

B∇•()∇∇

B–()=

∇

B µκ

t∂

∂B

µε

∂

∂t

----------

ρ 0=

g 0=

κ 0=

∇

E µε

∂

∂t

----------

∇

B µε

∂

∂t

----------

EEzt,() E

zt,()E

zt,(),,〈〉==

BBzt,() B

zt,()B

zt,(),,〈〉==

ρ 0=

E∇•

z∂

∂

zt,()0==

B∇•

z∂

∂

zt,()0==

t()=

7.1 Wave Equations and their simplest Solutions 407

(7.17)

Later, we will find that neither nor may depend on t. It is possible that the

space contains a field or , neither of which depend on time or space, and

thus, do not interest us. We might as well assume that

(7.18)

. (7.19)

Fields which, when a coordinate system is chosen properly, depend only on one

Cartesian coordinate and time are called plane waves. Therefore, we note that

plane waves have no field components in propagation direction (here the z-

direction), that is, they are inevitably transverse waves. This results from above

assumption, namely the absence of any volume charges. In the presence of volume

charges, it is entirely possible to have plane waves with field components in

propagation direction, the so-called longitudinal waves. The so-called “plasma

waves”, which are important in plasmas and solid state physics, are of this kind.

We will limit our discussion to transverse waves. In this case, one only needs to

consider the transverse field components , and :

(7.20)

. (7.21)

Almost obvious is that arbitrary functions and satisfy the associated wave

equation

(7.22)

This is d’Alembert’s solution to the wave equation, where

(7.23)

is the speed of light in the observed medium.

The proof is simple. Starting with the derivative

and

t()=

,, B

∂

∂z

------------ εµ

∂

∂t

------------

∂

∂z

------------ εµ

∂

∂t

------------

zct–() g

zct+() +=

εµ

----------=

z∂

∂E

f '

∂

∂z

------------ f ''

g''

t∂

∂E

cf '

– cg'

408 Time Dependent Problems II (Electromagnetic Waves)

from which is obvious that this satisfies (7.20). This applies similarly to and the

components of B ( and ). On the other hand, the components of B and E are

not independent of each other. From (7.2) follows that

A consequence thereof is that has to be constant in time. Anticipating this, we

have used this fact already above. Furthermore

or when integrating over the time

(7.24)

. (7.25)

On the other hand, using (7.1) for an insulator with , we get

This means, as claimed before, that and thus are also independent of t.

Moreover

∂

∂t

------------ c

f ''

g''

+ c

∂

∂z

------------

εµ

------

∂

∂z

------------

===

E∇×

z∂

∂

∂E

∂z

---------

–

∂E

∂z

---------

0,,〈〉

∂B

∂t

---------

∂B

∂t

---------

∂B

∂t

---------

,,〈〉–== =

∂B

∂t

---------

∂E

∂z

--------- f

' zct–() g'

zct+()+==

∂B

∂t

---------

∂E

∂z

---------– f

' zct–()– g'

zct+()–==

---

zct–()–

---

zct+()F

z()++=

---

zct–()

---

zct+()– F

z()+=

g 0=

H∇×

z∂

∂

∂H

∂z

----------–

∂H

∂z

----------

0,,〈〉

∂D

∂t

----------

∂D

∂t

----------

∂D

∂t

---------

,,〈〉== =

∂B

∂z

--------- εµ

∂E

∂t

---------

-----

c f

' zct–()– cg

' zct+()+[]==

---

' zct–()–

---

' zct+()+=

∂B

∂z

--------- εµ

∂E

∂t

---------

–

-----– c f

' zct–()– cg'

zct+()+[]==

---

' zct–()

---

' zct+() .–=

7.1 Wave Equations and their simplest Solutions 409

Integrating for z gives

(7.26)

. (7.27)

Comparing eqs. (7.24) through (7.27) with each other provides a condition for the

integration constants, which is

(7.28)

Consequently, these may not depend on either z or t, i.e., they have to be constant in

space and time. Disregarding such constant fields, the field of a plane wave has to

be:

(7.29)

This plane wave has two parts, one which moves in positive z-direction without

changing its shape:

(7.30)

Its propagation direction is

Expressed in a different form, one writes

(7.31)

---

zct–()–

---

zct+()G

t()++=

---

zct–()

---

zct+()– G

t()+=

z() G

t()=

z() G

t()=







zct–() g

zct+()+=

zct–() g

zct+()+=

---

zct–()–

---

zct+()+=

---

zct–()

---

zct+()–=











zct–()=

---

zct–()–=

---

zct–()=











001,,〈〉=

E×

---------------

410 Time Dependent Problems II (Electromagnetic Waves)

(7.32)

where

(7.33)

is the so-called characteristic impedance of the medium. The other part propagates

also without changing its shape, but in the other direction, that of the negative z-

axis:

Writing

allows to write this in the form of eqs. (7.31) through (7.33). These equations are

generally applicable for any plane wave. Since they are in vector form, they are

independent of a coordinate system. In a rotated coordinate system, the

propagation direction would not be along the z-axis anymore, but eqs. (7.31)

through (7.33) would still apply. Conversely, it is also true that

(7.34)

i.e., the three vectors E, B (or H), and (in this order) form a right handed

system. To show this, one multiplies (7.31) in a vector product with to obtain

Applying

and

completes the proof.

The characteristic impedance for vacuum is

E×

µc

---------------

E×

---

---------------

E×

---------------

===

--- =

zct+()=

---

zct+()=

---

zct+()–=











00 1–,,〈〉=

E c Be

×()Z He

×() ==

E×

---------------

× e

E⋅()

------------------

–

⋅()

------------------------+==

E⋅ 0=

⋅ 1=

7.1 Wave Equations and their simplest Solutions 411

(7.35)

If in eq. (7.29) (and therefore also ) or if (and therefore

also ), then the electromagnetic field oscillates only in a plane. These

waves are called linearly polarized. A general plane wave can be constructed from

two waves which are polarized perpendicular to each other. The result is then just

the wave described in (7.29). Fig. 7.1 and Fig. 7.2 illustrate examples of both types

of linearly polarized waves.

7.1.3 Harmonic Plane Waves

The waves of the kind depicted in Fig. 7.1 and Fig. 7.2 are also referred to as

“wave trains” or “wave packets”. They can be formed by superposition of

appropriate sets of harmonic waves with specific wave lengths. The underlying

------

4π 10

7–

⋅

--------

8.855 10

12–

⋅

--------

---------------------------------------- 3 7 7 Ω 120πΩ≈≈== =

0= B

0= E

Fig. 7.1 Linearly polarized plane wave train or wave packet (i)

-----=

Fig. 7.2 Linearly polarized plane wave train or wave packet (ii)

412 Time Dependent Problems II (Electromagnetic Waves)

formal concept is the possibility to represent any function as a Fourier integral, i.e.,

by superposition of so-called harmonic waves. For instance

(7.36)

This wave is sketched in Fig. 7.3.

Here, and are the amplitudes of the fields; is a phase angle which

depends on where the origin of the chosen coordinate system is, and the choice of

when the time is zero. is the angular frequency of the wave and is the wave

number. If is its frequency, its period, and its wave length, then we may

write

(7.37)

. (7.38)

The “phase velocity” of the wave is

(7.39)

This is a result of the wave equation (7.10) when using it in (7.36). We obtain

which, again, gives (7.39). Moreover, the wave expressed in (7.36) can be regarded

as a special case of (7.22). The reason is that

zt,() E

ωtkz– ϕ+()cos=

zt,() H

ωtkz– ϕ+()cos=

--------=











Fig. 7.3 Harmonic Plane Waves

ω k

f τλ

ω 2πf

2π

------==

2π

------=

εµ

---------- λ f

2π

------

2π

------

⋅

----=== =

–

-----

–=

7.1 Wave Equations and their simplest Solutions 413

i.e., a function of .

From (7.39) one obtains

(7.40)

The relation between and is called the dispersion relation, here for the case of

a plane electromagnetic wave in an ideal insulator. In other circumstances the

relation between and could be different, i.e., the dispersion relation could be

of the form:

(7.41)

In this general case, we have to associate different velocities with the wave. The

velocity with which the phase propagates is still the phase velocity. The phase

remains constant for

The constant phase is then

Therefore, the phase velocity in general is

(7.42)

Dispersion accounts for the fact that the phase velocity according to (7.42) may be

a function of the frequency (wave length). In the special case when the dispersion

relation is of the form given in (7.40), then the phase velocity is the same for all

frequencies (wave lengths), i.e., this is the dispersion free case.

In addition, the so-called group velocity is also of great importance. It is

defined by

(7.43)

For the specific case of the dispersion relation as of (7.40), both velocities coincide,

both are equal to c:

(7.44)

In later sections, we will find dispersion relations for which this is not true. The

significant quantity in these cases is not the phase velocity, but the group velocity,

which describes the transmission of signals or the energy transfer. It relates to a

group of waves (a wave packet), which is composed of individual waves having

different wave lengths. In the case of (7.40), all individual waves travel with the

same phase velocity . Under these circumstances, the wave packet

ωtkz– ϕ+()cos ϕ kz

----

t–





–cos ϕ kz ct–()–[]cos==

zct–()

ω ck =

ω k

ωωk() =

ω k()tkz– ϕ+

ω k()

-----------

t+=

ω k()tkz

– ω k()t– ϕ+ ϕ kz

–=

ω k()

-----------

ω k() =

---- c

dω

=== =