Lehner G. Electromagnetic Field Theory for Engineers and Physicists

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614 Appendices

(A.5.31)

If the boundary coincides with an equipotential surface, then the dipole layer is

uniform and there are no vortices, i.e., the fields must vanish. We will illustrate

this. Consider the very simple field depicted in Fig. A.5.3, which vanishes both,

inside and outside the sphere. The potential shall be constant on the surface of the

sphere:

(A.5.32)

This corresponds to the case when in (A.5.14) only the third term is non-vanishing.

Then one gets for the potential

Using (A.5.22) and because of , this results in

(A.5.33)

This again, confirms one of our previous results, namely that of Sect. 2.5.3, eqs.

(2.72) and (2.73) in particular. The potentials are constant and the fields vanish,

just like we had assumed. Nevertheless, inside the dipole layer, there exists an

infinitely strong field, which creates exactly the potential difference C.

τε

φ r

()–=

Fig. A.5.3

W 0=

φ r

() C=

4π

------

∂

∂r

-------

–

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

∫

–=

4π

------–

∂

∂r

--------

----















4πr

⋅⋅=

-----–==











A.5 The Helmholtz Theorem 615

A.5.2.2 Point Charge inside a Conducting Hollow Sphere

Here, we will revisit the problem of a sphere which we have repeatedly discussed

and solved, in Sect. 2.6.1 via the method of images, and in Sect. 3.8.2.3 by the

separation of variables. The sphere shall have a radius of . The potential on the

surface shall be and the charge shall be on the z-axis at . Then by

(3.336)

(A.5.34)

and the radial field at the sphere’s surface is by (3.340)

(A.5.35)

The surface of the sphere constitutes an equipotential surface, that is, there are no

tangential components (vortices).

The Helmholtz theorem under these conditions gives

(A.5.36)

The volume integral results with

(A.5.37)

in the first part of the potential obtained in eq. (A.5.34). On the other hand, the

surface integral, together with (A.5.35) provides the other part of that potential,

which can be shown by means of eq. (A.5.22). This second part is nothing else than

the potential of the image charge. However, in the context of this reflection, notice

that this does not constitute a method to solve the problem. After all, we had to use

the initially unknown field at the surface. Rather, the purpose here was to illustrate

the content of the Helmholtz theorem by means of an example.

It shall also be noted that only compatible quantities may be substituted into

Helmholtz theorem. In conjunction with the point charge Q, it would be possible to

consider fields different from the one given in (A.5.35), which would constitute a

different problem. In any case, the field prescribed at the surface has to possess a

flux which fits to the totality of all sources. Consider a field with radial components

at the surface of the following form:

(A.5.38)

then the total flux created by this field at the surface is

(A.5.39)

ϕ 0= zr

4πε

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

----





----





n 1+











2n 1+

---------------– P

θcos()

n 0=

∞

∑

4πε

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

2n 1+[]

n 2+

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

θcos()

n 0=

∞

∑

∇ Er'()•τ'd

4π rr'–

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

∫

r '()da'

4π rr'–

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

∫

–=

∇ Er'()•

-----

δ rr'–()=

θcos()

n 0=

∞

∑

r'()da'

∫

4πr

616 Appendices

If the total charge inside the volume is Q, then it has to be

(A.5.40)

which determines the first term of the series expansion (A.5.38) and agrees with

(A.5.35).

A.6 Maxwell’s Equations and Relativity

A.6.1 Galilean and Lorentz Transformation

Classical physics as coined by Newton, is based on an understanding which

appears self-evident to us, namely that time and space are absolute realities. The

result of this understanding is the supposition that the laws of physics are invariant

under the so-called Galilean Transformation. Consider a physical process in two

different reference frames, we shall call one Σ and the other Σ’, where Σ’ moves

with a constant velocity relative to the rest frame Σ. Both are inertial systems,

i.e., there shall be no inertial forces present. We will not discuss the important

implications that come along with this restriction. The relation between the

location vector r and the time t in system Σ on one hand, and the location vector r’

and the time t’ in system Σ’ is as follows

(A.6.1)

The coordinate axes of the two systems shall be parallel to each other and they

shall coincide for . The two equations (A.6.1) constitute the so-called

Galilean Transformation. One of its consequences is the familiar addition of

velocities. If a point particle moves with the velocity in system Σ, then its

velocity in the other system (Σ’) is and thus

(A.6.2)

We had assumed, and this is important here, that the velocity is constant. The

acceleration is therefore the same in both reference frames

(A.6.3)

Consequently, if there is a force F, then the equation of the motion is the same in

both frames

(A.6.4)

having assumed that the mass is the same in both reference frames (which will not

be the case once we consider relativistic effects). Though simplified and

abbreviated, this is precisely what is meant when saying, that the laws of classical

physics are invariant under the Galilean Transformation.

After having theoretically derived the speed of the propagation of electromagnetic

waves in vacuum and identified it as the speed of light in vacuum,

4πr

-----=

r ' rvt–=

t ' t=







t 0=

v–=

v+=

··

F mr

··

mr'

··

A.6 Maxwell’s Equations and Relativity 617

(A.6.5)

and then had also proven experimentally that waves really existed (Heinrich Hertz,

1887/88), the question became inevitable, for which reference frame this statement

applied, i.e., in which reference frame does light actually propagate with that

speed. From a classical perspective, there is only one reference frame with this

property. The speed of light according to eq. (A.6.2) is different for all other

reference frames. Initially, it seemed quite natural, yes downright self-evident, to

suppose that this unique reference frame (where the light had the given velocity)

was the surrounding absolute space in which the fixed stars were at rest (not the

earth, however). (From our current knowledge of space, with its vastness and the

incredible dynamic of the behavior of these fixed stars within their spiral nebulas

etc., this was a very naive conception.) In order for waves to propagate in this

absolute space, it was assumed to be filled with a suitable medium, which was

called ether. This ether was to play the same role for light waves as, for example,

gas plays for the propagation of sound waves. Since the earth moves in the ether,

the speed of light as seen from the earth should be different, exactly by its speed

through the ether, which means that eq. (A.6.2) should be applicable. By means of

this equation, it should be possible to determine the speed of the earth within the

ether (and thereby its speed relative to the absolute space). These chains of

reasoning triggered an intensive discussion and hectic experimental activity. Many

experiments were performed with the intention to confirm the just described

understanding (the famous Michelson interference experiment was one of them),

but all failed. The final result shall be presented now without going into further

details of these discussions. The result, which on one hand, is simple, but on the

other, vigorously objects our immediate conception, was first formulated by

Einstein (1905) in this form. It says that the speed of light in vacuum has the same

value in any uniformly moving reference frame (in all inertial systems), regardless

of their speed v, relative to each other. This forms the basis for the theory of special

relativity (not to confuse with the later formulated theory on general relativity,

which is not subject of this discussion). Therefore, Maxwell’s equations are not

Galilei invariant.

The theory of relativity that emerged from this understanding was mandated

by the experimental results and inevitable. It was the compulsory result of the

experimental experience, i.e., it would have emerged, even without Einstein.

Nevertheless, it rightly is connected to his name because he was the first to face the

necessary and inevitable consequences, which required to radically part with our

conception, and he formulated the results of the research of his time, and

furthermore, because he has done this in an ingenious and clear manner.

The theory of relativity has caused discussions, even beyond the realm of

Physics. Many of these discussions were more or less useful, frequently even

bizarre and senseless, sometimes initiated by people who did not even understand

it. Many of the debates, then and now, are distracted by the term relativity, which in

---------------=

618 Appendices

a naive way is taken out of context and the theory of relativity is rejected because

of some alleged philosophical reasons by which it is not permissible that

everything is relative. On the other hand, people misuse the theory and claim that

certain statements are relative, i.e., not absolute, even though they are entirely

unrelated to the theory. Feynman has commented on this in an ironic way in his

famous textbook [37, Section 16-1, “Relativity and the Philosophers”]. On the

other hand, let’s note for the record, that Einstein’s theory is ultimately a very clear,

even simple theory, which replaced the theory that preceded it historically (this

preceding theory could be called Galilean theory of relativity). Furthermore, it

shall be noted that Einstein’s theory, by no means contradicts its predecessor but

generalizes it because it includes it as a limit.

The fundamental statement of the theory of relativity is that the quantity

has the same value in every inertial frame, or that

(A.6.6)

is invariant. The mathematical transformation which ensures this, is the so-called

Lorentz transformation. In contrast to the Galilean Transformation, it also includes

the time. Time is now also dependent on the reference frame. To derive this

transformation is not difficult. It only requires the orthogonal transformation of the

coordinates of an n-dimensional space (here four-dimensional), which is well

known in mathematics.

To illustrate the content of eq. (A.6.6), consider an arbitrary point in the

laboratory frame Σ, from which a spherical wave emerges at time . In the

reference frame Σ’, this wave emerges from the point at the time . By

suitable translation in space and time, it is always possible to achieve a situation

where , , , and . This does not impose a limit on its

generality.

A.6.2 Lorentz Transformation as an Orthogonal Transformation

Consider an n-dimensional Euclidean space where two Cartesian coordinate

systems coexist, but are rotated against each other. A particular location is

represented in one system by and in the other by , where

(A.6.7)

This transformation must possess the property

----------------------------=

–++ x

i 1=

∑

xyzict,,,〈〉x

,,,〈〉x

〈〉==











' t' t

0= r

'0= t

0= t

'0=

〈〉 x

'〈〉

' L

k 1=

∑

= ik, 1 …n,=()

A.6 Maxwell’s Equations and Relativity 619

(A.6.8)

that is, the distance must be invariant. This is the case if the matrix operator

(A.6.9)

is orthogonal. Its properties can easily be derived

It is therefore

(A.6.10)

Of course we have

i.e., unity and therefore

(A.6.11)

Comparing eqs. (A.6.10) and (A.6.11) yields

(A.6.12)

This is exactly what represents the property of the orthogonal transformation

operator, namely that its inverse operator is equal to its transposed operator

(where ). Obviously, one also has

(A.6.13)

The two eqs. (A.6.10) and (A.6.13) express that both, the column vectors as well as

the row vectors of an orthogonal operator are unit vectors which are orthogonal to

each other.

With this mathematical preparation, we now turn our attention to the Lorentz

transformation. For that purpose, as before in eq. (A.6.6), we will use the four-

dimensional vector

(A.6.14)

where now, the four-dimensional distance has to be invariant. Formally, this

is the same problem as that of a four-dimensional Euclidean space. The difference

is merely that one of the coordinates ( ) and its corresponding matrix

i 1=

∑

i 1=

∑

invariant==

()=

i 1=

∑

k 1=

∑







l 1=

∑







i 1=

∑

k 1=

∑

l 1=

∑

k 1=

∑

===

i 1=

∑

1–

L 1

1–

i 1=

∑

1–

,= L

1–

i 1=

∑

i 1=

∑

xyzict,,,〈〉x

,,,〈〉x

〈〉==

i 1=

∑

ict=

620 Appendices

elements are now complex valued. This is therefore also called pseudo-Euclidean

four-dimensional space-time, which is also called Minkowski space.

Initially, we shall assume that the relative velocity v has only an

component :

(A.6.15)

This suggests to suppose that and remain unchanged in this case,

while only and will be changed by the transformation. This

assumption is not necessary but simplifies matters in the following discussion. The

transformation may then be written in the following form:

(A.6.16)

When , then the origin of the moving reference frame Σ’ is at ,

which allows one to write

Thus

(A.6.17)

Because of the orthogonality of L, one has

(A.6.18)

(A.6.19)

. (A.6.20)

Using eqs. (A.6.17) and (A.6.18) gives

(A.6.21)

Furthermore, from this and eq. (A.6.20) follows

and by (A.6.19) this yields

v v

00,,〈〉=

= zx

= ict x

'=L

'= x

'=L











t= x'0=

' fv

()x

t–()fv

()x

()

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

ict– fv

()x

()

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

–== =

-----

– i

-----

+1=

+0=

-----–

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

= L

-----

-----–

-------------------=,

--------

–

-------

–==

A.6 Maxwell’s Equations and Relativity 621

(A.6.22)

Introducing the frequently used abbreviation

(A.6.23)

finally, the Lorentz transformation emerges

(A.6.24)

or expressed with and

(A.6.25)

One can easily verify that all requirements are met and, in particular, eq. (A.6.6) is

invariant.

The case of an arbitrary velocity can be reduced to the just

derived result. For this purpose, decompose into components parallel and

perpendicular to v,

and obtain

-----–

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

= L

-----

-----–

-------------------–=,

----==

iβx

1 β

–

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

' x

iβx

– x

1 β

–

---------------------------=

tt'

t–

1 β

–

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

' x

t '

-----

– t+

1 β

–

-----------------------=

v v

,,〈〉=

rvt–()

rvt–()v•[]

-----

rvt–()

⊥

rvt–()rvt–()v•[]

-----

–=

622 Appendices

(A.6.26)

Of course, letting gives the result of our previous transformation

(A.6.25). Also easy to verify is that (A.6.26), indeed, is invariant as required, i.e.

The L operator (when using ) is obtained from (A.6.26) and in the

following form:

(A.6.27)

Its inverse transformation is obtained simply by replacing v by -v. It becomes

immediately obvious that the so obtained inverse matrix is . Letting

results in the Galilean Transformation. This means that the theory of

relativity includes classical physics in this limit and thereby generalizes it.

We have, thereby, gained the Lorentz transformation, first in its simple form

of eqs. (A.6.24) or (A.6.25) for and then in its more complicated

form of eq. (A.6.27) for . Next, we want to consider some of its

fundamental consequences. For the most part, we will use the simpler form, which

is sufficient for many purposes.

A.6.3 Some Consequences of the Lorentz Transformation

A.6.3.1 Lorentz Contraction

Consider a rod, oriented parallel to the and the axis, respectively. Its length

in system Σ is . What is its length in system Σ’? The

task requires caution. One has to ensure that the coordinate points of both ends in

rvt–()

1 β

–

-------------------- rvt–()

⊥

+ r

rv•()v

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

1 β

–

-------------------1–

1 β

–

-------------------–+==

-----

rv•()– t+

1 β

–

----------------------------------=











v v

00,,〈〉=

– r

–=

β vc⁄=

-----

1 β

–

------------------ 1–+

----------

1 β

–

------------------ 1–

----------

1 β

–

------------------ 1–

c 1 β

–

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

----------

1 β

–

------------------ 1–1

-----

1 β

–

------------------ 1–+

----------

1 β

–

------------------ 1–

c 1 β

–

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

----------

1 β

–

------------------ 1–

----------

1 β

–

------------------ 1–1

-----

1 β

–

------------------ 1–+

c 1 β

–

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

–

c 1 β

–

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

–

c 1 β

–

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

–

c 1 β

–

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

1 β

–

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

1–

β 0→

v v

00,,〈〉=

v v

,,〈〉=

–= l' x

' x

'–=

A.6 Maxwell’s Equations and Relativity 623

Σ’ are determined at the same time t’. This needs to be recognized because

simultaneousness in Σ does not imply simultaneousness in Σ’. Therefore, one

requires that , and obtain with (A.6.25)

Furthermore

(A.6.28)

From the moving system Σ’, one “observes” the rod shortened by the factor

. This phenomenon is the so-called Lorentz-contraction. The word

observes in the previous sentence was used in quotation marks because caution is

of order when interpreting this result, in order to avoid misconception – which has

frequently occurred.

The Lorentz-contraction causes, for example, that a sphere in motion for an

observer at rest changes to a flattened ellipsoid. However, it does not mean that a

visually observed sphere is seen by an observer as a flattened ellipsoid, nor that a

photographic picture would show a flattened ellipsoid, as it was frequently stated in

the past. This mistake was cleared only relatively late by Penrose [38] and Terrell

[39]. Since the speed of light is finite, visual or photographic observation of an

object gives a distorted picture, because it shows parts with different distances to

the observer at different times. In case of a large distance to the object (or small

solid angle) the object appears in its natural shape and length, but rotated. If the

object is a sphere, then the observer sees a rotated sphere, but not an ellipsoid. If

the object is a rod, then the observer sees a rod with its natural length but rotated.

The angle is such that the rod’s projection onto the direction of the motion

represents the Lorentz-contraction. We pass on a detailed discussion here. Those

details are found in the mentioned publications [38, 39]. It shall be emphasized that

there is no doubt on the reality of the Lorentz contraction. This is a real and

experimentally verified effect. The described facts simply mean that a visual

observation is not a suitable method to experimentally verify the Lorentz

contraction. The fact that a visually observed object, precisely because of the

Lorentz contraction remains visible in its natural shape is very remarkable. The

reason is that the distortion is compensated for by the different optical paths. In this

sense, the theory of relativity restores the “vividness” of objects in motion, which

were otherwise lost due to the finite speed of light.

' t

-----

– t

-----

–= t

–

-----

–()=,

l ' x

' x

'–

– v

–()–

1 β

–

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

–

-----

–()–

1 β

–

------------------------------------------------------------== =

l ' x

–()1 β

– l 1 β

– ==

1 β

–