Lehner G. Electromagnetic Field Theory for Engineers and Physicists

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574 Appendices

(A.1.36)

we find the Yukawa potential

(A.1.37)

from which we started. For an arbitrary distribution of volume charges we

find

(A.1.38)

Compare this to the classical result (2.20)

(A.1.39)

which agrees with the Yukawa potential for the special case of .

All this has peculiar ramifications for field theory, which can be demonstrated

via several simple examples. Furthermore, comparison of these theoretical

consequences with experimental results allows to gain further insight into the

nature of light quanta.

A.1.2 Examples

A.1.2.1 Uniformly Charged Spherical Surface

Consider a simple electrostatic problem, the field of a uniformly charged spherical

surface. The solution shall be just given, not derived. That this is the correct

solution can easily be verified. The fields are purely radial (Fig. A.1.2) and the

electric field is

(A.1.40)

(A.1.41)

and the potentials are

ρ Qδ r()=

4πε

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

κr–

ρ r()

ρ r'()

4πε

rr'–

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

κ rr'––

τ'd

∫

ρ r'()

4πε

rr'–

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

τ'd

∫

κ 0=

Fig. A.1.2

κr

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

κr

–

κr κr()cosh κr()sinh–

κr()

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

κr

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

κr–

κr

()sinh

1 κr+

κr()

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

A.1 Electromagnetic Field Theory and Photon Rest Mass 575

(A.1.42)

. (A.1.43)

What is extraordinary here, is that is not constant and therefore .

Essentially, this is evident: It is a unique characteristic of the Coulomb field, i.e., of

the quadratic distance relation, that there are no forces inside of a uniformly

charged spherical surface (this is also true for gravity). Any other field law does not

result in vanishing forces. Fig. A.1.3 makes this obvious. Search for a field inside

of a uniformly charged spherical shell is therefore one of the oldest methods

to verify Coulomb’s law. From today’s perspective, we can interpret these

experiments as an effort to measure the rest mass of light quanta.

Now, let’s consider another electrostatic problem, the field and the

capacitance of an ideal plate capacitor.

A.1.2.2 The Plane Capacitor and its Capacitance

According to Fig. A.1.4, we distinguish 5 regions, indicated by the indices 1

through 5. The regions outside the capacitor are 1 and 5. The conducting plates are

in the regions 2 and 4. The region 3 is inside, between the plates. The potentials are

(A.1.44)

(A.1.45)

(A.1.46)

(A.1.47)

, (A.1.48)

-----------

κr

–

κr()sinh

κr

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

⋅=

-----------

κr–

κr

()sinh

κr

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

⋅=

0≠

Fig. A.1.3

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

-----------

∼∼∼

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

-----------

∼∼∼

0≠

---

κ zb+()

–=

---–=

---

κz[]sinh

κa[]sinh

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

---

κ zb–()–

576 Appendices

where V is the voltage on the capacitor. One can easily verify that (A.1.11) and all

required boundary conditions are satisfied. The result is peculiar in several

respects.

a) and are location dependent, i.e., there are non-vanishing electric fields

in regions 1 and 5.

b) There is no field inside the plates and the potential is constant. This is chiefly

the reason for the need of volume charges, which result from (A.1.25):

(A.1.49)

c) The gradient of the potential has discontinuities at and [, i.e.,

the electric field is discontinuous, which means that there are surface charges

on both surfaces, the inner and outer surface of the plate. They are:

(A.1.50)

. (A.1.51)

From this, we find the capacitance to be

(A.1.52)

where

(A.1.53)

is the classical capacitance and A the area of the capacitor plates. For ,

and therefore . One obtains the same result

when calculating the total energy in all 5 regions according to eq. (A.1.29) and then

adding

Fig. A.1.4

12 3 4 5

z=-b z=+a

z=+b

z=-a

d=2a

p = b-a

z=0

ρρ

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

±== =

za±= zb±=

a±

κV

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

κd

------coth±=

b±

κV

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

±=

----

κV

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

κV

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

κd

------coth d

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

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

κd

------

1 κd

κd

------coth++=

---------=

κ 0→

κd 2⁄coth 1 κd 2⁄()⁄→ CC

→

A.1 Electromagnetic Field Theory and Photon Rest Mass 577

(A.1.54)

because of symmetry

(A.1.55)

and

(A.1.56)

This simple example makes it obvious, that one must part with the familiar

pictures.

A.1.2.3 The Ideal Electric Dipole

As a further example, consider the field of an electric point dipole, located at the

origin and pointing in the positive z-direction. Then

(A.1.57)

and the related field is

(A.1.58)

, (A.1.59)

. (A.1.60)

Of course, yields the classical result (Sect. 2.5, eq. (2.60)):

(A.1.61)

A.1.2.4 The Ideal Magnetic Dipole

Consider a magnetic dipole m at the origin and oriented in the positive z-direction,

then we get for the vector potential (in spherical coordinates)

(A.1.62)

where

(A.1.63)

and for the magnetic field

(A.1.64)

---

++++ 2W

++==

p θcos

4πε

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

1 κr+()e

κr–

p θcos

4πε

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

22κr κ

++()e

κr–

p θsin

4πε

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

1 κr+()e

κr–

κ 0=

p θcos

4πε

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

A 00A

,,〈〉=

m θsin

4πr

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

1 κr+()e

κr–

m θcos

4πr

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

22κr+()e

κr–

578 Appendices

(A.1.65)

. (A.1.66)

In the classical situation ( ), it is permissible to calculate the magnetic dipole

field outside of the origin from the gradient of a scalar magnetic potential,

eq. (5.55)

(A.1.67)

It has the same form as the classical potential of the electric dipole (A.1.61). From

this results a magnetic dipole field that has the same form as the electric dipole

field. Formally, a distinction between them is not possible. Not so for . Both

dipole fields, (A.1.58) through (A.1.60) and (A.1.64) through (A.1.66) exhibit

different forms. This is necessary, since because of (A.1.24), the magnetic dipole

field is not irrotational, even in the static case:

(A.1.68)

Therefore, to obtain it from a scalar potential is impossible. Eq. (A.1.68) applies to

all magnetostatic fields in current free regions.

The magnetic dipole field can be expressed as superposition

(A.1.69)

where

(A.1.70)

, (A.1.71)

(A.1.72)

and

(A.1.73)

, (A.1.74)

. (A.1.75)

The part of the field at the surface of a sphere with a fixed radius r exhibits the

characteristics of a classical dipole field, where the dipole moment appears

changed by a factor . The additional field has the same

magnitude everywhere on the surface of the sphere but has only a component

parallel to the axis.

m θsin

4πr

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

1 κr κ

++()e

κr–

κ 0=

m θcos

4πµ

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

κ 0≠

H∇×

------

A–=

4π

------

2 θcos

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

⋅ 1 κr

-----------++





κr–

1θ

4π

------

θsin

-----------

1 κr

-----------++





κr–

⋅=

1ϕ

4π

------–

θcos

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

---

κr–

⋅⋅=

2θ

4π

------

θsin

-----------

---

κr–

⋅⋅=

2ϕ

1 κr κ

3⁄++()e

κr–

A.1 Electromagnetic Field Theory and Photon Rest Mass 579

(A.1.76)

This fact is one to which we will return later.

A.1.2.5 Plane Waves

Interesting are also questions related to wave propagation. Here also, there are

significant discrepancies to the classical theory. We restrict ourselves to plane

waves in the infinite homogeneous space without currents or charges. For this case,

(A.1.33) and (A.1.34) read

(A.1.77)

. (A.1.78)

For plane waves travelling in the positive z-direction we have

(A.1.79)

. (A.1.80)

Substituting these in the wave equations (A.1.77) and (A.1.78) yields the

dispersion relation

(A.1.81)

From a strictly formal point of view, it has the same form as e.g. that of a plasma

wave

(A.1.82)

However, the reason is of an entirely different nature. Because of the Lorentz

gauge (A.1.21) the potential is

(A.1.83)

and one obtains the following fields:

(A.1.84)

, (A.1.85)

4πr

------------–

---

κr–

⋅=

∇

-----

∂

∂t

---------

– κ

ϕ–0=

∇

-----

∂

∂t

----------

– κ

A–0=

ϕϕ

i ωtkz–()

ik–()

-----

iω()

– κ

–0=

------ k

-------

,+= ω

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





ωkA

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

i ωtkz–()

+ikA

i ωtkz–()

ikA

i ωtkz–()

–=

580 Appendices

(A.1.86)

and

(A.1.87)

, (A.1.88)

. (A.1.89)

This is a remarkable result. We obtain three (instead of the classically two)

independent solutions.

a) If only : We obtain one linearly polarized TEM wave (with and

) as in the classical theory.

b) If only : We obtain a second linearly polarized TEM wave (with

and ) as in the classical theory.

c) If only : There is only an electric field in propagation direction but no

magnetic field at all. This is a longitudinal wave, which does not exist in this

form within the classical theory. The classical theory allows for longitudinal

waves only if there are volume charges. Classically we have

and with we obtain

For a plane wave

and also

i.e., k and D are perpendicular to each other. Likewise, because of ,

for the magnetic field of a plane wave we have

However, because of (A.1.25) this is no longer the case if . Now, for

we have

which allows longitudinal waves to exist.

Furthermore, the dispersion relation (A.1.81) bears remarkable consequences.

The phase velocity becomes

(A.1.90)

ikA

i ωtkz–()

–=

ikA

i ωtkz–()

–=

ωκ

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

i ωtkz–()

–=

0≠ E

0≠

D∇• ρ=

ρ 0=

D∇• 0=

i ωt kr•–()

D∇• ikD•–0==

B∇• 0=

kB• 0 .=

κ 0≠

ρ 0=

D∇• ε

ϕ–=

----

------ κ

–

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

-----------–

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

c≥== =

A.1 Electromagnetic Field Theory and Photon Rest Mass 581

Conversely, the group velocity is

(A.1.91)

Furthermore

(A.1.92)

Contrary to the classical case, we now have dispersion, even for plane waves in

lossless, uniform media (e.g., even in a vacuum), and furthermore, it is no longer

. Another important point is that the frequency may no longer be

arbitrarily small. We obtain the lowest possible frequency from the limit

(cutoff frequency)

(A.1.93)

to which we will return. The relations are illustrated in (Fig. A.1.5). After

multiplication by and using (A.1.10), the dispersion relation (A.1.81) can be

written in the form

(A.1.94)

Now, the energy of a light quantum is

(A.1.95)

and its momentum is

(A.1.96)

thus, we obtain

(A.1.97)

which is nothing else than the relativistic energy of a particle with a momentum of

p, which was our starting point (eq. (A.1.7).

dω 1

ωd

------- c 1

-----------– c≤===

⋅ c

k 0→

cκ

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

Fig. A.1.5

---------=

Dispersion of a plane wave

in vacuum

κc

-------1=





-------

-----

ωκc()⁄

1234

c⁄

+()=

W Ñω=

p Ñk=

582 Appendices

The wave equation is basically nothing else than the energy theorem, which is

the reason why the dispersion relation stemming from the wave equation leads

back to the energy theorem. This is also true in the classical case, . In this

case we have

We do not intend to increase the number of examples any further. It has been

shown that on the basis of the current theory, many familiar concepts from the

classical field theory are no longer valid. It remains the question, what the actual

rest mass of light quanta really is and whether or not we have to give up Maxwell’s

equations in favor of the Proca equations. We have to start by stating that based on

our current knowledge, we can not definitely answer this question. However, all

measurements taken so far, and all interpretations of the known electromagnetic

phenomena do not suggest that the light quanta’s rest mass is anything but

zero. Every arbitrarily accurate measurement is limited by its own precision in its

ability to make statements on upper limits of the rest mass . We will conclude

by elaborating some more on this issue.

A.1.3 Measurements and Conclusions

A.1.3.1 Magnetic Fields of Earth and Jupiter

Known from measurements of the Earth’s magnetic field (including satellite data)

is that, if at all, this field deviates only very little from that of a classical dipole

field. It is safe to say, that at the equator, the additional field of given by

(A.1.76) is much smaller than the field in (A.1.70) through (A.1.72), at least by

a factor of . This means that

Therefore,

and

κ 0=

ω ck=

Ñω cÑk=

Wcp=

410

3–

⋅

4πr

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

2κ

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

κr–

⋅ 410

3–

⋅ B

---=





≤=

4 10

3–

4πr

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

⋅ e

κr–

.≈

---

410

3–

⋅≤

---------

610

3–

⋅

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

≤=

A.1 Electromagnetic Field Theory and Photon Rest Mass 583

where r is the earth’s radius and with the values

this gives

(A.1.98)

that is, should the mass differ from zero, it may only be as large as this rather

tiny value.

An even smaller limit can be found analyzing satellite data from the magnetic

field of Jupiter [32], namely

(A.1.99)

A.1.3.2 Schumann-Resonances

According to (A.1.93), the frequency of electromagnetic, plane waves may not

become arbitrarily small but its lower limit depends on . Of course, low

frequency goes with long wave length, which prohibits laboratory experiments to

answer this question.

However, the earth with the lower limit of the ionosphere constitutes a rather

large resonant cavity whose resonant frequencies are know as Schumann-

resonances. The lowest resonance is at about 8 Hz. Suppose (A.1.93) were

applicable unchanged to resonant cavities (which is not really precise because the

dispersion relation for a resonant cavity depends on parameters describing its

geometric shape), then we obtain.

and

(A.1.100)

A more accurate calculation, taking into account the mentioned geometry factor of

the system [33] yields a slightly larger upper limit:

(A.1.101)

A summary of various values and methods to determine this limit is listed in

Fig. A.1.6, which is based on a similar figure by Goldhaber and Nieto [32].

610

3–

⋅ Ñ

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

≤

2π

------

2π

------

6.6 10

34–

Js⋅==

r 6.4 10

m⋅=

c 310

----

⋅=

4.2 10

51–

kg⋅≤

810

52–

kg⋅≤

cκ

------------== 2π 8⋅≤

------

610

50–

kg⋅≈≤

410

49–

kg⋅≤