An Elementary Electron Model

for Electron-Electron Scattering

Based On Static Magnetic Field

Energy

ARL-TR-2332 February 2001

Harry J. Auvermann 

Approved for public release; distribution unlimited.
The findings in this report are not to be construed as an

official Department of the Army position unless so

designated by other authorized documents.

Citation of manufacturer’s or trade names does not

constitute an official endorsement or approval of the use

thereof.

Destroy this report when it is no longer needed. Do not

return it to the originator.
ARL-TR-2332 February 2001

Army Research Laboratory

Adelphi, MD 20783-1197

An Elementary Electron Model

for Electron-Electron Scattering

Based On Static Magnetic Field

Energy

Harry J. Auvermann 

Computational and Information Sciences Directorate

Approved for public release; distribution unlimited.
Abstract

Electron motion paths that exhibit zero radiation in a Maxwells equation

solution have been reported. Such paths, require a radiationless model of

the electron itself, such as the charged hollow sphere. When the electricfield energy of this model is set equal to the rest mass energy of the electron, the radius of the resulting sphere is called the ”classical electron radius.” Analysis reveals that the static magnetic-field energy of the classical

model is many times the electron rest mass energy when the sphere is given

an angular velocity large enough to exhibit the electron magnetic moment.

The necessary angular velocity produces a peripheral velocity many times

the speed of light. A classical model with a peripheral velocity near the

speed of light is a loop whose radius is the Compton wavelength divided

by 2π; such a loop has a very small dimension perpendicular to the loop

plane. Experiments reveal point-like electron scattering properties down to

at least 1/100 of the classical radius. The small transverse dimensions of

the loop model indicate similar scattering results. Recently, a proposal was

submitted to investigate the scattering properties of interacting loops. Because of limited proposal length, derivation of loop model equations could

not be included. This report contains the details of the analysis.

ii
Contents

1 Introduction 1

2 Equations for Physical Constants 3

3 Magnetic-Field Energy of a Spherical Shell Electron Model 5

4 Properties and Configuration of Loop Electron Model 7

5 Loop Electron Model Related to Experimental Findings 9

6 Potential Energy Results for Two Interacting Charged Loops 10

7 Summary and Discussion of Results 13

References 15

Appendixes 17

A FY01 Director’s Research Initiative Proposal 17

References 21

B Magnetic Moment of a Spinning Charged Sphere 23

C Potential Interaction Energy of Two Charged Loops 25

D Properties of a Charged Hollow Torus 29

Distribution 35

Report Documentation Page 39

iii
Figures

1 Relative interaction potential energy for two electron models . . 12

C-1 Interaction diagram for two charged loops . . . . . . . . . . . . 26

D-1 Three-dimensional diagram of torus . . . . . . . . . . . . . . . . 29

D-2 Sectional diagram of torus through z-axis . . . . . . . . . . . . . 29

D-3 Patch dimensions in x-z plane . . . . . . . . . . . . . . . . . . . . 31

D-4 Patch dimensions in x-y plane . . . . . . . . . . . . . . . . . . . . 32

iv
1. Introduction

When the Bohr atom is introduced in a physics curriculum, the derivation

of the constants of motion for the electron proceeds by assuming this moving charge does not radiate as it traverses the circular or elliptical orbit,

even though an accelerating charge is known to radiate energy according

to Maxwell’s equations. I have always been intrigued by the problem of

finding radiationless charge-current distributions but did no serious investigation until I discovered an article published in 1964 [1]. The following

quote is a transcription of the first two paragraphs of the introduction from

this article:∗

“It still seems a fairly common belief that there exist no nontrivial chargecurrent distributions which do not radiate, according to classical electromagnetic theory retarded potential solutions. However, early in this century Sommerfeld,1 Herglotz,2 and Hertz3 considered extended electron

models, and established the existence of radiationless self-oscillations. In

1933, Schott4 showed that a uniformly charged spherical shell will not radiate while in orbital motion with period T, provided the shell radius is an

integral multiple of cT/2; the orbit need not be circular nor even planar. In

1948, Bohm and Weinstein5 found several other rigid spherically symmetric distributions that can oscillate linearly without radiating.

In this paper we derive a simple exact criterion for absence of radiation,

and apply it to moving rigid extended charge distributions.6 We find that

there are many such distributions, some of which may “spin,” and others

which need not be spherically symmetric. The allowed types of such distributions are severely restricted by the condition of no radiation; further,

it must be true that the finite radius of a rigid volume distribution be an

integer multiple of cT, where T is the period of orbital motion. This last

restriction implies that the perimeter of the orbit is less than the extent of

the distribution.4”

Encouraged by these published results, I began an effort to find a radiationless alternative to the “Solar System-like” orbits of the Bohr atom during

the year (1968–1969) that I held a temporary appointment in the Physics

*The following are the references cited in Goedecke‘s article. 1A. Sommerfeld, Nachr. Akad. Wiss. Goettingen, Math.-Physik, Kl. IIa Math.-Physik.

Chem. Abt. 1904, 99 and 363; 1905, 201. 2G. Herglotz, Nachr. Akad. Wiss. Goettingen, Math.-Physik, Kl. IIa Math.-Physik. Chem.

Abt. 1903, 357; Math. Ann. 65, 87 (1908). 3P. Hertz, Math. Ann. 65, 1 (1908). 4G. A. Schott, Phil. Mag. Suppl. 7 15, 752 (1933). 5D. Bohm and M. Weinstein, Phys. Rev. 74, 1789 (1948). 6In Bull. Am. Phys. Soc. 9, 148 (1964), “which I [Goedecke] received while writing this

paper, there appears an abstract by S. M. Prastein and T. Erber which implies that some of

the content of this paper has been worked out independently by these authors.”

1
Department of The University of Texas at Arlington. One can quickly conclude that a necessary part of producing a radiationless model of the hydrogen atom is a radiationless model of the electron. I therefore began a study

of the properties of the electron with the simple derivation of the classical radius. The semiclassical derivation for this quantity equates the static

electric-field energy of a hollow sphere, bearing the electron charge to the

rest mass energy. As a part of my education, I examined the static magneticfield energy of a spinning sphere whose radius is the classical radius, expecting that the magnetic-field energy would be much less than the classical electric-field energy. Using my own approximation for the magneticfield energy, I was quite surprised to find that the magnetic-field energy

was some 900 times the electric-field energy. Additionally, the peripheral

velocity of the sphere had to be many times the speed of light to achieve

the required magnetic moment. Backing off from the classical radius requirement, I sought the radius of a charged loop whose peripheral velocity

was the speed of light but still had the correct magnetic moment and field

energy. My calculation, although approximate, resulted in the model loop

radius being what I call the “Compton radius,” the latter being the radius

of a circle whose circumference is the Compton wavelength.

At this juncture, I made one further calculation. I determined approximately

that the required “charged wire” radius to give an electric-field energy for a

Compton radius-sized loop to be equal to the electron rest mass was a number so small that it defied imagination. Other pressures caused me to abandon the work on this project, but I have continued to think about it. One

significant event in this connection occurred perhaps in the early 90s. This

event was the report that a colliding beam experiment, I believe the European Organization for Nuclear Research (CERN), showed that the cross

section for electron-electron scattering was spherical to at least a factor of

100 less than the classical size. I have not relocated the reference to this report to verify this result, but I have been assured by two colleagues that this

result is currently universally accepted in the particle physics community.

Because of my “charged wire” radius calculation, I wondered when the experiment report came out if this might somehow indicate that I was on the

right track with my Compton radius loop model.

I have described the events up to when the notice of the current director’s research initiative (DRI) was circulated. Since I have been intrigued by

this problem of finding radiationless charge-current distributions, I derived

anew my previous results before submitting my proposal on 14 July 2000

to investigate the scattering properties of interacting current loops. A copy

of the proposal has been included as appendix A. This report contains the

derivation including an improved magnetic-field energy expression. Also

included in this report, but not in the proposal, is a classical calculation

of the potential energy necessary to bring two Compton radius loops to a

separation distance equal to a small fraction of the classical radius.

2
2. Equations for Physical Constants

A recent article [2] presents the latest information on the fundamental constants. Equations for physical constants in terms of fundamental constants

are presented here. The fine structure constant [2, p 448] is

α =



μ0c2

4π

 (2π)e2

hc 

=

(2π)e2

4πε0hc

, (1)

where e is the electronic charge, h is Planck’s constant, and c is the speed of

light in vacuum. The Bohr radius [2, p 448] is given as

a0 =



μ0c2

4π

−1 h2

(2π)2mee2



, (2)

where me is electron rest mass. The classical electron radius [2, p 449] is

r0 =



μ0c2

4π

 e2

mec2



. (3)

The Bohr magneton [2, p 448] is

μB =

 eh

4πme



. (4)

The Compton wavelength of the electron [2, p 449] is

λC =

 h

mec



. (5)

For this derivation, a quantity that I call the “Compton radius” is defined

as the radius of the circle whose circumference is the Compton wavelength,

rC =

 h

2πmec



. (6)

Note that these three radii are related through the fine structure constant

rC

ao

= r0

rC

= α . (7)

The energy represented by the electron rest mass is

we = mec2 . (8)

From the electromagnetic wave equation, the magnetic constant μ0 and

electric constant ε0 are related to c [2, p 448],

c2 = 1

μ0ε0

. (9)

3
The potential of a point of charge q at a distance r [3, p 312] is

φs = q

4πε0r . (10)

By multiplying equation (10) by the charge and setting the left-hand side

equal to the rest mass energy of the electron and then the charge on the

right-hand side equal to the electronic charge, one can define the resulting

radius to be the classical radius of the electron.

4
3. Magnetic-Field Energy of a Spherical Shell Electron Model

The next step is to spin the electron modeled as a charged shell and to

determine the magnetic moment of this configuration. An approximation

for the magnetic moment of a spinning charged sphere has been derived

in appendix B in terms of an equivalent current loop. Equation (11) is an

adaptation of equation (B-6) in the appendix, where the superscript s indicates that the variables are those of the sphere and vs is the peripheral

velocity of the sphere:

μs = 2reqsvs

3 . (11)

The free electron g-factor [2, p 449], or electron magnetic moment μe in Bohr

magnetons, is

gj

2 = μeμB

. (12)

Using this relation and setting the sphere magnetic moment equal to the

electron magnetic moment allow the rotational velocity to be isolated as

vs = 3gjμB

4ree . (13)

Substitution for the known quantities from section 2 produces the velocity

in terms of fundamental constants as

vs =

3

4

 gj

2

  c

α



. (14)

It is instructive to estimate the multiplier of c, which here gives the peripheral velocity of this model. The fine structure constant in the denominator

produces a factor of about 137 in the numerator, and the free electron gfactor is about 1. Thus, the peripheral velocity is upward of 100 times the

speed of light, a truly astonishing result. The sense of this calculation is

that a current loop with a radius equal to the classical electron radius must

have a velocity inconsistent with relativity theory to exhibit the measured

electron magnetic moment.

Another comparison may be made with this equivalent current loop model.

This comparison calculates the magnetic-field energy of the equivalent current loop. The general expressions for the inductance of a loop [3, p 326] and

for the energy stored in the inductance of a circuit [3, p 327] are

Lg = b



μ



ln 8b

a



− 2



+ μ

4



; Let μ = μ

Lg = Cg(ρg)μb ; Cg(ρg) = ln(dgρg)

ρg = b

a

; dg = 8 exp 

− 7

4



≥ 1.39

wcircuit = LcircuitI2

circuit

2 , (15)

5
where Lg is the inductance of a general loop, b is the radius of the loop, a

is the radius of the wire, μ is the medium permeability, and μ is the permeability of the wire. Also wcircuit is the circuit energy, Lcircuit is the circuit

inductance, and Icircuit is the circuit current. Assuming that the wire permeability is the same as that of the medium permits the simplification shown

in the second line of equation (15). Replacing the general loop dimensions

of equation (15) with the dimensions of the equivalent loop gives the inductance Leq

s of the loop equivalent to the spinning sphere

Leq

s = Cg(ρs)μ0r0 ; ρs = r0

rt

. (16)

The quantity Cg(ρs) involves the natural logarithm of the ratio of the classical electron radius to the radius rt, which is an unspecified radius of the

torus tube containing the charge and corresponds to the wire radius a. This

quantity is thus related to the assumed configuration of the loop and does

not change with the loop radius if the configuration remains the same. The

minimum value of the parameter ρs is one so that the minimum value of

this factor is ln(1.39) = 0.329.

The field energy expression of equation (15) may now be used wherein the

energy of the equivalent loop weq

s is substituted for wcircuit, the equivalent

inductance from equation (16) is substituted for Lcircuit, and the equivalent

current Ieq

s is substituted for Icircuit. With all relevant quantities brought

together, the magnetic energy of the spinning shell model of the electron

will be

weq

s = Leq

s (Ieqs )2

2 ; Leq

s = Cg(ρs)μ0r0; Ieqs =

 eνs

2πr0



=

Cg(ρs)μ0r0

2

  3gjec

8(2π)r0α

2

; νs = 3gjc

8α

(17)

=



9Cg(ρs)μ0(gj )2r0

128(2π)2

  ec

αr0

2

.

Rearranging and substituting give the result

weq

s =



9Cg(ρs)(gj )2μ0r0

128(2π)2

  ec

αr0

2

= Λ(ρs)

 μ0r0

4(2π)2

  ec

αr0

2

= Λ(ρs)

μ0e2c2

(4π)2α2r0



= Λ(ρs)

e2

(4π)2ε0α2r0



(18)

= Λ(ρs)

 we

4πα2



; Λ(ρs) = 

9CL

s (ρs)

8

 gj

2

2

.

Here, the magnetic field energy with the use of the spherical model (estimated in eq (18)) is very much more than the experimentally determined

rest mass energy of the electron. Aside from the dimensionless quantity

Λ(ρs), which is about 3/8 as a minimum, the numerical factor (4πα2)

−1 is

about 1494.

6
4. Properties and Configuration of Loop Electron Model

In this section, the formalism presented in section 3 is used to determine

the radius of a current ring whose energy is equal to the rest mass energy

of the electron under the condition that the magnetic moment of the ring

is equal to the magnetic moment of the electron. The radius of the ring as

determined will be identified with the symbol rcl

m with the other variables

used in the derivation identified with a similar subscript-superscript pair.

The eventual equation to be solved is

Lcl

m = Cclm(ρm)μ0rclm ; ρm = rcl

m

rm

wcl

m = Lcl

m(Iclm)2

2 ; wcl

m = mec2 (19)

where rm is the torus tube radius for the magnetic loop.

The magnetic moment equation is

μcl

m = μe = gjμB

2

μcl

m = AclmIclm ; Aclm = π(rclm)

2 ; Icl

m =

evcl

m

2πrcl

m



. (20)

From equations (19) and (20), the current is eliminated as in

mec2 = Cg(ρm)μ0rcl

m(Iclm)2

2 =



Cg(ρm)μ0rcl

m

2

  gjμB

2π(rcl

m)2

2

. (21)

The solution for the radius is obtained by rearranging equation (21) as

rcl

m =



Cg(ρm)μ0(gjμB)2

2(2π)2mec2

1/3

=



Cg(ρm)μ0(gj )2(eh)2

2(2π)2mec2(4πme)2

1/3

. (22)

One way to analyze the expression on the right in equation (22) is to extract

a factor equivalent to the Compton radius as

rcl

m =



Cg(ρm)μ0(gj )2(eh)2

2(2π)2mec2(4πme)2

1/3

=

 h

2πmec

 

Cg(ρm)(gj/2)2α

2π

1/3

. (23)

The magnetic radius is thus seen to be the Compton radius times a factor

that is not a function of the loop radius but rather is a function of the quantity ρm, which is the ratio of the loop radius to the torus radius. The latter is

an indicator of the extent of the loop dimension parallel to the loop axis. To

7
gain insight to the magnitude of this torus radius, determine the velocity of

the loop charge for the case of rcl

m = rC. This is obtained from the relation

μe = rcl

mevclm

2 ; vcl

m = 2μe

rcl

me

vcl

m = gjμB

rcl

me =

 gj

rcl

me

  eh

4πme



. (24)

With the substitution for the Compton radius, the result is

vcl

m =

 hgj

4πme

  1

rC



=

 hgj

4πme

 2πmec

h



= (gj/2)c . (25)

This shows that the resulting velocity of the loop charge is near the velocity

of light if the loop radius is the Compton radius. Assuming this calculation

is accurate enough, one may determine the loop configuration parameter

ρm from

Cg(ρm)α

2π

1/3

= 1

Ccl

m(ρm) = 2π

α = ln(dgρm) (26)

ρm = exp 2π

α

+

7

4 − ln(8)

∼= exp(2π137) ∼= e861 .

What is astonishing about this estimate is that the indicated torus tube radius is so incredibly minute.

8
5. Loop Electron Model Related to Experimental Findings

In section 4, I have shown that a current loop structure whose radius is the

Compton radius satisfies the magnetic moment and the rest mass energy

requirements necessary to model an electron. Furthermore, such a structure has the potential to reproduce the electron-electron scattering crosssection results reported in the literature. I make this statement because the

indicated dimension of the current loop model in the direction perpendicular to the loop plane is quite small. The scattering geometry for two colliding loops is so complex that just what happens as they approach each

other is not readily apparent for the general case with arbitrary orientation

of the two magnetic moments and arbitrary displacement of the loop center paths. However, for coaxial center paths with orientation such that the

paths are perpendicular to the planes of the two loops and for only classical electric-field interaction, a distance of closest approach equal to that

reported in the experiment can be achieved for the two loops. Thus, in the

backscatter direction, the scattering is appropriate within the framework

imposed by the provisos in the previous sentence.

From my previous approximate calculation of the “charged wire” radius

alluded to in the introduction of section 1, the torus radius is so small that

the electric-field energy necessary to compress the charge into a torus of

this size may rival the computed magnetic energy. This then indicates that

the distribution of the electron rest mass energy between the electric and

magnetic fields may be useful in suggesting further modifications of the

configuration of a classical model of the electron. Accounting for relativistic effects more completely will be necessary in further modifications of the

loop model also. It may be necessary to introduce periodic motions of the

loop to realize the radiationless restriction. One possible motion is the rotation of the loop about an axis inclined with respect to the magnetic moment.

Additionally, accounting for the spin angular momentum may require further modifications.

These thoughts are presented to emphasize that a current loop model of

the electron is just that, a model. Elaborations are expected to be necessary.

In this same spirit, it is not prudent to attempt these elaborations before

the more elementary configurations are investigated as to their scattering

properties. Thus, the objective of the DRI proposal was to investigate the

scattering properties of progressively more complex models in the geometry of the recorded experiment. The a priori expectation is that some of the

general features of the experimental findings will be present with the use of

the simplest model. On the other hand, a full relativistic treatment, including both electric and magnetic interactions, may be needed. As far as what

is known at present, no exposition of the theory for loop-loop scattering

exists.

9
6. Potential Energy Results for Two Interacting Charged Loops

This section compares the potential energy of the current loop electron

model in the interaction geometry described in appendix C with the potential energy of the spherical shell electron model. The potential energy of

the spherical shell model We

AB(p) is given as

We

AB(p) = −

 p

∞

dr q2

a

4πε0r2



=

q2

a

4πε0p



. (27)

In the equation, the radius of the shell rs is not specified. If rs <p<r0,

which is the condition imposed upon the shell model by the experimental

results, the field energy must be larger than the measured rest mass energy

of the electron. With this in mind, one can base the comparison upon the

shell model energy for a separation of the Bohr radius rB given by [2]

rB =

4πε0h2

(2π)2meq2

a



; α =

(2π)q2

a

4πε0hc

. (28)

Also shown in equation (28) is the expression for the fine structure constant

α [2]. The constants used in this derivation are shown in relation to the Bohr

radius as

Compton radius = rC = rcl

m = a = αrB

Classical radius = r0 = α2rB

Nearest approach distance = pn = α3rB

Nearest approach parameter = qn = α2/2 . (29)

The reference energy WR is defined in equation (30) by the insertion of

equation (28) into equation (27):

WR = me



(2π)q2

a

4πε0h

2

= mec2α2 . (30)

In equation (31), both equation (C-13) from appendix C for the interaction

potential energy for the loop model Wm

AB(p) and equation (27) are rewritten

in terms of equation (30):

Wm

AB(p) = 

2k1/2q2

aK(k)

(2π)4πε0a



= WR



2k1/2K(k)

(2π)α



We

AB(p) =  p

∞

dr q2

a

4πε0r2



= WR

 p

rB

−1

. (31)

10
I have defined several symbols in equation (32) to simplify notation for

presentation of the data:

Ξm( ̄p) = Wm

AB(p)

WR

=



2κ( ̄p)1/2K(κ( ̄p))

(2π)α



; κ( ̄p) =  1

1 + (2α)−2p ̄2



Ξe( ̄p) = We

AB(p)

WR

=  ̄p−1 ;  ̄p =

 p

rB



. (32)

One more step is taken to place the data in a proper perspective, such as

Wm(ξ)=Ξm( ̄p) = 

2 ̄κ1/2K( ̄κ)

(2π)α



;  ̄κ =

 1

1 + (2α)−2ξ−2 log(α)



We(ξ)=Ξe( ̄p) = ξ− log(α) ; ξ =  ̄p−1/ log(α) ; α = 10log(α) . (33)

A new variable ξ is introduced that represents the distance of nearest approach. This variable is designed to emphasize the sweep of the distances

being considered. With this choice of the abscissa variable, the relation between the two relative energy functions at the separation points of p =

rB, rC, r0, pn occurs at the variable values of ξ = 1, 10−1, 10−2, 10−3, respectively. Results calculated from the expressions in equation (33) are shown

in figure 1. The essence of these results is displayed on the left side of the

figure that corresponds to a minimum separation of pn from equation (29).

The graph shows that the energy required to bring the spherical model

electron to this distance of nearest separation is some 4 orders of magnitude greater than that required to bring the loop model of the electron to

the same distance of nearest separation. It is instructive to record some of

the absolute numbers associated with the interaction under investigation.

The fundamental constants [2] are

α = 0.00729735 = fine structure constant

me = 9.10939 × 10−31 kg = electron rest mass

c = 299792458 m/s−1 = speed of light

e = 1.60218 × 10−19 C = electronic charge

rB = 5.29177 × 10−11 m = Bohr radius . (34)

The value of the electronic charge is only needed to convert from Joules

to electron volts. Derived quantities and the calculated relative potential

energies are

pn = α3rB = 2.056350 × 10−17 m

WR = mec2α2 = 4.3597 × 10−18 J = 27.2114 electron volt

We(rB)=1.00 ; We(pn)=2.573380 × 106

Wm(rB)=9.999468 × 10−1 ; Wm(pn)=5.199462 × 102 . (35)

11
Figure 1. Relative

interaction potential

energy for two electron

models.

10–3 10–2 10–1 100 10–2

100

102

104

106

108

Position parameter (ξ)

Relative potential energy

Spherical model

Loop model

Absolute potential energies are

We

AB(rB)=4.3597 × 10−18 J = 27.2114 electron volt

Wm

AB(rB)=4.3594 × 10−18 J = 27.2100 electron volt

We

AB(pn)=1.1219 × 10−11 J = 70.0253 × 106 electron volt

Wm

AB(pn)=2.2668 × 10−15 J = 1.4148 × 104 electron volt . (36)

Of particular interest in the results shown in equation (36) is the order

of magnitude of the energy that must be supplied to bring two electrons

to within the separation distance of a fraction of the classical radius of

r0 = 2.8179 × 10−15 according to the two models. Whereas, the spherical model indicates 70,000,000 electron volts, the loop model shows some

14,000 electron volts.

12
7. Summary and Discussion of Results

The classical physics calculations of this report started with the spherical

shell model of the electron for which the classical radius is calculated by

equating the electric-field energy to the measured electron rest mass energy. Assuming this sphere was spinning fast enough to generate a magnetic moment equal to its measured value, I showed that the peripheral velocity required would be many times the speed of light. Furthermore, I also

showed that the magnetic-field energy of such a magnetic moment would

be some 1000 times the rest mass energy. The quantity then sought was the

radius of some object (probably a loop) whose peripheral velocity was near

the speed of light (equal to it in the derivation) and, at the same time, giving the correct magnetic moment and the correct magnetic-field energy. The

resulting object was a torus whose radius was proportional to the Compton wavelength divided by 2π—the latter quantity is being referred to as

the magnetic radius rcl

m. The proportionality factor was the cube root of the

natural logarithm of the torus radius divided by the toroidal tube radius

rm. Selecting a value for this proportionality factor implied a rudimentary

structure. If the value selected is unity, then the tube radius is uncommonly

small. If, on the other hand, a value is selected less than unity, while the

tube radius is increased, the loop radius is decreased from the magnetic radius, which in turn requires a peripheral velocity greater than the speed of

light. I calculated further results using the proportionality factor of unity.

At this point, I include modern experimental results in the calculations. It

is widely understood that high-energy electron-electron scattering experiments show that the scattering pattern is that of a point like entity down

to several orders of magnitude less than the classical radius. Since the indicated tube radius rm is so small, the speculation arises as to the scattering

pattern of such a loop. The first step in such an investigation was performed

in this report. The potential interaction energy was calculated under the

assumption of only electric-field forces. The arrangement of the coaxial interacting loops was that their planes were parallel and that the direction

of approach was perpendicular to these planes. Calculations showed that

the potential energy of the spherical model was about 4 orders of magnitude greater than that of the loop model at the nearest approach distance

chosen. Thus, to the limited extent suggested by the primitive interaction

scenario used in this report, the loop model results are in accordance with

the experimental results.

While no definitive theory has yet been applied to calculate interaction parameters for alternate (and more realistic) scenarios, one can reach certain

conclusions by starting from the nature of the force fields involved. Some of

these conclusions concerning alternate scenarios pertinent to further work

follow:

13
• Scenario 1: Loops approach in planes perpendicular to approach line

(electric and magnetic forces). Magnetic forces will have dramatic influences, especially since the interaction force is attractive if the magnetic moments are in the same direction. The relation between electric

and magnetic forces as a function of separation is the priority calculation needed because the results will show if magnetic forces must be

used in the following scenarios. An aspect of this relation was indicated by an approximate calculation done some time ago. This calculation was used to judge the radius of a charged torus tube necessary

for the electric-field energy to be the rest mass energy. The rough results showed this radius to be very small also. An improved calculation of this effect has been included in appendix D, where it is indeed

shown that the electric-field energy of a hollow torus is the electron

rest mass energy. This information could not be used in this report

because an appropriate partition of the energy between the electric

and magnetic fields has not been determined.

• Scenario 2: Loops approach “edge one” (electric field). The nearest

approach distance will be nearly twice the loop radius. Since the interaction force is higher for the nearest arc segments of the loops than

it is for the farther separated arc segments, the scenario is unstable

because of the torque involved. Any alignment error will tend to increase the error until the loops are parallel. This unstable approach

alignment will have a very low probability and therefore contribute

little to an averaged scattering cross section.

• Scenario 3: Loop centers on approach line—planes are not perpendular to approach line (electric forces). Here also torque will be exerted

on each loop by the field of the other because of the different distances

between loop segments. The torque will tend to rotate the loops into

parallel plane alignment at the point of nearest approach after which

the torque will reverse the rotation. There is a dynamic effect here,

since the torque depends upon the relative orientation angle, which

is the angle being changed by the integral of the torque times the moment of inertia.

• Scenario 4: Loop center not on same approach line—planes have general orientation (electric field). This is the most general case for electricfield interactions. The dynamic effect here is enhanced because the

loop offset greatly increases the torques. It is possible that in the latter stages of the approach, each loop will be spinning. If this does

occur, then the net force will be a function of time different from that

given by the static field variation as a function of separation only.

14
References

1. Goedecke, G. H., “Classically radiationless motions and possible implications for quantum theory,” Phys. Rev. 135: (1B), 13 July 1964, pp B281–

288.

2. Mohr, P. J., and B. N. Taylor, “CODATA recommended values of the

fundamental physical constants: 1998,” Rev. Mod. Phys. 72 (2), April

2000, pp 351–495.

3. Menzel, D. H., ed., Fundamental Formulas of Physics, Dover Publications,

Inc., New York, 1960.

15

Appendix A. FY01 Director’s Research Initiative Proposal

File Number: FY01-CISD

Directorate: Computational and Information Sciences Directorate

Title: Radiationless Moving Charge and Current Distributions Using Clues

From Semiclassical Static Electro-Magnitism

Principal Investigator: Dr. Harry J. Auvermann (hauverma@arl.army.mil),

U.S. Army Research Laboratory, Computational and Information Sciences

Directorate, Battlefield Environment Division, Atmospheric Acoustics and

Electro-Optics Propagation Branch, AMSRL-CI-EP, Adelphi Laboratory

Center, bldg 202, rm 4G102, ph 301-394-2088, fax 301-394-4797.

Objective: Determine if a magnetic model of the electron has scattering

properties matching those measured in electron-electron colliding beam

experiments.

Technical Challenge/Background: When the Bohr atom is introduced in a

physics curriculum, derivation of the electron motion proceeds under the

assumption that this moving charge does not radiate as it traverses the circular or elliptical orbit, even though an accelerating charge is known to

radiate energy according to Maxwell’s equations. I became aware of an article published in 1964 [1] on radiationless charge-current distributions. The

author found that many such distributions exist. A necessary part of producing a radiationless model of the hydrogen atom is a radiationless model

of the electron.

I began a study of the properties of the electron starting with the derivation of the classical radius. The semiclassical derivation for this quantity

equates the static electric-field energy of a hollow sphere bearing the electron charge to the rest mass energy. For my own benefit, I decided to investigate the static magnetic-field energy of a spinning sphere whose radius

is the classical radius, thinking that the magnetic-field energy would be

much less than the classical electric-field energy. Using my own approximation for the magnetic-field energy, I was nevertheless very surprised to

find that the magnetic-field energy was some 900 times the electric-field

energy. Additionally, the peripheral velocity of the sphere had to be many

times the speed of light to achieve the required magnetic moment.

Backing off from the classical radius requirement, I sought the radius of

a charged loop whose peripheral velocity was the speed of light but still

had the correct magnetic moment and field energy. Using my calculation,

although approximate, I determined the answer for the model loop radius being what I call the Compton radius, or radius of a circle whose

17
circumference is the Compton wavelength. I determined also that the required ”charged wire” radius (to give an electric-field energy for a Compton radius-sized loop to be equal to the electron rest mass) was a number

so small that it defied imagination.

In the intervening years, an article appeared on a colliding-beam experiment (I believe at the European Organization for Nuclear Research) that

showed the cross section for electron-electron scattering was at least a factor of 100 less than the classical size. I have not verified or found the reference to this article, but two colleagues have assured me that this result

is currently universally accepted in the particle physics community. When

the experiment report was published, in light of my ”charged wire” radius

calculation, I theorized that I may be on the right track with my Compton

radius loop model.

Recently, I checked my past calculation. The effort proved surprisingly easy

with the benefits of hindsight. The derivation used the commonly available

expressions for the various physical constants [2–4]: the fine structure constant, the Bohr radius, the classical radius, the Bohr magneton, the Compton wavelength, the rest mass energy, speed of light in terms of the permeability and permittivity of free space, the magnetic moment of a current

loop, the free electron g-factor, the field energy of an inductor, and the inductance of a current loop. The radius was shown to be the Compton radius

multiplied by the cube root of a factor consisting of the natural logarithm

associated with the inductance times the g-factor squared times the fine

structure constant divided by 2π. Because this factor contained the logarithm of the ratio of the current loop radius to the ”wire radius” and was

to the one-third power, I set it equal to unity and solved for the ratio. The

approximate value obtained was the natural base raised to the power 861.

As I recall, the ratio I obtained previously for the charge loop was the natural base raised to the power 234. These incredible exponent numbers come

about through the reciprocal of the fine structure constant.

Relationship to ARL Mission: The relationship of this effort to the ARL

mission is long-term but clearly established. When the U.S. Army fires

nuclear weapons, radioactive cleanup is a vital activity on the battlefield.

Therefore, the Army needs scientific backup in training and in equipment

procurement to responsibly develop operational capability in this area. To a

physicist, development of this capability is a natural result of investigations

such as is contained in this proposal. The three-element trail is a new understanding of the (1) electron, (2) the atom, and (3) the nucleus. The new

understanding arises from incorporating the idea of nonradiating charge

or current distributions into fundamental theory. Exploitation of the new

idea of the magnetic energy of the electron begins a process whereby the

Army, besides attending to its own needs, gives back to the civilian and

scientific community something potentially quite significant. To state this

long-term mission in a single paragraph is in no way intended to minimize

the difficulty expected when performing the work.

18
Approach: The first requirement is to study the appropriate literature relevant to the high-energy collider experiment result, which will provide the

parameters for the theoretical analysis. The next step will be to establish the

differences to be expected when a loop is substituted for one of the interacting spherical particles accounting only for electric-field effects. Step by

step, the theory will be expanded to the advanced relativistic scattering theory. Indications exist in the literature scanned so far that later experiments

took place with polarized particles [5].

It is almost certain that all the steps described here cannot be accomplished

in the one year allotted to projects funded under this program. These details

are given to establish the overall progression of the intent and mind-set for

the work.

Tasks Milestones:

• Oct 00: Complete financial matters. Develop a detailed work plan.

• Jan 01: Complete scenario design. Complete electric-field study.

• Jun 01: Complete magnetic-field study. Begin high-energy study.

• Oct 01: Write final report with recommendations for follow-on work.

Benefits of Successful Completion: Successful completion implies that the

theoretical scattering studies are corroborated in some manner by the experimental results. To the extent of this corroboration, a revised model of

an electron will be indicated. This revised model can then be used in a

more elaborate theoretical scattering investigation than was possible during the limited time period originally allotted to a DRI project. Success in

this more elaborate venture will bring the postulation of a radiationless

electron model nearer. Success can thus have a ripple effect on models for

atoms, leptons other than the electron, baryons, and ultimately the nucleus.

I would like to emphasize that these latter effects are complete speculation at present. Ultimate success would be knowledge of the nucleus elaborate enough to suggest methods of radioactivity decay-rate modification

[6], which would contribute to more efficient radioactive waste disposal.

Collaborations: Dr. George H. Goedecke, Department of Physics, New Mexico State University, Las Cruces, New Mexico, has agreed to collaborate on

this project. Dr. Goedecke is the author of the first article cited in the references to this proposal and has published later findings on the subject. He

is a recognized authority on scattering theory, both electromagnetic and

quantum mechanical. He is also an author of many papers on other aspects

of modern physics. Dr. David A. Ligon of ARL, whose dissertation was in

the field of Quantum Electro Dynamics, has agreed to review progress of

the work and provide insight within the limits of his other commitments.

19
Budget:

Salary: $60,000 Dr. H. J. Auvermann, six man months

Equipment: $0

Travel: $0

OGA: $0

Contract: $15,000 Dr. G. H. Goedecke, short-term analytical

services

Other external: $0

Total cost: $75,000

Qualifications of Principal Investigator: Auvermann was awarded the

Ph.D. degree in Physics/Math in 1957 by the University of Texas at Austin.

During most of his career, he has been working in the field of optics. Since

1990, he has been Project Leader for the 6.1 Work Unit Battlefield Acoustic

Propagation Research. He initiated the creation of the Turbule Ensemble

Model (TEM) of turbulence [7,8], supervised the insertion of TEM into the

existing two-dimensional Fast Field Program acoustic propagation model,

and supervised the development of the Two-Way Wave Equation Model

(TWWEM), a three-dimensional acoustic propagation model.

Literature Search: A search with the use of the elements electron-electron

and scattering found some 1500 articles, 90 percent of which deal with

semiconductor physics. A search through Physical Review failed to locate

the article mentioned in the Technical Challenge/Background section. A

review article was found [9] as well as several articles on polarization effects other than the one in the reference section [5].

20
References

1. Goedecke, G. H., “Classically radiationless motions and possible implications for quantum theory,” Phys. Rev. 135(1B) (1964), p B281.

2. Abramowitz, M., and I. A. Stegun, Handbook of Mathematical Functions,

U.S. Government Printing Office, Washington, DC 20402 (1970).

3. Wolfe, William L., and George J. Zissis, The Infrared Handbook, U.S. Government Printing Office, Washington, DC 20402 (1978).

4. Menzel, D. H., ed., Fundamental Formulas of Physics, Dover Publications,

Inc., New York (1960).

5. Shieh, S. Y., “Electron-electron scattering cross section including polarization corrections,” Phys. Lett. B B26(4) (1968).

6. Anderson, J. L., J. Phy. Chem. 76 (1972), p 3603.

7. Goedecke, G. H., and H. J. Auvermann, “Acoustic scattering by atmospheric turbules,” J. Acoust. Soc. Am. 102(2), Pt. 1 (August 1997).

8. Goedecke, G. H., R. C. Wood, H. J. Auvermann, V. E. Ostashev, D. I.

Havelock, and Chueh Ting, “Spectral broadening of sound scattered

by advecting atmospheric turbulence,” J. Acoust. Soc. Am. (May 2001).

9. Panofsky, W.K.H., “Some remarks on the early history of high-energy

electron-electron scattering,” Int. J. Mod. Phys. A 13(14) (1998), p 2429.

21

Appendix B. Magnetic Moment of a Spinning Charged Sphere

This appendix will be concerned with deriving an approximation to the

magnetic moment of a spinning charged sphere. The method will be to

consider the sphere as being made up of a series of loops positioned on

the surface of the sphere and add up the contributions to the magnetizing

force-field contributions at the center of the sphere. The equivalent current

in a loop whose radius is the classical electron radius will be defined as

the current that gives the same magnetizing force at its center. The field HL

along the axis of a current loop is given by [1]

HL = ILr2

2 [z2 + r2]

3/2 , (B-1)

where IL is the current in the loop and r is the radius of the loop.

The distance from the plane of the loop is the variable z. If the radius of the

sphere is r0 and qa is the total charge, the surface charge density σ is

σ = qa

4πr2

0

. (B-2)

If the sphere center is at z = 0 and the angle φ is measured from the positive

z-axis, quantities of interest are

r = r0 sin(φ); z = r0 cos(φ); I(φ)dφ = σ(2πr)(r0dφ)(Ωr)

(2πr)

HS =

 π

0

dφHL(φ) =  π

0

dφ I(φ)r2

2 [z2 + r2]

3/2



. (B-3)

The variable HS is the field at the center of the sphere, and Ω is the angular

velocity. The constituents are brought together as

HS =

 π

0

dφHL(φ) = qaΩr0

4πr2

0

  π

0

dφ r3

0 sin(φ)3

2r3

0 [cos(φ)2 + sin(φ)2]

3/2



=

 qaΩ

6πr0



(B-4)

and the integration performed. With equation (B-1) for z = 0 and the equivalent loop current Icl, the two are equated and solved for Icl as



Iclr2

0

2r3

0



=

 qaΩ

6πr0



Icl =

qaΩ

3π



. (B-5)

[1] Menzel, D. H., ed., Fundamental Formulas of Physics, Dover Publications, Inc., New

York (1960), p 321.

23
Equation (B-5) gives the equivalent loop current for a sphere that carries a

charge qa that rotates at an angular velocity of Ω. “Equivalent” means that

the magnetic field at the center of the loop is the same as at the center of the

sphere. The magnetic moment [2] is shown to be the current times the loop

area. Thus the magnetic moment of a spinning charged sphere is

μs = 2πr2

0Icl =



2r2

0qaΩ

3



=



2r0qavcl

3



, (B-6)

where vcl is the peripheral velocity of the equivalent current loop to the

spherical shell.

[2] Menzel, D. H., ed., Fundamental Formulas of Physics, Dover Publications, Inc., New

York (1960), p 323.

24
Appendix C. Potential Interaction Energy of Two Charged Loops

The interaction studied in this appendix is that of two charged coaxial circular loops positioned with their planes parallel. The only force considered

is the static electric-field repulsion. The background suggesting the study

of the interaction of circular loops has been given in the main text. There the

discussion acknowledges the complicated nature of the total interaction of

charged and current loops. The restricted interaction equations developed

here are intended as a precursor to an extended development to be accomplished at a later date.

Consider the interaction diagram shown in figure C-1. In the figure, the

A loop is in the x-y plane and the B loop is in a plane parallel to the x-y

plane, with its center a distance zb = d above the origin O. The various

vectors marked by bold symbols are defined as

*ra = element position vector for A loop = ˆxa cos(α)+ˆya sin(α)

*rb = element position vector for B loop = ˆxb cos(β)+ˆyb sin(β)+ˆzd (C-1)

*rab = *rb − *ra ,

where a is the radius of A loop and b is the radius of B loop. The differential

force vector on the charge element at point *rb caused by the charge element

at point *ra is

dF*ab =

1

4πε0r2

ab  qaa dα

2πa  qbb dβ2πb 

rˆab =

qaqb dα dβ

(2π)24πε0r2

ab  *rab

rab



rab = |*rb − *ra| =

	

a2 + b2 + d2 − 2ab cos(β − α)


1/2

. (C-2)

The total charges on the loops are qa and qb. After integration over the two

angles, the total force on B loop will be in the z direction. This is obtained

by forming the dot product of equation (C-2) and zˆ with the result shown

as

FAB =

 qaqbd

(2π)24πε0

  2π

0

dα  2π

0

dβr−3

ab . (C-3)

After the α integration is done, the result will be the same for each β so

that the β integration will be trivial and β can be set to any value in the α

integration. The value is set to zero with the result shown as

25
Figure C-1. Interaction

diagram for two

charged loops (A and

B). O = origin.

x y

ra

rab

α

B

A

z

β

rb

O

→

→

→

FAB =

 qaqbd

(2π)4πε0

  2π

0

dα r−3

ab ; Let  ̄b = b

a

;  ̄d = d

a

=

 qaqbd

(2π)4πε0

  2π

0

dα 	a2 + b2 + d2 − 2ab cos(α)


−3/2

(C-4)

=

2qaqb  ̄

d

(2π)4πε0a2

  π

0

dα 	1 +  ̄b2 +  ̄d2 − 2 ̄b cos(α)


−3/2

,

along with additional manipulation. The above integral is beginning to

look like a standard Elliptic Integral [1], so the variable of integration is

changed after use of a trigonometric identity with the result shown as

FAB =

2qaqb  ̄

d

(2π)4πε0a2

  π/2

0

dθ 	(1 −  ̄b)

2 +  ̄

d2 + 4 ̄b sin2(θ)


−3/2

. (C-5)

At this point, the identity of the two loops is recognized with the resulting

form

FAB =

4q2

a

(2π)4πε0d2

  π/2

0

dθ 	1 + D2 sin2(θ)


−3/2

; D = 2  ̄d−1 . (C-6)

If this force is integrated from infinity to a particular distance p, then the

result will be minus the potential energy Wm

AB(p) of B loop in the field of

A loop and is shown as

Wm

AB(p) =  p

∞

ddFAB = −

4q2

a

(2π)4πε0

  p

∞

dd

d2

  π/2

0

dθ 	1 + D2 sin2(θ)


−3/2

. (C-7)

It appears that the distance 2a has a special role to play, so the variable of

the integration will be changed to a unitless quantity with the use of 2a as

a parameter as

26
Wm

AB(p) = −4q2

a

(2π)4πε0

  p

∞

dd

d2

  π/2

0

dθ 	1 + D2 sin2(θ)


−3/2

; η = d

2a

Wm

AB(q) = −2q2

a

(2π)4πε0a

  q

∞

dη

η2

  π/2

0

dθ 	1 + η−2 sin2(θ)


−3/2

; q = p

2a

(C-8)

= −CAB  q

∞

dη

η2

  π/2

0

dθ 	1 + η−2 sin2(θ)


−3/2

; CAB =

2q2

a

(2π)4πε0a



.

With the anticipation of later investigations, the above integral deserves

special attention because the value of the parameter q is to be chosen so

that the point of closest approach is to be in the range to match the experimental results. With α now representing the fine-structure constant and the

recognition of the loop radius to be the magnetic radius rcl

m or the Compton

radius, the classical radius is α rcl

m. The closest approach should be some

100 times smaller than the classical radius. Therefore, for theoretical purposes, q will be thought of as equal to α2. It is necessary to break up the

integration into two intervals as

Wm

AB(q) = Wm

AB(q)

CAB

= −(L1 + L2)

L1 =

 1

∞

dη

η2

  π/2

0

dθ 	1 + η−2 sin2(θ)


−3/2

(C-9)

L2 =

 q

∞

dη

η2

  π/2

0

dθ 	1 + η−2 sin2(θ)


−3/2

,

where the relative potential energy Wm

AB(q) has been defined also. Solution

of the integral L1 proceeds by an inversion of the order of integration as

L1 =

 1

∞

dη

η2

  π/2

0

dθ 	1 + η−2 sin2(θ)


−3/2

=

 π/2

0

dθ  1

∞





dηη




η2 + sin2(θ)

3/2





= −

 π/2

0

dθ 	1 + sin2(θ)


−1/2

= −K(−1) = −2−1/2K

1

2



, (C-10)

where K(m) is the Complete Elliptic Integral of the First Kind [1]. Solution

of integral L2 proceeds as

L2 =

 q

1

dη

η2

  π/2

0

dθ 	1 + η−2 sin2(θ)


−3/2

=

 π/2

0

dθ  q

1





dηη




η2 + sin2(θ)

3/2





= −

 π/2

0

dθ 	q2 + sin2(θ)


−1/2

−

	

1 + sin2(θ)


−1/2 

(C-11)

= −

 1

(1 + q2)1/2

  π/2

0

dθ q2

(1 + q2)



+

sin2(θ)

(1 + q2)

−1/2

− L1 ,

[1] Wolfram Research, Mathematica 3.0 Standard Add-On Packages, Wolfram Media,

Champaign, IL 61820 (1996).

27
with additional manipulations such as

L2 = −

 1

(1 + q2)1/2

  π/2

0

dθ 1 −

 1

(1 + q2)



cos2(θ)

−1/2

− L1

= −

 1

(1 + q2)1/2

  π/2

0

dθ 1 −

 1

(1 + q2)



sin2(θ)

−1/2

− L1 (C-12)

= −

 1

(1 + q2)1/2



K(k) − L1 ; k = (1 + q2)

−1 .

Equation (C-13) presents the major result of this derivation

Wm

AB(p) = 

2k1/2q2

aK(k)

(2π)4πε0a



; k = (1 + q2)

−1 ; q = p

2a , (C-13)

which is the expression for the potential energy of two loop model electrons

positioned as in figure C-1.

28
Appendix D. Properties of a Charged Hollow Torus

The derivation in this appendix is shown so as to obtain the equation for

the electric field of a charged hollow torus and, from this electric field, find

the potential of the torus surface with respect to a zero potential reference

at infinity. With the potential and the total charge known, the capacitance

is immediately available. The capacitance and the charge then can be used

to obtain the field energy of the torus. The approximation is made that the

torus is an equipotential surface. Everywhere, the toroidal cross section is

a circle. The locus of the centers of the cross-section circles is also a circle.

Figure D-1 shows a three-dimensional diagram of the torus. As indicated

in the sectional view of figure D-2, the symbol b represents the radius of the

loop and the symbol a represents the radius of the torus cross section.

Figure D-1.

Three-dimensional

diagram of torus.

–1.0

0

1.0

2.0

2.0

1.5

1.0

0.5

0

–0.5

–1.0

0

1.0

2.0

x

z

y

Figure D-2. Sectional

diagram of torus

through z-axis.

b a

x

z

29
The formula for the surface area of a torus is given in equation (D-1) [1]. In

the formula

At = π2



b2 − a2



(D-1)

from the reference, the symbol a is the inner radius of the torus and the

symbol b is the outer radius of the torus. With the translation of the equation (D-1) formula to the parameters of figure D-2, the equation (D-2) formula is obtained:

At = (2π)

2ab . (D-2)

Letting the symbol qt represent the total charge on the torus, the surface

charge density σt is

σt =

 qt

At



=

 qt

(2π)2ab

. (D-3)

For the electric field to be found, the electric-flux density D is related to the

surface charge density and the direction is assumed normal to the surface.

Since the surface is assumed equipotential and since the field is a conservative one, the potential can be obtained by integration of the field along any

path from infinity to the surface. The path along the positive x-axis is chosen for its comparative simplicity. By symmetry, this path represents any

straight line path in the x-y plane. If the symbol r represents the position

along the x-axis, then the integration is from r = ∞ to r = a + b.

To find the flux density at a point r on the x-axis, assume that a rectangular

detector D ̄ centered on the x-axis (and perpendicular to it) has a dimension

w in the y direction and a dimension h in the z direction. The object is to

find the dimensions of the patch on the near surface from which issues the

flux intercepted by D ̄. This flux will be augmented by the flux that issues

from the patch on the far surface. The far surface is the inner part of the

torus, which is on the other side of the z-axis from the near surface. Symbols with the subscript n will apply to the near patch, and symbols with

the subscript f will apply to the far patch. Only these two patches need

be considered. Justification for this conclusion is discussed later in this appendix. Determination of patch dimensions is facilitated by two diagrams.

The first is shown in figure D-3, which depicts the situation in the x-z plane.

The rays from the near patch appear to come from the center, marked Cn,

of the right-hand trace of the surface in this plane. The detector subtends

an angle αn from this point. The rays from the far patch appear to come

from the center, marked Cf , of the left-hand trace of the toroidal surface.

The detector subtends an angle αf from this point.

Equation (D-4) shows the calculation of these two-patch dimensions:

αn =

 h

r − b



; hn =

 ha

r − b



αf =

 h

r + b



; hf =

 ha

r + b



. (D-4)

[1] Spiegel, M. R., Mathematical Handbook of Formulas and Tables, Schaum’s Outline Series,

McGraw-Hill, Inc., New York (1968), 34th Printing (1995), p 10.

30
Figure D-3. Patch

dimensions in x-z plane.

Cf

hf

αf

z

hn

αn

D

h x

x = r Cn

The second patch dimension diagram is shown in figure D-4, which depicts

the situation in the x-y plane. The rays from both patches appear to come

from the loop center, which is the coordinate system origin. This is seen

easily for the near patch. It is also true for the far patch, because the normal

rays tend to focus the flux at the origin in this plane. The detector subtends

an angle βn from this point for the near patch. The detector subtends an

angle βf for the far patch. Although for βn = βf , the patch dimensions are

different.

The calculation of these two-patch dimensions is

βn =

w

r



; wn =

w(b + a)

r



; wf =

w(b − a)

r



. (D-5)

The flux from the entire charge in a patch proceeds outward, as justified

at the end of this appendix. The partial flux at the detector from the two

patches designated by ∆FD ̄ is

∆FD ̄ = σt(hnwn + hfwf ) =

=

 qt

(2π)2ab  hbr − a

 w(a + b)

r



+

 hb

r + a

 w(a − b)

r

 . (D-6)

The electric-flux density is then obtained by dividing the flux by the detector area as

D(r) =  qt

(2π)2ar   a + br − a



+

 a − b

r + a

 . (D-7)

Dividing by the electric constant ε0 gives the electric field as

E(r) =  2qt

(2π)2ε0r

  r + b

r2 − a2



=

2[1 + (b/r)]

π

  qt

4πε0(r2 − a2)



. (D-8)

At great distances where r is much greater than a or b, the electric field is

seen to be the same as that from a point charge except for the 2/π factor in

the first bracket. To obtain the potential φt, assume that the charge is positive and accumulate the energy necessary to move a unit positive charge

from infinity to r = a + b through the field of equation (D-8) as

φt =

 2

π

  qt

4πε0

  a+b

∞

dr  r + b

r(r2 − a2)



. (D-9)

31
Figure D-4. Patch

dimensions in x-y

plane.

Cf

Wf

y

Cn

Wn

βn

D

x

w

x = r

To solve the integral, substitute for the variable of integration a tan(θ) as

φt = Qt

 θt

π/2

dθ 

sec(θ)2[tan(θ) + b/a]

tan(θ)[tan(θ)2 − 1] 

; θt = arctan(1 + b/a)

=

Qt

a

  θt

π/2

dθ [a sin(θ) + b cos(θ)]

sin(θ) cos(2θ)



; Qt =

 2

π

  qt

4πε0a



. (D-10)

Further manipulations are carried out by

φt =



Qt



a2 + b21/2

a

  π

4 +δ

π/2

dθ  sin(θ + γ)

sin(θ) cos(2θ)



; θt = π

4 + δ

tan(γ) = b

a

; tan π

4 + δ



=1+ b/a ; tan(δ) =  b/a

2 + b/a

. (D-11)

The integration result is shown as [2]

I =

 π

4 +δ

π/2

dθ  sin(θ + γ)

sin(θ) cos(2θ)



=

1

2

 

cos(γ) ln 

cos(δ)

sin(δ)



+ sin(γ) ln 

(cos(δ) + sin(δ))2

4 cos(δ) sin(δ)

  (D-12)

=

 1

2(a2 + b2)1/2

 

a ln 

a

b



+ b ln 

(2a + b)(a + b)

2ab  

.

Substituting in equation (D-11), the final expression for the potential is

φt =

 1

π

  qt

4πε0a

 

ln 

a

b



+

 b

a



ln 

(2a + b)(a + b)

2ab  

=

 1

π

  qt

4πε0a

  

1 +

b

a



ln 

a

b



+

 b

a



ln 1 +

b

2a

 1 +

b

a

 

. (D-13)

For large a/b (and small b/a), the ln(a/b) term is dominent so that an approximate potential is given by

φ ̃t =

 1

π

  qt

4πε0a



ln 

a

b



. (D-14)

[2] Wolfram, S., MATHEMATICA, 2nd ed., Addison-Wesley, New York (1991).

32
The equation for the capacitance C is [3]

C = q

∆φ; Ct = qtφt

(D-15)

in terms of the charge q and the potential difference ∆φ. The capacitance

Ct of the torus is also shown in equation (D-15). The energy W stored in a

capacitance [3] is given in two forms as

W = C(∆φ)2

2 = q22C . (D-16)

With the use of the second form and substituting quantities defined in

equation (D-13), the formula for the electric-field energy of the torus Wt

is shown as

Wt = q2

t

2Ct

=

 1

2π

 q2

t

4πε0a

  

1 +

b

a



ln 

a

b



+

 b

a



ln 1 +

b

2a

 1 +

b

a

 

. (D-17)

It is instructive to use the approximate formula for the torus potential,

equation (D-14), to obtain an idea of the magnitude of this electric-field

energy for the torus configuration calculated in the main text of this report.

The approximate value for the electric-field energy of the torus W ̃ t is

W ̃ t =

 1

2π

 q2

t

4πε0a



ln 

a

b



. (D-18)

From equation (23) in the main text, the loop radius is given as the Compton

radius, and from equation (26), the approximate value of the loop parameter ρm is given. These values are shown as

rC

r ̃t

=  ̃ρm = exp 2π

α



. (D-19)

The variable rC is identified with the variable b in equation (D-18), and

the variable r ̃t is identified with the variable a in equation (D-18). With the

appropriate substitutions made in equation (D-18), the result is shown as

W ̃ t =

 1

2π

 q2

t

4πε0rC



ln 

rC

r ̃t



=

 1

2π

 q2

t

4πε0rC

 2π

α



. (D-20)

Consolidating equation (D-20) and noting the relation between the Compton radius and the classical radius r0 from equation (7) in the main text, one

can show the result as

W ̃ t =

q2

t

4πε0r0



= we . (D-21)

[3] Menzel, D. H., ed., Fundamental Formulas of Physics, Dover Publications, Inc., New

York (1960), p 318.

33
The final identification is made with we of equation (8) in the main text,

which is the rest mass energy of the electron. Thus we find that the approximate electric-field energy of the present configuration of the hollow torus

is the same as that of the classical electron model. This information has only

been mentioned in the main text of this report because an appropriate allocation of the rest mass energy between the electric and magnetic fields has

not been determined.

That only two patches are needed for finding the electric field of a torus

may be concluded by considering the situation wherein charge radiates

electric flux both outward (as with the derivation just shown) and inward.

The reduction in flux by a factor of two is made up by the inclusion of

four patches instead of two. In reference to figure D-3, the focusing effect

in the z-direction causes the flux from the left-hand patches of the tubes to

be redistributed onto the right-hand patches, thus effectively augmenting

the outward-moving flux. In figure D-4, this same effect occurs in the ydirection. The inward-moving flux from the left-hand portion of the torus

moves through the focus point (the coordinate origin) and redistributes itself on the right-hand portion of the torus. A further point can be made

concerning the net flux within the torus: With flux being focused in both

directions inside the torus, the net flux at each point balances to zero in

figure D-3, indicating zero field, considering only one circle. The effect of

the flux from one circle on the other circle has not been included. Thus, the

assumption that the surface is equipotential only becomes true as the ratio

a/b approaches zero. In figure D-4, in the x-y plane, the focused flux from

both sides balances to zero everywhere inside the loop, again indicating

zero field in this plane.

34
35

Distribution

Admnstr

Defns Techl Info Ctr

ATTN DTIC-OCP

8725 John J Kingman Rd Ste 0944

FT Belvoir VA 22060-6218

DARPA

ATTN S Welby

3701 N Fairfax Dr

Arlington VA 22203-1714

Mil Asst for Env Sci Ofc of the Undersec of

Defns for Rsrch & Engrg R&AT E LS

Pentagon Rm 3D129

Washington DC 20301-3080

Ofc of the Secy of Defns

ATTN ODDRE (R&AT)

The Pentagon

Washington DC 20301-3080

Ofc of the Secy of Defns

ATTN OUSD(A&T)/ODDR&E(R) R J Trew

3080 Defense Pentagon

Washington DC 20301-7100

AMCOM MRDEC

ATTN AMSMI-RD W C McCorkle

Redstone Arsenal AL 35898-5240

US Army TRADOC

Battle Lab Integration & Techl Dirctrt

ATTN ATCD-B

FT Monroe VA 23651-5850

US Military Acdmy

Dept of Mathematical Sci

ATTN MAJ L G Eggen

West Point NY 10996-1786

US Military Acdmy

Mathematical Sci Ctr of Excellence

ATTN MADN-MATH MAJ M Huber

Thayer Hall

West Point NY 10996-1786

Natl Ground Intllgnc Ctr

Army Foreign Sci Tech Ctr

ATTN CM

220 7th Stret NE

Charlottesville VA 22901-5396

Natl Security Agcy

ATTN W21 Longbothum

9800 Savage Rd

FT George G Meade MD 20755-6000

Dir for MANPRINT

Ofc of the Deputy Chief of Staff for Prsnnl

ATTN J Hiller

The Pentagon Rm 2C733

Washington DC 20301-0300

Sci & Technlgy

101 Research Dr

Hampton VA 23666-1340

SMC/CZA

2435 Vela Way Ste 1613

El Segundo CA 90245-5500

TECOM

ATTN AMSTE-CL

Aberdeen Proving Ground MD 21005-5057

US Army ARDEC

ATTN AMSTA-AR-TD

Bldg 1

Picatinny Arsenal NJ 07806-5000

US Army Corps of Engrs

Engr Topographics Lab

ATTN CETEC-TR-G P F Krause

7701 Telegraph Rd

Alexandria VA 22315-3864

US Army CRREL

ATTN CRREL-GP R Detsch

72 Lyme Rd

Hanover NH 03755-1290
Distribution (cont’d)

36

US Army Dugway Proving Ground

ATTN STEDP 3

Dugway UT 84022-5000

US Army Field Artillery Schl

ATTN ATSF-TSM-TA

FT Sill OK 73503-5000

US Army Infantry

ATTN ATSH-CD-CS-OR E Dutoit

FT Benning GA 30905-5090

US Army Info Sys Engrg Cmnd

ATTN AMSEL-IE-TD F Jenia

FT Huachuca AZ 85613-5300

US Army Natick RDEC

Acting Techl Dir

ATTN SBCN-T P Brandler

Natick MA 01760-5002

US Army OEC

ATTN CSTE-AEC-FSE

4501 Ford Ave Park Center IV

Alexandria VA 22302-1458

US Army Simulation Train & Instrmntn

Cmnd

ATTN AMSTI-CG M Macedonia

ATTN J Stahl

12350 Research Parkway

Orlando FL 32826-3726

US Army Soldier & Biol Chem Cmnd

Dir of Rsrch & Techlgy Dirctrt

ATTN SMCCR-RS I G Resnick

Aberdeen Proving Ground MD 21010-5423

US Army Tank-Automtv Cmnd RDEC

ATTN AMSTA-TR J Chapin

Warren MI 48397-5000

US Army TRADOC

Anal Cmnd—WSMR

ATTN ATRC-WSS-R

White Sands Missile Range NM 88002

US Army TRADOC

ATTN ATCD-FA

FT Monroe VA 23651-5170

Nav Air War Cen Wpn Div

ATTN CMD 420000D C0245 A Shlanta

1 Admin Cir

China Lake CA 93555-6001

Nav Surfc Warfare Ctr

ATTN Code B07 J Pennella

17320 Dahlgren Rd Bldg 1470 Rm 1101

Dahlgren VA 22448-5100

Nav Surfc Weapons Ctr

ATTN Code G63

Dahlgren VA 22448-5000

USAF Rsrch Lab Phillips Lab

Atmos Sci Div Geophysics Dirctrt

Hanscom AFB MA 01731-5000

US Air Force

ATTN Weather Techl Lib

151 Patton Ave Rm 120

Asheville NC 28801-5002

USAF Rome Lab Tech

ATTN Corridor W Ste 262 RL SUL

26 Electr Pkwy Bldg 106

Griffiss AFB NY 13441-4514

NASA Marshal Spc Flt Ctr

Atmos Sci Div

ATTN Code ED 41 1

Huntsville AL 35812

Hicks & Assoc Inc

ATTN G Singley III

1710 Goodrich Dr Ste 1300

McLean VA 22102

Natl Ctr for Atmos Rsrch

ATTN NCAR Library Serials

PO Box 3000

Boulder CO 80307-3000
37

Distribution (cont’d)

NCSU

ATTN J Davis

PO Box 8208

Raleigh NC 27650-8208

Pacific Mis Test Ctr Geophysics Div

ATTN Code 3250

Point Mugu CA 93042-5000

US Army Rsrch Lab

ATTN AMSRL-IS-EW

White Sands Missile Range NM 88002-5501

Director

US Army Rsrch Lab

ATTN AMSRL-RO-D JCI Chang

ATTN AMSRL-RO-EN W D Bach

PO Box 12211

Research Triangle Park NC 27709

US Army Rsrch Lab

ATTN AMSRL-D D R Smith

ATTN AMSRL-DD J M Miller

ATTN AMSRL-CI-AI-R Mail & Records

Mgmt

ATTN AMSRL-CI-AP Techl Pub (2 copies)

ATTN AMSRL-CI-LL Techl Lib (2 copies)

ATTN AMSRL-CI-EP H J Auvermann

(15 copies)

Adelphi MD 20783-1197

1. AGENCY USE ONLY

8. PERFORMING ORGANIZATION

 REPORT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

12a. DISTRIBUTION/AVAILABILITY STATEMENT

10. SPONSORING/MONITORING

 AGENCY REPORT NUMBER

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

6. AUTHOR(S)

REPORT DOCUMENTATION PAGE

2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

11. SUPPLEMENTARY NOTES

14. SUBJECT TERMS

13. ABSTRACT (Maximum 200 words)

Form Approved

OMB No. 0704-0188

(Leave blank)

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources,

gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this

collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson

Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503.

12b. DISTRIBUTION CODE

15. NUMBER OF PAGES

16. PRICE CODE

17. SECURITY CLASSIFICATION

 OF REPORT

18. SECURITY CLASSIFICATION

 OF THIS PAGE

19. SECURITY CLASSIFICATION

 OF ABSTRACT 20. LIMITATION OF ABSTRACT

NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18

298-102

An Elementary Electron Model for Electron-Electron

Scattering Based On Static Magnetic Field Energy

February 2001 Final, June to August 2000

Electron motion paths that exhibit zero radiation in a Maxwell’s equation solution have been

reported. Such paths, require a radiationless model of the electron itself, such as the charged

hollow sphere. When the electric-field energy of this model is set equal to the rest mass energy of

the electron, the radius of the resulting sphere is called the “classical electron radius.” Analysis

reveals that the static magnetic-field energy of the classical model is many times the electron rest

mass energy when the sphere is given an angular velocity large enough to exhibit the electron

magnetic moment. The necessary angular velocity produces a peripheral velocity many times the

speed of light. A classical model with a peripheral velocity near the speed of light is a loop

whose radius is the Compton wavelength divided by 2π; such a loop has a very small dimension

perpendicular to the loop plane. Experiments reveal point-like electron scattering properties down

to at least 1/100 of the classical radius. The small transverse dimensions of the loop model indicate

similar scattering results. Recently, a proposal was submitted to investigate the scattering

properties of interacting loops. Because of limited proposal length, derivation of loop model

equations could not be included. This report contains the details of the analysis.

Current loop, classical scattering, rest mass energy

Unclassified

ARL-TR-2332

OFEJ60

61110253A11

B53A

61102A

ARL PR:

AMS code:

DA PR:

PE:

Approved for public release; distribution

unlimited.

42

Unclassified Unclassified UL

2800 Powder Mill Road

Adelphi, MD 20783-1197

U.S. Army Research Laboratory

Attn: AMSRL-CI-EP hauverma@arl.army.mil

U.S. Army Research Laboratory

39

Harry J. Auvermann

email:

2800 Powder Mill Road

Adelphi, MD 20783-1197