artículo electro

8/16/2019 Artículo electro

1/19

IS CLASSICAL ELECTRODYNAMICS ANINTERNALLY INCONSISTENT THEORY?

Jos6 A. Heras

Departam,en,to de F{si,ca, Escuel,a Superior de Ffuica y Matem,dticas, In,st,ituto Polif,6cn,icoNacion,al, Apartado Postal 21-081, 04021, Mdri,co D'i,stri,to ederal, M6.rico.

The existence of runaway solutions in the classical radiation reaction theory of the elec-tron has tradit ionally put in doubt the internal consistency of classical electrodynamics.The responsible of introducing such undesirable solutions is the centenary Abraham-Lorentz equation, which describes the non-relativist ic motion of a point charge underthe combined action of an external force and of i ts induced radiation reaction force.Although several attempts to replace this equati


2/19

Is classical el,ectrodyn,am"ics an' ' ' ' I57

where r" is the characteristic time defined by

r - 2 e 2 e ) 3rnc3 '

with m being the mass of the particle, e its electric charge and thedot indicates

time differentiation. unfortunately, the Abraham-Lorentz equation isshown to be

unsatisfactory in at least six aspects:

(a) It involves the time derivative of the acceleration which means that theequa-

tion is of third order in the time derivative of the particle position' Therefore,

the specification of the initial position and Velocity of the particleis not suf-

ficient to uniquely determine the solr-rtionof the equation.

(b) It predicts unpirysical runaway solutions, w[ich may be eliminated whena par-

ticr-rlar asymptotic condition is imposecl but then unphysicai preaccelerations

are introdrrceci 2] .

(c) In the absence of external forces the self-force survives which disagreeswith

the iclea hat when the extelnal folce vanishes he self-force sirould alsovanish,

i .e. , Ec1. (1) ctoes not reflect the expectecl dea that the self-force sinclucecl

by the exterual force.

(d) In the absence of an external force the equation predicts tirnevariations of

velocity, rvhich violates the lar'vof inertia'

(e) It exhlbits problems of nonr-ruiqtreness 3] '

(f) For a point electron the characteristic ime in Eq (2) takes the value:

n ^ n , , t n - 2 4 -7 e - O . Z O X l U r .

(3)

which is distinctly quantr.rtn mechanical not classicai' The uncertaintyin

energy associated with such a short time, A,E x (312)a-rnrc2= 206mc2,

exceecls much the rest enelgy of the electron aud then quantum effects,as a

pair production, cannot be ignorecl at this energy scale' At least in principle'

the efl'ects of the radiation reaction force on an electron seem to be outsideof

the validity domain of classical electrodynamics'

It is really sr.rrprising o find that clespite of all these defects, the Abraham-Lorentz

equation still renains in our cument textbooks of electrodynamics'

O'e fu'darne'tal reason fbr rvl-rich he Abral-ram-Lorentz equatiotl has sun'ived

during one centttry is that it is usually obtainecl from energy conservation using

the totai powel radiated by the charge [1]. This por.ver s derived in turn from

Maxwell's equatiols. It is really diffrcult to put in doubt a result derived frorn


3/19

158 Josd A. Heras

energy conservation and Maxwell's equations. Nevertheless, having in mind thepathological solutions of the Abraham-Lorentz equation as well as the smallness ofits characteristic time for electrons, we have to admit the possibility that classicalelectrodynamics may be internally inconsistent.

In this paper we address the radiation reaction problem in electrodynamics bydiscussing four alternative expressions for the equation of motion of a point par-ticle including the radiative effects. These alternative equations do not predictrunaway solutions and therefore they are candidates for appropriately replacing theAbraham-Lorentz equation. The first equation is the well-known integro-differentialequation obtained from the Abraham-Lorentz equation using the condition thatthe acceleration vanishes at infinite times

[4-6].The second equation is the Ford-

O'Connell equation [7,8] which has recently been claimed as the correct equationof motion [9,10] of a point charge. The third equation has recently been presentedby the author [11]. This equation additionally includes the reactive eIl 'ects f themagnetic moment of the electron. The four equation has also been derived by thepresent author [12,13] and considers only the radiation from the magnetic momentof the electron. Although the discussion given here is for non-relativistic particles,

it may be extended to relativistic particles.

2. The Abraham-Lorentz equation

We begin this section with a quotation by Griffiths [14]: "Accord'ing o the laws o.fclassical eLectrodEnamics, n accelerated charge radiates. Thi.s rad'iation carries offenergA, uhi,ch must come at the erpense of the parti,cle's ki,neti,cenergA. Under thein,fluence o.f a g'iuen .force, here.fore, a charged part'icle accelerates ess than a neutralone o.f the same mass. The radiation eu'idently ererts a .force back on the charge(F."a) -a recoil .force, rather like th,at o.f a btLlleton a gnn." This is the currentpicture of the radiation reaction force. The fundamental idea is the following: Theradiation reaction force behaves as a recoil-like force, i.e., a force suddenly acting

against the particle motion. Nevertheless, his reasonable dea is not reflected, at

least in principle, by the current theory based on the Abraham-Lorentz equation.

En fact, in absence of external forces the solution of Eq. (1) satisfying the initial

condition a(0) : a6 is a runaway acceleration:

which leads, by assuming he initial condition o(0) : tr6, o the runaway velocity:

a(t) : us - asT" asr.et/ ' ' '

( A \

(5 )

According to this expression, the velocity changes n

forces. prediction that violates the inertia law. i.e. the

time in absence of external

runaway solution in Eq. (5 )


4/19

[s cl,assical electrodyn,amics o,n, 159

violates Newton's first law. Furthermore, the self-force does not damp the particle'smotion, but just the opposite, his force actually increases uch a rnotion The run-away solutions together with the smallness of the characteristic time for an electronconstitute the strongest objections to the equation of Abraham-Lorentz. Actually,they put in doubt the validity of this equation, and moreover) they question th einternal consistency of classical electrodynamics.

However, to create the radiation reaction force the particle needs o be externallyaccelerated. This seems o indicate that Eq. (1) should not be considered withoutthe external force. We consider, for example, a charge inside the electric fieldE(t) : F,se-t/7, where ? is the associatecl haracteristic ime. This charge s thenurged by the external force F.*6(l) :

eEoe-t/T and thus Eq. (1) takes the form:

a ( t 1 * o" - t 1 '

* r u h .?'rL

The solution of this equation under the initial condition a(0)

/ , \ e E g T r _ t / . F | / _a( t ) : - ; - - - : \ ( " - ' r ' - e " ' ' , ) .m \ ] t T e )

For times t > T this acceleration becomes

a ( t ) x - : P o ' - e t / ' ' ' . ( 8 )m ( T a r. ) -

Equation 7) may be integrated with u(0) :0 to obtain he velocity

a( t ) =eE0T2 ' |1 ; t / r \ + "PT" . , ( l - e t / ' , .1 , (g)

n ( 1 - + r " ) ( 1 - e

- i rn , ( r + r " 1

For times t > T this velocitv becomes

eEnTr. t-a \ t ) =

nl l f + Te

(6)

: 0 is given by

(7 )

(10 )

The runaway behavior of the acceleration and velocity is evident despite we haveapplied an exponentially decreasing electric field to the particle. This prediction isclearly unacceptable from a physical point of view. Therefore, the presence of theexternal force in Eq. (1) does not seem to eliminate the runaway behavior of thesolutions. iFrom the above results we can extract the following conclusion:

The Abraham-Lorerttz equation predicts that for times t > T the velocityof the electron caused by the external force F.*t(t) : eEoe-t/" increases

infinitely on account of the action of the radiation reaction force.However, Dirac 12] provided an ingenious procedure to eliminate the runaway

solutions of Eq. (1). The general solr-rtion f Eq. (1) [with F.*t * 0, in at least ina finite time interval] is given by

I r l t la( r ) l " (O)

- - | a r ' - t ' /n F"* , ( l ' ) " , / , , (11)L * " J o I


5/19

160 Josd A. Heras

For an arbitrary choice of a(0), the particle's acceleration ncreases as etlr ' , andtherefore Eq. (11) is in general a runaway expression. However, in Eq. (11) weobserve hat if the time approaches o infinity the solution also approaches o infinity

at least that the square bracket approaches zero as tire time approaches to infinity.This means that the runaway behavior vanishes when the acceleration vanishes attime infinity, i.e, a -- 0 as f ---+oo. This asymptotic condition can be achieved byassuming he particular initial condition

a(o) .1- [* 4, r r" - t ' l r, 'F"*r( t ' ) .T J O

iFromEqs. (11) and (12) t follows hat the solution of Eq.

1 / Ga(t) :

'I ,17r"-(t '

-t) l '"F"*r(t ') .r" Jt

This is evidently a nonrunaway expression which seems oexcept because t can be written as f4-61:

a(t)* lr*

ds e- 'F"" (t + sr"),

(r2)

(1) is given by

(13 )

be physically acceptable

(r4)

where s : (f ' - t)1r". According to Eq. (14), the acceleration a(t) depends on th e

force F"*s (t + sr") at times greater than l. This means that the particle acceler-

ates before the force acts on it, i .e. , the particle undergoes preaccelerations, which

violates the principle of causality. In other words: via a particular choice of theinitial acceleration, the unphysical runaway behavior was changed by the equally

unphysical acausal behavior

In an attempt to understand the disturbing solutions of the Abraham-Lorentzequation, Dirac [2] pointed out that in the case of electrons, the so-called preaccel-

erations actually occur in times distinctively quantum mechanical (r" - 10-2as) nor

classical and therefore no observable acausality is predicted at the classical level.

Except for brief periods of preacceleration, the integro-difl'erential Eq. (14) seems

to be an appropriate alternative for the Abraham-Lorentz equation.

Horvever, it is certainly diflicult to accept the existence of an acausal behavior

(even when it occurs in extremely brief periods of time) in an internally consistent

classical theory of point charges. On this point Bennet wrote[15]: "Sttch

acausal

behaui,or seems so philosophically di,staste.fu,l hat thi,s feature o.f clussi,cal electro-

dynam'ics has tr,suallE been regarded as crn arti,fact o.f the class'ical reatment, and

merely sEmptomatic o.f an incomplete heory."

On the other hand, Rohrlich [9] has recently pointed out that the unphysical

result of preacceleration is not a defect of the equation of motion but is due to the

unjustified use of an external force that changes too fast in time. According to


6/19

Is classi.cal el,ectrodyn,am,ics an 1 6 1

Rohrlich the point particle approximation used in the derivation of the Abraham-

Lorentz equation forbids external forces that rises or falls too fast (a step function

is an extreme example). It is interesting to note that after a century the validity

limits of the Abraham-Lorentz equation continue being still discussed [16].Following to a certain extent the analogy outlined by Griffiths [14] between

the particle-field system and the gunman-bullet system, the radiation reaction force

seems o be an internal force induced by an external force. A causal relation between

these forces is then expected. However, in the Abraham-Lorentz equation such a

causal relation is obscure because he internal force survives even when the external

force is vanished. The analogy suggests also that both the internal and the external

forces are sudden forces, .e., forces acting on the particle in relatively short periodsof time. To see what means "relatively short" in the case of electrons, we ca n

consider Jackson's estimates [1], which are based n energy considerations, or th e

characteristic times during which the radiative efl'ects on electrons are important.

If an external force causes a particle (initially at rest) of mass m to have an

acceleration of typical magnitude a for a period 7 then its typical energy ts is its

kinetic energy (after the period of acceleration) which is of order of

to - *r2 ( 1 5)

where u is the velocity. If this particle has also a charge e then it simultaneously

radiates during the period T an electromagnetic energy of the order of

according to Larmor's formula. When both energies t6 and tr.4 are comparable,

i.e., ts - trad, we expect that the radiative effects, ike the radiation reaction force,

become important. But how to compare t6 and trua? We need to assume a relation

between the magnitude of the velocity and the magnitude of the acceleration. In

the case of impulsive forces such relation is given by

- 2e2 2Td r a d - - 3 a 3 '

) ,e2T - - --

3mc3'

( 1 6 )

( 1 7 )

( 1 8 )

u( l * -

T

iFrom Eqs. (15) and (16) we find that only when the external force is applied so

suddenly and for such a short time

that the radiative effects will modify the motion of the particle appreciable. In other

words: for those forces that are applied duringtimes T - Te he reactive eft'ects will

be important. This semiquantitative argument is strongly based on the validity of

Eq. (17). As above mentioned, a family of forces satisfying Eq. (17) is the family


7/19

162 Jos6 A. Heras

of sudden or impulsive forces. This result disagrees with Rohrlich's statement thatsudden forces (those that change too fast in time) are excluded from the validitydomain of the Abraham-Lorentz equation.

Unexpectedly, he characteristic ime (r. - 10-24 s) of the self-force urns outto be a typical time of the quantum domain. This seems to indicate an internalinconsistency f the theory, which seems o be even more serious han the disturbingexistence of runaway solutions of the Abraham-Lorentz equation. The reason issimple: the time r" is outside of the classical domain, which is roughly characterizedby an inferior limit of the order of the Compton time q : \.1(2rc) = 1.3 x10-21s, where )" is the Compton wavelength of the electron. The fact is thatquantum effects, as pair productions,

enter at scale times(

T1 and thus classicalelectrodynamics seems o be only appropriate for scale imes ) rr. We can showthat ru = .005rr. Therefore, the effects of the self-force will be appreciable whenthe external force is applied during times T - re br,rt he point is that none forcecan be applied during this time without the intervention of quantum effects Toavoid quantum effects the force should be applied during times T ) rs x 206r". Itfollows that the classical effects of the self-force predicted by the Abraham-Lorentzare,, b5, consistency, necessarily nsignificant.

Furthermore, it is interesting to note that the smallness of the time r" for elec-trons (r. - 10-24 s) has led to the general belief that the self-force Fself : r"ntit inthe Abraham-Lorentz equation nra: Fuxt { r.ntit is a small term compared withthe other terms. This explains to a certain extent why the self-force is frequently

handled as a small perturbation. Rohrlich emphasizes his idea [9]: "In .fact, theself-.force hi,ch inclttdes radiation reaction i,s always only a small correct'ion to theeqtt"ati ,on ,f mot'ion rna: F"*t, in which radi,ation ,s neglected." This idea is, ofcourseT nconsistent with the runaway character of the solutions of Eq. (1). In fact,when Eq (1) is solved as an initial-value problem, we find that self-force s runawayand therefore it grows overcoming quickly to the external force. Therefore, the ideathat the self-force s small compared with the external force s not naturally reflectedby the direct solutions of Eq. (1) However, if we consider the integro-differentialEq. (14) instead of the differential Eq. (1), the situation changes ubstantial ly. Forexample, we consider a charged particle inside the electric field E(r) : F,se-t/T,where ? is tlie associated characteristic time. This particle is urged by the external

force F.*1(r):

eBse-t/T. With this force, Eq. (1a) gives [17]:

a(r) #n*t/r.This is integrated with o(0) : 0 to obtain the velocity

eF,nTu( t1; ff i ( r-e - t i r ;

(1e)

(20)


8/19

For times t > T this velocity reduces to

Is cl,assical el,ectrodyn,antics an' . . . 163

(21)EsTo = , 4 t + r J T '

It should be noted that if T { r" then the velocity of the particle caused by the

external force increases, as ong as he time z" dominates to the time 7, on account of

the action of the radiation reaction force. If T > r" then the velocity of the particle

caused by the external force reduces, as long as the time 7 dominates to the time r" ,

on account of the action of the radiation reaction force. Therefore, the idea that the

effects of the self-force are always small when compared to those of the external force

is notgenerally

validwithin the general

contextof the Abraham-Lorentz theory of

point charges. However, if this particle is identified with a classical electron then

the force should be applied during times T )_ q x 206r.. This implies that the

eft'ects of self-force are necessarily small by arguments of classical consistency. Fo r

the classically allowed highest time, T - rx = 206r., Eq. (21) gives

u = . 9 g 5 1 6 e E 0 ? .?TL

lFrom the above equation we can extract the following conclusion:

(22)

The integro-differential Eq. (14) predicts that for times t > T the velocity

of the electron caused by the external force F.*.(l) : eEoe-t/? reduces less

than ,\Vo on account of the action of the radiation reaction force.

3. The Ford-O'Connell equation

In a provocative paper entitled [9]: "The correct equation of motion of a classi-

cal point charge," Rohrlich has claimed that the Lorentz-Abraham-Dirac equation

(the relativistic generalization of the Abraham-Lorentz equation) is not the correct

equation of motion of a classical point charge. He then proceeds to derive what he

calls the correct equation, which in the non-relativistic limit, is given by

m a : F. * t * Te F e x t r (23)

where now the radiation reaction force is represented by F".1g: TeFext ' Equation

(23)was first derived by

Ford andO'Connell [7]

in i991. However, it should be

mentioned hat previously Eliezer [18]obtained Eq. (23) with Fext : eE, being E an

electric field, as a solution of another equation invoh'ing a series of approximations.

Furthermore, Eq. (23) was also obtained by Landau-Lifshitz [19] in the particular

case n wlrich the external force s the Lorentz force: Fext: eE* (elc)u xB.

Jackson [1] has pointed out that the Ford-O'Connell equation: ".. . is a ualid

equati,on o.f motion wi,thout runaw&A solttt'ions or acaL;sal ehaui'or. It is a sensible


9/19

166 , losd, A. Heras

where 7 is a time scale over which the acceleration appreciably changes, This means

that in certain time scale, he effects of Eqs. (28) and (29) can compensate between

themselves n such a way that the radiations emitted by the electric charge and the

magnetic moment are comparable. To find such a time scale we require to calculate

the power radiated by the magnetic moment of the electron. Of course, the power

radiated by the electric charge is given by the well-known Larmor formula:

q ^ 2p": f to2 .The radi ation electric field of a suddenly accelerated, on-relativistic particle with

a magnetic moment is given by [11-13, 26]:

in x t(r_ra) l (31)rad:L o" ' . l ' " t

The associated Poynting vector Sp : @lar)lE..a12n takes the form

sr, n ( n x p ) 2 ( n . h ) ' (32)4trR2c5

This is used into the instantaneous radiated power per unit solid angle: dPrf dQ:

(S, .n)ft2 to obtai n, after an integration over all solid angle, he total power

(33)

(34)

powers radiated by the

(30)

(35)

(36)

(37)

P,,:ffiu' ;ro" a)'.If p and & are instantaneously perpendicular then Eq. (33) reduces o

P t , : { " '15cb

iFrom Eqs. (30) ancl (34) it follows tirat the rate of the

magnetic moment and the electric charge is given by

P,, 2u 2P. - iaeP

where the approximation it - a,lT was used. It follows lhat P," - P" for times

T - \mplk"). It is useful to define the characteristic ttme T by the expression

T: \pUU o e c

Therefore, for times T - T both the radiated power by the electric charge and the

radiated power by the magnetic moment are of the same order: Pp - P". If the

particle is identified with the electron ltt: eftl(2mc) :9.27 x 10-21 stc cm] then

the time 7 takes the value

T: 4 . 0 7 x 1 0 - 2 2 s = . 3 r r .


10/19

Is classical elect'rodyn'amics a'n' " ' 767

During this time the light travels the distance d"= 43.3rs, where 16 is the classical

radius of the electron. It follows that for times T < T the power radiated by the

magnetic moment of the electron dominates to that due to its electric charge'

By energy consiclerations we can also estimate the characteristic times during

which the radiative effects originated by the magnetic moment of the electron be-

come important. If a sudden external force causes a particle (initially at rest) of

mass m to have an acceleration of typical magnitude o for a period ? then its typical

energy ts is its kinetic energy (after the period of acceleration) which is of order of

to - ma2T2, (38)

where a - ulT being o the velocity. If this particle has also a magnetic moment

of magnitude pr hen it simultaneously will radiate during the period 7 an electro-

magnetic energy of the order of

t r* ) -# , (39)

according to Eq. (3a). When both energies 6 and t,'6 are comparable, we expect

that the radiative effects, ike the radiation reaction force' become important' From

Eqs. (38) and (39) we conclude that only when the external force is appliedso

sucldenly and for such a short time

/ , r ' , 1 /3r - ( , l p - - ) (40 )

\ Ibmc" /

that the radiative effects will modify the motion of the particle appreciable'It is

convenient to define the characteristic time of the moment pl by theexpression

/ , r r 1 / 3

, , , ( . r- \ (41)\ rDmc"

Therefore, for those forces applied during times T - T, the reactive effectsof the

magnetic moment will be important. If the particle identifies with anelectron then

the characteristic time r, takes the value

Tr": L01 x 10-22 s, 142)

which may be approximated by rr = .08rs. During this time the light travels thedistance d = 10.6 s, where 16 s the classical adius of the electron' The

value of the

characteristic time in Eq. (a2) is really surprising because the characteristictime

associatecl o the electric charge of the electron (" : 6'26x 10-24 s') issignificantll'

smaller compared',vith that due to its magnetic moment:

rrr = 16r". (43)


11/19

168 Josd A. Heras

5. An extension of the Abraham-Lorentz equation

In 2003 the present author derived [11] another alternative equation of motion

for a non-relativistic point particle possessing an electric charge and a magnetic

dipole moment:

ma: F**r. * rrrnit , - rfm"ti,

where r, is the characteristic time defined by Eq. (41) More exactly, Eq. (44)

is a non-relativistic equation of motion of a suddenly-accelerated point particle

including the radiation reaction force due to its magnetic moment (F""rr : -rflm'ri)

in addition to that due to its electric charge (F""u : r"mir). In obtaining trq. (44)

the vectorsp

and

'Zi

are assumed to be instantaneouslyperpendicular. Equation

(44) does not exhibit most of the defects (u)-(f) present in the Abraham-Lorentz

equation. Therefore Eq. ( a) also emerges ike an appropriate candidate to replace

the unsatisfactory Eq. (1). On the basis of trq. (aa)) we can conclude hat classical

electrodynamics of the point electron s internally consistent.

The radiation reaction theory associated to Eq. (44) starts by considering

suddenly accelerated, non-relativistic particle having both an electric charge and

magnetic rnoment. The electric radiation field is given by [11]:

(44)

(45)

(46)

(47 )

(4e)

(50)

a

a

| ( e (n x o ) , p (n . a ) l , lt,ud,1" t E +-;F-.fJ..,,

from rvhich the associated Poynting vector is obtained

q _ n e 2 ( n x c - ) 2 -"eF 4r R2 C3

'n ( n x p ) 2 ( n . i r ) 2 n e { n ' ( o t p ) } ( ' a )fw

If a and h are collinear and pe and a are perpendicular then Eq. (47) reduces to

2 e 2 , 4 t r 2 . ,P"t,: -#" ' t + #d'(iB)

Via de usual argument based on energy conservation 12,13], his power implies th e

radiation reaction force

4rR2c5

This yields the total instantaneous radiated power

F " p :

The associated equation of motion is given by

a - r e a + r ' z i : 0 .

P.t.,'$" ' - #r. @, +#a' - #0, ' a) '

2e2 4Lt' ,. .k t"

-t r rs

a '


12/19

Is classical electrodunarn'ics a,n, . . 169

The general solution of Eq. (50) is a linear combination of three complex solutionswith runaway and nonrunaway behaviors. Fortunately, under the initial conditionsa(0) : as, &(0) : -aolr and ii(O) -- aolr2, where the time r is given by

€ ' / t +a r? 2 r, { /3 ( 5 1 )6{ 1 / 3

)

with { : t082,?8r +12 3rf(27rfl - 4" ), t he solution f Eq. (50) s nonrunaway:

(52)

It should be noted that this solution was obtained under the point particle approx-

imation. This means that the existence of runaway solutions lies on the structureof the equation of motion rather than on the point particle approximation.

For arbitrary values of rr and r", the time r in Eq. (51) is complex and then

Eq. (52) is a complex solution. However, f either of the equivalent conditions

,rlr" > (4l2nr/3 or LLle (401243)112,

where r : e2 l(mc2) is the classical radius of the particle, is satisfied then the time

r in Eq. (51) is real and so Eq. (52) becomes eal. In particular an electron satisfies

the condition in Eq. (53): ,plr" = 16 or ptfe = 68.516, where 16 is the classical

radius of the electron. For an electron the time r in Eq. (52) takes the value

r : . 9 9 x 1 0 - 2 2 ,

(53)

(54)

(56)

which differs about 27o rom the t ime zu given n Eq. (41) , i .e . , , r : .98rr. This

result indicates that in sudden processes he self-accelerations of the electron are

predominantly caused by the radiation from its magnetic moment.

In the presence of an external force F"*1, Eq. (50) takes the form

t " ' F . * ta - T e a + T i O : ; . l .oo/

In particular, we consider again an electron urged by the external force F.*1(r) :

eF,se-tlT, where T ) rs. The associated equation of motion is given by

3 . . . e E o - t l Ta - r e a t T i a : ; e ' ' - .

In this equation we have the three forces: the self-force of the electric charge, the

self-force of the magnetic moment and the external force. Again the general solution

of Eq. (56) is a li near combination of three complex solutions exhibiting runaway

and nonrunaway behaviors. However, under the initial conditions

eEsT2 T - ,) eEsT(T2 - t2 )a ( 0 ) 3 , a(0)

* ( 7 3 * r u T 2 - r f i ) r ' a(0)-

*(73 * r"T2 - r | ) r z '(57)


13/19

I70 Josd A. Hera,s

the solution of Eq. (56) is not runaway:

, , \ e E g T 3 r - t / T - t / t ta( t ) ff i ( e - " ' - e - " ' ) . (58)

This is integrated with the initial condition o(0) :0 to obtain the velocity

. /+\ _ eEsT3 ["(1- e- t l r ) - r ( l - e- t /") )

J \ L )-

This velocity reduces to

, = 'EoT (1 r T )n7for t ) T, t > r, T )) r" and when 7 is significantly

T3 > r,l. For the classically allowed highest time, T - r^

, = 923"E.f .

iFrom the above equation we can extract the following conclusion:

Equation (55) predicts that for times t > T the velocity of the electron

caused by the external force F.*r(l) : eBoe-t/" reduces less than 7.77o on

account of the action of the radiation reaction force.

It should be emphasizecl hat the introduction of the magnetic moment of the

electron has increased about sixteen imes (from .57o o 7.7%) the maximum reduc-

tion of the velocity of the electron producecl by the external force. In other words:

For tlre external force F"*1(t) : eBse-t/T in the classical domain (T > zq), th e

maximum velocity of the self-force (t'""tr) predicted by both the integro-differential

Eq. (14) and the Ford-O'Connel l Eq. (23) :

Oself -.005o.*1 '

is only a tenth sixth part of the velocity of the self-force predicted by the generalized

Abraham-lorentz Eq. (44):

uself - '08t1"*,,

where o"*1 is the velocity predicted by the external force.

6. The effective equation of motion of the point electron

The small difference between the time rr, : l.0l x 10-22 of the magnetic mo-

ment of the electron and the time r : .99 x 10-22 of the self-force due to both

the electric charge and the magnetic moment of the electron has an unambiguously

interpretation: In sudden processes he self-accelerations of the electron are essen-

tially caused by the radiation from its magnetic moment and not by the radiation

(5e)

(60)

greater than rr so that= 132, Eq. (60) gives

( 6 1 )


14/19

Is classical elect'rodynamics an ' ' ' 17I

from its electric charge. This result changes undamentally the background of the

classical heory of the radiation reaction of the point electron in the sense ha t

during a century the reactive efi'ects of a suddenly-accelerated electron have been

associatecl o its electric charge by ignoring entirely its magnetic moment. The ai m

in this section is to reverse his traditional idea, i.e., to show that the reactive effects

of the suddenly-accelerated electron are essentially due to its magnetic moment.

The radiation electric field a suddenly accelerated, non-relativistic particle with

a magnetic moment ;,1 s given by [11,13]:

E-- , [nx P( : r a ' ) l (62)r a d -

L R " ' l ' u ' '

The associated Poynting vector Sp: @lafllE,.al2n is specifically given by

d n ( n x p ) 2 6 ' q 2eu -4nR2c5

This is usecl nto the instantaneous racliated power ctPL,ldQ: (Sr, 'r,)R2 to obtain,

after an integration over all solid angle, the total instantaneous radiated power

Pr":ffio'-#0".i,) 'via de usual argument based on energy conservation (see Refs. 12 and 13), Eq .

(64) implies the radiation reaction force

4u2 . . z . . . \Fp - I s " , o* - lS ;E lp . a )p .

In the case n which p and'zi

are perpendicular, Eq. (65) reduces o

Fp: ffi.a,which originates the following equation of motion:

a r r r

a -f r i o , : U,

where r, is the characteristic time defined by Eq. (41). The general solution of Eq'

(67) is a linear combination of runaway and nonrunaway solutions' However, if we

assume he initial conditions: a(0) - a6 a(0)- -a6fr, a(0) - aslrfi then th e

solution of Eq. (67) is nonrunaway:a ( t ) : a g e - t / r r "

For an electron the value of the characteristic time r* is given by Eq. (42)'

In the presence of an external force, Eq. (67) takes the general form:

(68)

a r , . Fa - f r I a - -- m

(63)

(64)

(65 )

(66)

(67)

(6e)


15/19

I72 Josd A. Heras

We consicier again an electron inside the electric field E(r) : F,se-t/T, where ? )

4. This electron is urged by F(t) : eEoe-t/" and then Eq. (69) takes the form

3 . . . e E o - t / Ta + r ; & - - e, ]TL(70)

(73

the solution of Eq. (70) is given by

, , \ e E 6 T 3 r - t / T - t / - ,a( t ) ;@ _A(u-" ' - e - ' , ' ) .

This is integrated with t'(0) : 0 to give the velocity

Under the initial conditions:

a(0) 0, a(0)eEsT2 a(0) - eF,s(72 - Trr)

mrr (72 Tr r+ r3 ) ' mrfr(T2Trr+rfr) '

eEsT3 ["(1 - e- ' l r ) - rp(7 - "- t / ' , ' ) ]m(T'3 - ri)

eE,"Tu = - ( 1 - r r l T ) .

ll L

when ? is significantly greater than r, so that ?3

lrighest time, T - r^ = 12.75rr, Ec1. 4) gives

( 7 1 )

(72)

a ( t ) :

which reduces o

f o r t ) T, t > r r a n d

the classically llowed

(74)

> rj, For

(75 ) x .g2r"F.T

iFrom the above equation we arrive ut tfrJruffowing conclusion:

Equation (69) predicts that for times t > T the velocity of the electron

caused by the external force F.*r(t) --eBse-t/? reduces less than 7.8Vo on

account of the action of the radiation reaction force.

It is important to note that the difference between the velocity in Eq. (61)

predicted by the "electric-magnetic" Eq. (56) and the velocity in Eq. (75) predicted

by the "magnetic" Eq. (70) is of order of 10-2. Therefore, with an error of only 2T o

we can ignore the racliative effects due to the electric charge Or in other worcls:

the reactive effects on a suddenly accelerated electron caused by its electric charge

and its magnetic moment can be approximated (module an error of 2%) by those

due only to its magnetic moment.

Therefore, we propose that Eq. (69) describes effectively the non-

relativistic motion of a classical electron under the combined action of a

sudden external force and of its induced self-force.


16/19

Is classical elect'rodyn,am'ics an, . . . 173

7. Conclusions

In this paper we have discussed our equations of motion for a non-relativistic

point electron under the combined influence of an external force as well as of its in-

duced radiation reaction force or self-force. Specifically, w€ have studied an electron

inside the electric field E(r) : E6s-tlT, where Z is the associated characteristic

time satisfying the condition of T 2 rs that guarantees hat the associated force

F"*t : eEge-tlT '

can be considered in the classical domain. The four equations and the relevant

conclusions are the following:

(A) The Abraham-Lorentz equation:

n7a: F"* t * T"miL,

lvhich is derived from energy conservation and the Larmor formula. This

equation predicts that for times t > T, the velocity of the electron caused

by the external force ('u : eEoTlm) increases ndefinitely on account of the

action of the radiation reaction force:

u(t )=- "P 2-" ' /" .n t \ l * T e )

The runaway behavior of this velocity is unacceptable from a physical point

of view, circumstance that leads to r-rs o pose the question: Is classical elec-

trodynamics an internally inconsistent theory?

(B) The integro-differential equation obtained from the Abraham-Lorentz equation

by assuming a suitable asymptotic condition:

o'lt) : ds e-"F"*1(t + sr") ,

where s: (t' _ t)1r". This equation predicts that for times t > T the velocity

of the electron caused by the external force (o : eEoTlm) reduces less than

.5Y on account of the action of the radiation reaction force:

= . 9 9 5 1 6 " E o ? .

The integro-clift'erential equation elimin",.rin" runaway accelerations but in -

troduces pre-accelerations.

(C) The Ford-O'Connell equation:

* ,*

ma: F.*t * T"Fu*t ,


17/19

I74 Jos6 A. Heras

which apparently describes an electron with structure. This equation predicts

that for times t > T the velocity of the electron caused by the external force

(u : eEsT/m) reduces ess han .5% on account of the action of the radiation

reaction force:

a = .gglL|,-E'r

The Ford-O'Connell equation does not p..I.t runaway accelerations.

(D) The generalization of the Abraham-Lorentz equation proposed by the author:

F"*t * r.md,-rf ;m'r i ,

which is derived from energy conservation and the powers radiated by theelectric charge and the magnetic moment of the electron. Actually, in this

equation the self-force due to the magnetic moment of the electron, the last

term of the equation, has been introduced. The generalized equation predicts

that for times t > T the velocity of the electron caused by the external force

(a : eEgT/nz) reduces ess than 7 7To n account of the action of the radiation

reaction force:

, = .g2J'EoT .

The generalized equation cloes not or"dl:, runaway accelerations provided

suitable initial conditions are assumed.

(E) The effective equation clerived by the author:

which is derived rrom enerJ:--:.::.";re power radiated the mag-

netic moment of the electron. This equation predicts that for times t ) T th e

velocity of the electron caused by the external force (tt : eBoTlm) reduces

less han 7.87o on account of the action of the radiation reaction force:

a x .g2rq ' r

The effective equation does not preclic, .llu*u, accelerations providecl suit-

able initial conditions are assumed.

(F) From the comparison of the two equations proposed by the author, it followsthat in sudden processes he self-accelerations of the electron are predomi-

nantly caused by the radiation from its magnetic moment and not by the ra-

diation from its electric charge. Therefore, the reactive effects on the electron

predicted by both the integro-dift'erential Eq. (14) and the Ford-O'Connell

Eq. (23) may be ignored (for a suddenly external force) with respect to those

predicted by the effective Eq. (69).


18/19

Is cl,assi,cal el,ectrodun,amics an, . . . 775

(G) For external forces acting on the classically allowed highest time ? - rr, both

equations proposed by the author restore reasonably the internal consistency

of classical electrodynamics and place the time scale of the electromagnetic

reactive effects on an electron about sixteen times greater than that of the

Abraham-Lorentz and also of the Ford-O'Connell equation rc.26 x 10-24) s,

i.e, at the order of a tenth of one zeptosecond:

Time scale for the radiation reaction on an electron - .7. ,

rvhere Iz : 70-2r s. For an external force produced by an electromagnetic

pulse at the scale of zeptoseconds, he reduction of the velocity caused by

the radiation reaction force willreach the "classical"

imitof 7.7%. Optical

scientists are now striving to produce, via the lasetron, pulses at the scale of

zeptoseconds [27]. Accordingly, with the lasetron the reactive effects of the

magnetic moment of the electron could be tested.

(H) Is classical electrodynamics an internally inconsistent theory? Response: ly'of,

rvhenever we consider the radiation emitted bv the magnetic moment of the

electron.

References

1. J. D. Jackson, Classi,cal Electrodynam'ics,3rd trdition, Wiley, New York, 1999.2. P. A. M. Dirac, "Classical heory of radiating electrons," Proc. Roy. Soc. London A

L67, 148-169 1938).3. R. Blanco, "Nonuniqiueness f the Lorentz-Dirac equation with the free-particle as -

ymptotic condition," Phgs. Reu. E 51, 680-688 1995).4. E. J. Moniz and D. H. Sharp, "Absence f runaways and divergent self-mass n nonrel-

ativistic quantum electrodynamics," hgs. Reu. D 4, 1133-1136 1974).5. E. J. Moniz and D. H. Sharp, "Radiation reaction n nonrelativistic quantum electlcr

dynamics," Phys. Reu. D 15 2850-2865 1977).6. H. Levine, E. J. Moniz and D. H. Sharp, "Motion of extended charges n classical

electrodynamics," Am. J. Phys. 45,75-78 (1977).7. G. W. Ford and R. F. O'Connell, "Radiation reaction in electrodynamics and the

elimination of runaway solutions," Phys. Lett. A L57,277-220 1991).8. G. W. Ford and R. F. O'Connell, "Relativistic orm of radiation reaction," Phgs. Lett.

A L57, 182-184 1993).9. tr. Rohrlich,"The correct equation of motion of a classical point charge," Phgs. Lett.

A 283,276-278 (2001).10. F. Rohrlich,"Dynamics f a classical uasi-point harge," Phys. Lett. A 303,307-310

(2002).11. J. A. Heras,"The radiation reaction orce on an electrorr eexamined," PhEs. Lett. A

3L4,272-277 (2003).12. J. A. Heras,"A new approach o the classical adiation fields of moving dipoles," Phys.

Let t . A 237,343-3 48 1998) .13. J. A. Heras,"Explicit expressions or the electric and magnetic ields of a moving mag-

netic dipole," Phys. Reu. E 58 ,5047-5056 1998).


19/19

176 Jos6. A. Heras

14. D. Griffiths, Introducti,on to electrodgnan'Lt,cs Prentice-Hall, Englewood, NJ, 1999)3rd. Ed., p. 465.

15. C. L. Bennet, "Evidence or macroscopical ausality violation," Phys. Reu. A 35 ,240e-241e 1e87).

16. F. Rohrlich,"The validity limits of physical heories: esponse o the preceding Letter,"Phys. Lett. A 295,320-322 (2003).

17. G. N. Plass, "Classical electrodynamic equations of motion with radiative reaction,"Reu. Mod. Phys. 33 ,37-62 (1961) .

18. C. J. Eliezer, "On the classical heory of particles," Proc, R. Soc. A I94,543-555(1e48) .

19. L. D. Landau and E. M. Lifshitz, The class'ico,l heorE of Fi,elds (Pergamon, NewYork, 1975) 4th Ed .

20. H. Spohn, "The critical manifold of the Lorentz-Dirac equation," Ettrophgs. Lett. 5O

287-2e2 2000).2L G, W. Ford and R. F. O'Connell, "Total power radiated by an accelerated harge,"Phys. Let t . A 158, 31-32 1991) .

22. G. W. Ford and R. F. O'Connell, "Structure effects on the radiation emitted from anelectron," Phys. Reu. A 44 ,6386-6387 1991).

23. R. F. O'Connell, "The equation of motion of an electron," Phys. Lett. A 3L3,497-497(2003) .

24. M. Ribaric and L. Sustersic, "Qualitative properties of an equation of motion of aclassical oint charge," Phgs. Lett. A 295, 318-319 2002).

25. W. E. Baylis and J. Huschilt, "Energy balance with the Landau-Lifshitz equation,"Phys. Lett. A 3OL,7-12 (2002).

26. J. A. Heras, "Radiation fields of a dipole in arbitrary motion," Am. J. Phys.6211 0 9 - 111 5 1 9 9 4 ) .

27. A. E. Kaplan and P. L. Shkolnikov, Lasetron: a proposed ource of powerful nuclear-time-scale lectromagnetic ursts," Phys. Reu. Lett.88 074301 2002).

artículo electro

Documents