Extended Electron in Constant Electric Field Part 1 : The net electric force Fe produced on the extended electron

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1 1 Exteded Electro i Costat Electric Field Part 1 : The et electric force Fe produced o the exteded electro Hoa Va Nguye hoaguye2312@yahoo.ca Abstract : Whe a exteded electro is subject to a exteral costat electric field E, the et electric force F e is produced o it. The aalysis o the exteded model of the electro shows that F e is the resultat of two opposite forces F ad F ( i.e., F e = F + F ), where F is the resultat of all elemetary forces f e which are produced o surface dipoles of the electro ( F = f e ), ad F is the electric force produced o the core (-q 0) of the electro. We will calculate fe, F ad F by applyig boudary coditios o the surface of the exteded electro. The effective electric charge Q of the electro will be deduced from the expressio Fe = Q E. Part 2 discusses the radiatio process of the exteded electro i the electric field. Keywords : electric dipoles, surface dipoles, the core, electric ad magetic boudary coditios, radiatio by forces, radiatio coe, coditios for radiatio. 1. Itroductio : The readers are recommeded to read the previous article 1(a) : "A ew exteded model for the electro " to have a view o the exteded model of the electro ad the assumptios for calculatios ; sice all the calculatios i this article will be based o this model ad the assumptios o the electric ad magetic boudary coditios. I a utshell, the exteded model of the electro is a spherical composite structure cosistig of the poit-charged core ( -q 0 ) which is surrouded by coutless electric dipoles ( -q,+q ) as schematically show by Fig.1 ad other figures i this article. Sice this article " Exteded electro i costat electric field " is rather log, it will be divided ito two parts : Part 1 : Determiatio of the et electric force Fe, Part 2 : The radiatio of the exteded electro i electric field. To reduce the legth of the mai text, log calculatios are put ito the appedices, oly the results of these calculatios are show i the mai text. ( Appedices A ad B are placed at the ed of part 2, the readers ca read them later if they wat to ). Whe a poit electro of electric charge e is subject to a exteral electric field E, the Loretz force F L = e E is a sigle force ; i.e., it caot be decomposed ito simpler compoets. But for a exteded electro, the et force Fe is more complicated because it cosists of coutless poit charges : the electric dipoles (-q, +q ) ad the core ( -q 0 ), ad hece lots of sigle forces are produced o it whe it is subject to the exteral field. To determie the et electric force Fe produced o the exteded electro, we will first calculate the "elemetary" force fe which is developed o a surface dipole, the the resultat F = fe of all forces fe produced o all surface dipoles of the electro ( sectio 2 ). The we will calculate the electric force F' which is produced o the core ( -q 0 ) of the electro ( sectio 3 ), ad fially the et force Fe = F + F' ( sectio 4). Sectio 5 discusses the variability of the relative permittivity ε of the exteded electro. Sectio 6 discusses the variability of the effective electric charge Q of the exteded electro. Sectio 7 discusses the equatio of motio of the exteded electro i the exteral electric field. 2. Determiatio of the elemetary force fe produced o a surface dipole ad the resultat F = fe.

2 2 Let E be the exteral electric field which exerts o the exteded electro ; E creates the electric field E' iside the exteded electro. Boudary coditios 1(a) o the electric field allow us to determie the electric field E' from the compoets of the exteral field E which is supposed to be a kow field. First, let us calculate two elemetary electric forces fe produced o two arbitrary surface dipoles M ad N o the upper ad lower hemispheres of the electro, respectively. ( Fig.1 ) For the surface dipole M o the upper hemisphere, the agle α is 0 α π /2 ( Figs. 4 ad 5 ). The force fe is the resultat of two forces f ad f which are developed o two eds -q ad +q of the dipole M, respectively. The calculatios are show i the Appedix A. The results are : * whe 1 ( Fig. 4), fe = f - f = ( 1/ - 1 ) q E cos, fe is cetrifugal, (1-a ) * whe 1 ( Fig. 5 ), fe = f - f = ( 1 1/ ) q E cos, fe is cetripetal, (1-b) Where is the relative permittivity of the exteded electro ( ' ) to the free space ( 0 ) ; i.e., = ' / 0. fe F.1 : Two surface dipoles M ad N o the upper ad lower hemispheres of the exteded electro. ( Error : permittivity of free space F.4: Whe 1 : F.5 : Whe 1 : outside the electro is 0, ot the resultat force fe actig the resultat force fe actig as show i F.1 ). o dipole M is cetrifugal. o dipole M is cetripetal. fe Now let us calculate the resultat force fe actig o the surface dipole N o the lower hemisphere of the electro ( Figs. 8 ad 9 ) : /2, ( hece cos 0 ) ( See Appedix A ). The results are : * whe 1 ( Fig. 8 ) : fe = f - f = ( 1/ - 1 ) q E ( -cos ), fe is cetripetal, (1-c ) * whe 1 ( Fig. 9 ) : fe = f - f = ( 1 1/ ) q E ( -cos ), fe is cetrifugal. (1-d) fe fe

3 3 F.8 : Whe 1 : the resultat force fe F.9 : Whe 1 : the resultat force fe actig o dipole N is cetripetal. actig o dipole N is cetrifugal. Now let us combie the results from four figures above to show the directios of the elemetary forces fe produced o two surface dipoles M ad N : * whe 1 : Fig. 10 shows two forces fe at M ad N ; ad Fig.10-a ( from Fig.10) shows all forces fe o the upper ad lower surface of the electro. * whe 1 : Fig. 11 shows two forces fe at M ad N ; ad Fig.11-a ( from Fig.11) showig all forces fe o the upper ad lower surface of the electro. F.10 : Whe 1: F.11 : Whe 1: - o the upper hemisphere, fe is cetrifugal - o the upper hemisphere, fe is cetripetal - o the lower hemisphere, fe is cetripetal - o the lower hemisphere, fe is cetrifugal F.10-a ( from F.10 ) Whe 1 : F.11-a ( from F.11 ) Whe 1 : - o the upper hemisphere : all fe are cetrifugal - o the upper hemisphere : all fe are cetripetal - o the lower hemisphere : all fe are cetripetal - o the lower hemisphere : all fe are cetrifugal

4 4 We have fiished determiig the magitude ad the directio of the elemetary forces fe i two cases whe 1 ad 1. Now we will determie the directio ad magitude of the resultat force F = fe which exerts o the surface dipoles of the exteded electro i these two cases. F.12: 1 : F.13 : 1 : Projectios of fe at M ad N oto Projectios of fe at M ad N oto the axis OE : f M ad f N poit upwards. the axis OE : f M ad f N poit dowwards. Because all forces fe are symmetric about the axis OE ( as show i two Figs.10-a ad 11-a ), their resultat force F = fe lies o OE ad is equal to the sum of all projectios of fe oto OE. So, to determie the magitude of F, we have to calculate first the projectio of fe oto OE. Let f M be the projectio of fe at M oto the axis OE ad f N, the projectio of fe at N oto OE. Whe 1 ( Fig. 12 ), f M ad f N poit i the directio of E ; their magitudes are : f M = fe cos = ( 1/ -1 ) q E cos cos = ( 1/ -1 ) q E cos 2 (1) f N = fe cos = fe (- cos ) = ( 1/ -1 ) q E (- cos )(- cos ) = ( 1/ -1 ) q E cos 2 (2) The magitudes of f M ad f N i two Eqs. ( 1 ) ad (2) have idetical form but i Eq.(1) : 0 /2 ad i Eq.(2) : /2. So, F = fe = f M if varies from 0 to ; that is F = i1 ( 1/ -1 ) q E cos 2 i = ( 1/ -1 ) q E i 1 cos 2 i (3) F poits i the directio of E ( Fig.14 ) where is umber of surface dipoles o the surface of the electro ; i is the agular positio of a surface dipole, varyig from 0 to ; is the relative permittivity of the electro : = / 0 ; q is electric charge at the ed of a surface dipole ; Whe 1 ( Fig. 13 ) f M ad f N poit i the opposite directio to E, so the resultat F = fe = f M also poits i the opposite directio to E. The same reasoig as above gives F = i1 ( 1-1/ ) q E cos 2 i = ( 1-1/ ) q E F poits i the opposite directio to E ( Fig.15) i 1 cos 2 i (4)

5 5 F.14 : 1: the resultat force F ( = fe ) F.15 : 1: the resultat force F ( = fe ) poits i the directio of E. poits i the opposite directio to E. I two equatios (3) ad (4), the sum i1 cos 2 i depeds o the umber of surface dipoles ad the agular positio i of the surface dipole, they represet the physical structure of the electro. The magitude ad directio of F thus deped o as show i Eqs. (3) & (4) ad two Figs.14 & 15. The magitude ad directio of F ( with respect to the directio of E ) is graphically show i Fig.16 We have fiished determiig the magitude ad directio of the resultat F ( = fe ). F.16 : The smooth curve shows the magitude ad directio of F versus : Eq. (3) - for 1 : F is positive ; i.e., F E as show by F for 1 : F is egative ; i.e., F E as show by F. 15

6 6 From two equatios (3) ad (4) we ote that F exists oly if 1 ; this meas that the absolute permittivity of the exteded electro is differet from the permittivity 0 of the surroudig free space ( 0 ) ; that is, they are two differet media. Otherwise, if = 1, the = 0 ad F = 0 ; this physically meas that the electro has o electric dipoles aroud its core ; i.e., oly the core exists : this is a poit electro. Therefore, mathematically, the poit electro is a particular case of the exteded electro whe = 1. For the exteded electro 1 as we will see below. 3. Determiatio of the et electric force produced o all iterior dipoles ad the electric force F produced o the core ( q 0 ) of the electro 3.1 The et electric force produced o all iterior dipoles is zero. The exteral electric field E produces the field E iside the exteded electro ; all iterior dipoles ad the core ( - q 0 ) are thus subject to the field E. Because the dipole is extremely small, two eds q, +q of a iterior dipole are exerted upo by the same field E ad hece two electric forces fe ad f e are equal ad opposite. The dipole may be slightly re-orieted ( or polarized ) but the resultat force fe + f e produced o the dipole is cacelled out. This explaatio applies to all iterior dipoles, which are possibly polarized by the field E, but the et electric force produced o all iterior dipoles is zero. 3.2 Determiatio of the electric force F produced o the core (-q 0 ) of the electro. The core is a poit charge ( -q 0 ) which is subject to the field E. To determie the electric force F' produced o the core, we have to calculate field E first. The calculatio is show i Appedix B. It shows that the electric field E' is parallel to the exteral field E ad has magitude equal to : E = ( 1/) E (12) So, the electric force F produced o the core (-q 0 ) is F = - q 0 E = - (1/) q 0 E (13) F is thus always egative ; i.e., it always poits i the opposite directio to E ( Fig.17). Fig.18 shows the variatio of F vs accordig to Eq.(13). F.17 : E is the exteral electric field ; F.18 : The curve represets the magitude E is the electric field produced o the core (- q 0 ), ad directio of the electric force F versus. F is the electric force produced o the core, F is always egative ; i.e., F E. it is always egative ; i.e., F E.

7 7 4. Calculatio of the et electric force F e produced o the exteded electro The et electric force Fe which is developed o the exteded electro whe it is subject to a costat electric field E is the sum Fe = F + F. F is give by Eq.(3) ad F' by Eq.(13). Sice F ad F are colliear, their algebraic sum is 1 1 Fe = ( - 1) q E cos 2 i - q0 E i (14) 1 1 or Fe = [ ( - 1) (q / q0 ) cos 2 i - ] q0 E (15) Let s set a (q / q 0 ) i i cos 2 i, ( it is a factor iside the above square bracket) (16) a is thus a dimesioless positive umber sice q, q 0 ad Eq.(15) becomes i cos 2 i are positive umbers ; Fe = ( a 1 - a) q 0 E ( Fig. 19 ) (17) Note : If we isert a from (16) ito the expressio of F i Eq.(3), we get a simpler expressio for F : F = ( 1-1) a q0 E, i which the parameter 'a' absorbs the charge q ad the sum i cos 2 i. Fe (+) E Fe F a1 -q oe -aq oe 0 1-1/a 1 (-) Fe F F = ( 1-1) a q0 E, ( F.16 ) F = - (1/) q 0 E, ( F.18 ) Fe = F + F' = ( a 1 - a) q 0 E, ( F.19 ) F.19 : Fe = F + F. For 1 : F ad F are both egative ; Fe is always egative, i.e., Fe poits i opposite directio to E For 1 1/a 1 : Fe is egative ad teds to zero as 1 1/a For 1 1/a : Fe becomes positive ; i.e., Fe poits i the directio of E : the exteded electro behaves like a positro. We have thus calculated the et electric force Fe that is produced o the exteded electro whe it is subject to the exteral electric field E. The magitude ad directio of Fe deped o two parameters ad 'a' ; their actual ( or physical ) itervals of variatio are discussed i the sectio 5 below.

8 8 We otice that for a poit electro = 1, F = 0, F = - q 0 E, ad hece Fe = F + F = - q 0 E This is the familiar value of the electric force produced o the poit electro : Fe = e E. But for a exteded electro the magitude of Fe is modified by the factor ( a 1 - a ) which depeds o two parameters 'a' ad. Ad thus the electric force Fe does ot vary liearly with E ( as the Loretz classical equatio Fe = e E ). The parameter 'a', as defied by the expressio (16), characterizes the physical structure of the electro, ad depeds o q, q 0, ad i. q, the magitude of electric charge o the ed of a dipole ;. q 0, the magitude of electric charge of the core ;., total umber of surface dipoles o the surface of the electro ;. i, represetig the agular distributio of the surface dipole i o the spherical surface of the electro; the agle varies from 0 to. 5. The actual ( physical ) itervals of variatio of ad a Let us examie Fig.19 which shows the variatio of the magitudes of F, F ad their sum Fe versus. If = 1, F = 0, F = - q 0 E, Fe = F + F = - q 0 E = F : the et force Fe is thus reduced to the force F which is produced o the core of the electro ; this meas that whe = 1 the exteded electro is mathematically equivalet to a poit charge which is the core ( as already oted at the ed of sectio 2 ). So, for the exteded electro, 1 ; that is, either 1 or 1. For 1, both F ad F are egative, ad hece Fe is always egative. This meas that the et electric force Fe produced o the electro is always differet from zero, ad as a cosequece, from the Newto s secod law of motio, the acceleratio is always differet from zero ; ad hece the velocity of the electro will icrease to ifiity with time : this is the ruaway behaviour. Sice the velocity of ay particle must be limited to c, so we reject the iterval of variatio of 1 For 1, F becomes positive while F is always egative ; Fe = F + F : ( Fig.19) * Fe 0 i the iterval (1-1/a ) 1 : Fe poits i the opposite directio to E ; this meas that the exteded electro behaves like a real electro. * Fe 0 for (1-1/a ) : Fe poits i the directio of E ; i.e., the exteded electro behaves like a positive electro or a positro, so we reject this iterval. Thus we accept the iterval (1-1/a ) 1, for which Fe is egative, it accelerates the electro i the opposite directio to the exteral field E. This iterval will be writte shortly as 1. Figs. 4, 8, 10, 10-a, 12, ad 14 satisfy this coditio ( 1 ). They show the directios of the elemetary forces fe ad the resultat F ( = fe ) : F is upward ( positive ). Fig. 17 shows F' always dowward ( egative ). Sice F ad F vary with, whe = 1-1/a, F = - F ad Fe = F + F = 0 ( Fig.19 ) : the electro reaches the speed limit c. We ote that the speed limit c is a postulate of the special theory of relativity : it has o physical iterpretatio. But here we ca explai the physical reaso why the exteded electro's speed teds to c : whe it is accelerated by a exteral electric field E, the magitudes of two opposite forces F ad F' ted to be equal whe = 1-1/a, ad the resultat force Fe 0. From the Newto's secod law of motio : if Fe = m dv /dt 0, the dv /dt 0 ad v teds to a costat that must be c because the electro is beig accelerated. So, the variability of two opposite forces F ad F' ( with ) helps explai why the speed limit c is reached. I short, sice varies i the iterval (1-1/a ) 1, we come to the followig results :

9 9 - The correctio factor ( a 1 - a ) of the electric force Fe i Eq.(17) varies from -1 to 0. - The magitude of the et electric force Fe varies from q 0 E to 0. - The velocity of the electro varies with ad teds to c as 1-1/a. - Sice the relative permittivity is a positive umber, 1-1/a meas that 1-1/a 0 i.e., 'a' is greater tha uity (a 1). Four remarks related to the values of : 1. The positro. I this article we oly ivestigate the properties of the electro ( ot the positro ) ; this is whe varies i the iterval (1-1/a ) 1, where a 1. I the followig, this iterval is writte shortly as 1. From Fig.19 we otice that whe 1-1/a, the electro is accelerated i the directio of the applyig field E ; i.e., the electro behaves like a positro. This trasitio suggests that e - ad e + ca be trasformed from oe to the other if its relative permittivity is beig chaged aroud the value (1-1/a) where F = - F, Fe = 0 ad its speed is about c. [ For compariso, let us recall that Dirac revealed from his relativistic equatio of the electro that the electro ca have two eergy states : positive ad egative. He wrote i his Nobel lecture 2 : if we disturb the electro, we may cause a trasitio from a positive-eergy state of motio to a egativeeergy oe... ; they correspod to the motio of a electro with a positive charge istead of the usual egative oe - what the experimeters ow call a positro ". Thus, for Dirac, e - ad e + are ot idepedet particles sice e - ca be trasformed to e + by a disturbace ( he did ot specify what kid of disturbace ) ; i other words, the positro has its origi from the electro ]. The calculatios of forces o the exteded electro lead to Fig.19 which shows the variatio of the et electric force Fe with ; it is the chage of about the value (1-1/a) that trasforms e - to e + ad vice versa, so the disturbace here is the abrupt chage of. Moreover, the positro is ot created aloe, but together with the electro i the pair productio e - - e +. This suggests that the positro may have the origi from the electro. 2. Why 1? We have come to the result 1 such that the velocity of the electro is limited to c whe it is accelerated i E ; ( otherwise, if 1, it will ruaway ; i.e., its velocity teds to ifiity). Moreover, whe 1, Fe is egative, it accelerates the electro i the opposite directio to the applyig field E, this is a feature of the real electro. This value of relative permittivity 1 is specific to the material of the exteded electro because commo materials such as water, glass, mica all have 1 ; e.g., for air = 1, water ( 81), glass ( 5-10 ), mica (6 ) ad o material with 1 is metioed or listed i the curret textbook 3. This may be because the exteded electro is composed of electric dipoles meawhile commo materials are composed of atoms or molecules. 3. The orietatio ( or polarizatio ) of electric dipoles Now from Eq. (12) : E = ( 1/) E, sice 1, E E. This meas that whe a exteral electric field E is applied o the exteded electro, it creates a electric field E parallel ad stroger iside the electro. The reaso for this is that the applyig field E polarizes all electric dipoles iside the electro, givig rise to the field E greater tha E. This polarizatio causes the permittivity of the electro to vary i the iterval (1-1/a ) 1. We ca say that is a measure of the sesibility of the structure of the electro to the applyig field E.

10 10 Sice electric dipoles cotai both charges ad +, the polarizatio of the esemble of electric dipoles i the exteded electro creates a electric dipole momet for the exteded electro 4, 5. A poit electro with oly egative charge caot have a electric dipole momet. 4. The arrow iterval of variatio of the relative permittivity The parameter a is a dimesioless positive umber defied by the expressio (16) : a (q / q 0 ) i cos 2 i The magitude of a icreases with the umber of surface dipoles because i cos 2 i is the sum of positive terms ( cos 2 i ). For a exteded electro, the umber of surface dipoles is expected to be large ( 1 ) ; ad hece, a is also expected to be a large umber (a 1). If a 1, the 1/a is a ifiitesimal umber, ad hece ( 1-1/a ) is ifiitesimally less tha 1 ; therefore, the iterval (1-1/a ) 1 becomes a very arrow iterval (1-1 ). Ad hece, we ca say that is approximately costat ad equal to 1-, ( 1- ). (This approximatio is eeded for calculatios i the Appedix C ). We coclude that whe the exteded electro is subject to the exteral field E, its permittivity varies i the arrow iterval (1-1/a 1) which causes the et electric force Fe to chage i the iterval ( -q 0 E, 0 ) ad the effective electric charge Q of the exteded electro varies i the iterval iterval ( - q 0, 0 ) as we see below. 6. The variability of the electric charge of the exteded electro From Eq.(17) which gives the et electric force Fe Fe = ( a 1 - a) q 0 E ( show i Fig.19 ) the effective electric charge Q of the electro ca be deduced as Q = ( a 1 - a) q 0 a 1 (18) From Fig.19, for the electro ( Fe is egative ), varies i the iterval (1-1/a ) 1 ad hece Q is egative ad varies i the iterval ( - q 0, 0 ). To fid the correspodece betwee the velocity v with ad Q, let us recall that o page 6 of the previous article etitled 1(b) A Foudatioal Problem i Physics : Mass versus Electric Charge ", we have obtaied the geeral expressio for the effective electric charge of the electro whe it is subject to a exteral field : Q = - ( 1 v 2 / c 2 ) N/ 2 q 0 (19) where N 0 is a real umber represetig the applyig electric or magetic field. A mius sig is added o the right had side of Eq.(19) to idicate the egative charge Q of the electro. From (18) ad (19) we get

11 11 a 1 - a = - ( 1 v 2 / c 2 ) N/ 2 (20) or = (a 1 ) / [ a - ( 1 v 2 / c 2 ) N/ 2 ] (21) So, whe v c, 1, Q - q 0 ad Fe - q 0 E (equivalet to a poit electro) ad as v c, 1 1/a, Q 0 ad Fe 0. So, whe it moves at v c, the exteded electro is mathematically equivalet to a poit electro (i terms of electric charge ad force, both are costat ). But as v c it behaves differetly from the poit electro : its effective charge Q ad the et force Fe chage with velocity, i.e., with time, while it is accelerated i the electric field E. We ote that Eq.(20) provides a relatioship betwee three parameters 'a', ad the velocity v of the electro. The Appedix C shows that this equatio ca suggest a way to relate the radiatio ad absorptio of light of the exteded electro to its velocity ; that is, whe the electro radiates, it slows dow ; ad whe it absorbs light, it speeds up. 7. Equatio of motio of the exteded electro i exteral electric field E Equatio of motio of the exteded electro i the form of the Newto's 2d law of motio is m v. = Fe = F + F = ( a 1 - a ) q 0 E from Eq.(17) (22) or m v. = - ( 1 v 2 / c 2 ) N/ 2 q 0 E from Eq.(20) (23) where v. (= d 2 x/ dt 2 ). Hece, the equatio (23) is a secod order differetial equatio which obeys the Newto s first law of motio ( law of iertia ) ad avoids the problems of ruaway ad preacceleratio. We should ote that the radiatio reactio force F rad ( which is the recoil force produced by the radiatio of a acceleratig poit charge ) does ot exist i the radiatio of the exteded electro. The force F which is produced o the core ( - q 0 ), is ot the radiatio reactio force produced by the radiatio of the exteded electro. Meawhile the equatio of motio of the poit electro ( Loretz-Abraham-Dirac equatio ) icludig the radiatio reactio force F rad is a third order differetial equatio m v. = F ext + F rad (24) where F rad = 2q 2 v.. / 3c 3 ( i cgs ) ; v.. (= d 3 x / dt 3 ) is the secod time derivatives of the velocity v. Eq.( 24) does ot obey the law of iertia ad is ivolved i may problems icludig the ruaway ad preacceleratio 6. Summary ad coclusio More tha 100 years ago, may physicists had proposed various exteded model for the electro : Loretz ( 1904 ), Poicaré ( 1906 )... All these models cotaied oly egative charges. The ew exteded model cotais both types of charges ( + ad - ) i its volume. It is a versio 1(a) of the image of the screeed electro by vacuum polarizatio such that calculatios ca be performed o it to determie its effective electric charge. We have come to the followig results :

12 12 Whe the electro is subject to a exteral electric field E, two opposite electric forces F ad F' are produced o it : F is the resultat of all elemetary forces fe, F' is produced o the core (- q 0 ). The et force Fe = F + F' accelerates the electro i the opposite directio to E, varies with velocity ad teds to zero as v c. The effective electric charge Q chages with the velocity ad teds to zero as v c. The ivestigatio o this exteded model leads us to followig ideas : 1/ The cocept of screeed electro by vacuum polarizatio 1(a) should be abadoed sice it does ot provide ay form of calculatio to determie the effective electric charge of the electro. The virtual pairs of ( e -, e + ) are fictitious, they must ot have real effect o the electro. Whereas electric dipoles or photos are real particles (as coceived by Feyma), which costitute the real part of the electro aroud its cetral core ( -q 0 ) istead of those fictitious pairs of ( e -, e + ). 2/ The old-fashioed cocept of mass varyig with velocity 1(b) should be abadoed ad replaced by the cocept of varyig effective charge which is derived from the ew exteded model of the electro. 3/ The exteded model of the electro ca help explore more experimetal properties of the electro, such as its radiatio ad spi i exteral electric ad magetic fields. Part 2 : Radiatio of the exteded electro i costat electric field. 1. Itroductio : radiatio by forces Radiatio process is a tough topic to discuss because of the dual ature of light ad the ukow structure of the electro that emits light. I this part 2, the exteded model of the electro 1(a) will be used i the discussio of the radiatio process. As for light, it is assumed that the electro emits light, ad light may be particles ( called photos), as coceived by Feyma* or it may be a chuk of self-sustaiig field** which detaches from the electro ad travels through space, as coceived by the classical theory of radiatio. I this article ad subsequet oes, we choose to cosider light as particles which are idetified as tiy eedles carryig two opposite charges -q ad +q at their two eds ; these "electric dipoles" form the outer part of the exteded electro 1(a). Whe the electro emits its electric dipoles ito the surroudig space, we say it is radiatig. The followig sectios will itroduce the readers to a ovel way of explaiig the radiatio process : this is radiatio by forces. 2. Radiatio of the exteded electro by electric forces Before discussig the radiatio process ad defiig the coditios for radiatio, let us review two electric forces fe ad G which are produced o surface dipoles of the exteded electro whe it is subject to a exteral costat electric field E. These two forces will help explai the radiatio process. First, the elemetary forces fe produced o all surface dipoles are show i F.10-a whe 1 : fe are cetrifugal o the upper hemisphere ad cetripetal o the lower hemisphere ; their magitude is * Feyma : " I wat to emphasize that light comes i this form particles. It is very importat to kow that light behaves like particles, especially for those of you who have goe to school, where you were probably told somethig about light behavig like waves. I m tellig you the way it does behave like particles. ( Optics, E.Hecht, p.138 ) ** " If the poit charge is subjected to a sudde acceleratio caused by some exteral force, the pieces of the electric ad magetic fields break away from the poit charge ad propagate outward as a selfsupportig electromagetic wave pulse. " ( Classical Electrodyamics, 1988, H. C. Ohaia, p. 411 )

13 13 fe = ( 1/ - 1 ) q E cos, 1, 0 Secod are the cohesive forces G which attract all surfaces dipoles toward the core of the electro ; i.e., they are cetripetal ad produced by the self-field E 0 of the electro ; their magitude is G = [(1/) 1] q E 0, Eq.(16) which had bee calculated i Fig.19 of the article 1(a) "A New Exteded Model for the Electro ". These two forces fe ad G are show together i Fig.20 ad Fig.21 below. O the upper hemisphere, while fe are cetrifugal, the cohesive forces G are cetripetal. So, if the magitude of fe is stroger tha that of G ( fe G ), these surface dipoles ca break away from the surface of the electro ; that is, the electro emits these electric dipoles upwards. This meas that the electro radiates upwards, from the upper hemisphere, while the electro is movig dowwards i the opposite directio to the applyig field E. Meawhile, o the lower hemisphere, sice both fe ad G are cetripetal, these surface dipoles caot break free from the surface of the electro ; that is, there is o radiatio from the lower hemisphere Fig. 20 shows whe 1 : - electric forces fe : cetrifugal o the upper hemisphere ad cetripetal o the lower hemisphere, - cohesive forces G : all are cetripetal, - electric force F actig o the core ( q 0 ), - the directio of the applyig field E. F.21 shows whe 1 : the radiat zoe limited by the agle o where fe 0 = G. The coe of radiatio is upward ( i the directio of E ) while the electro moves dowwards.

14 14 I short, the exteded electro does ot radiate from its etire surface, but oly from a limited zoe aroud the orth pole, o the upper hemisphere of the electro as show i Fig.21. I the followig, we will determie this radiat zoe aroud the orth pole ad defie the coditios for radiatio of the exteded electro i the electric field E. 3. The coe of radiatio - Coditios for radiatio. Now let us determie the radiat zoe o the upper hemisphere as show i Fig.21. Let us recall that the magitude of fe produced o a surface dipole M of the upper hemisphere has bee calculated i part 1 ( Fig.4 ) ; fe is cetrifugal, its magitude is fe = ( 1-1 ) q E cos, where 1 ad 0 /2 (25) fe thus depeds o the agular positio of the surface dipole. Let 0 be the value of the agle at which fe 0 = G ( i magitude ) (26) the the coditio fe G ( for the radiatio to occur as stated above) is rewritte as fe fe 0 or ( 1-1 ) q E cos ( 1-1 ) q E cos 0 (27) or cos cos 0 or 0 (28) Fig.21 shows the radiat zoe o the upper hemisphere, restricted i a zoe aroud the orth pole of the electro, limited by the agle 0 which is defied by the relatio (26). All rays radiatig from the radiat zoe are cotaied iside a radiatio coe of half- agle 0 at the vertex O. So, the existece of the agle 0 meas the existece of the radiatio coe. Now let us determie the coditio for the existece of the agle 0. From (26) we have ( 1-1 ) q E cos 0 = G where E is the applyig electric field. The magitude of G has bee calculated to be equal to G = ( 1-1 ) q E0 where E 0 is the self field of the electro. Hece ( 1-1 ) q E cos 0 = ( 1-1 ) q E0 or cos 0 = E 0 / E 1 (29) The coditio for the radiatio to occur is thus E E 0 (30) Therefore, i) if E E 0, cos 0 1 ; 0 does ot exist : o radiatio emits from the electro while it is accelerated i E ; this also meas that i free space ( E = 0 ) the electro caot radiate. ii) if E = E 0, cos 0 = 1, 0 = 0 : the radiatio coe reduces to a ray ( or a beam ) of light emaatig from the orth pole of the electro. iii) if E E 0, cos 0 1, 0 0 /2 : radiatio occurs aroud the orth pole of the

15 15 electro ; a radiatio coe emaates from the electro as show i Fig.21. iv) if E, cos 0 0, 0 /2 : the radiat zoe expads to cover the etire upper hemisphere ( this is oly a limit case sice E ca ever ted to ifiity ). Thus E E 0 is the coditio for the existece of the agle 0 (which implies the coditio for the emissio of radiatio ). So, while it is accelerated i E, the exteded electro ca or caot radiate *, ** depedig o the stregth of the applyig field E compared to that of its self- field E 0, but ot o its acceleratio ***. We ote that i the case i) E E 0 : the electro is accelerated by the applyig field E but it does ot radiate because cos 0 1 ( i.e., 0 does ot exist ) ; therefore, the radiatio of the exteded electro does ot deped o its acceleratio. ( The Larmor 's formula liks the power radiated (P = 2q 2 a 2 / 3c 3 ) by a radiatig poit electro to its acceleratio ; this formula does ot applied to the exteded electro ). Followig are statemets of three emiet physicists about the radiatio ad acceleratio : * Pearle 7 : A poit charge must radiate if it accelerates, but the same is ot true of a exteded charge distributio. **Jackso 8 : Radiatio is emitted i ways that are obscure ad ot easily related to the acceleratio of a charge. *** Feyma : We have iherited a prejudice that a acceleratig charge should radiate. We also ote that the radiatio reactio force F rad (which is the recoil force produced by the radiatio of a radiatig poit electro ) does ot exist i the radiatio of the exteded electro. The force F produced o the core ( - q 0 ), although poitig i the opposite directio to the radiatio, is ot produced by the radiatio of the exteded electro. As metioed i sectio 7 of part 1, if the radiatio reactio force F rad is icluded i the equatio of motio of the electro, the equatio becomes a third order differetial equatio ; this is Loretz-Abraham- Dirac equatio for the poit electro which is ivolved i such problems as ruaway ad preacceleratio. Coclusio This article presets the theory o a ew exteded model of the electro 1(a), it reveals some features which are quite differet from those of the poit electro. I part 1, the et electric force Fe developed o the exteded electro is ot a sigle force, but the resultat of two opposite forces F ad F' ; Fe ad the effective electric charge Q chage with velocity of the electro ad ted to zero as v c. Part 2 itroduces a ovel way of explaiig the radiatio process of the exteded electro : radiatio by electric forces. Whe the exteded electro is subject to a exteral field E which is stroger tha the selffield E 0 of the electro, the electro radiates i the directio of E by the elemetary forces fe that emaate from the upper hemisphere of the exteded electro ( Fig. 21 ) ; the radiatio does ot deped o the acceleratio of the electro ad produces o radiatio reactio force.

16 16 Appedix A : Calculatio of the elemetary force fe F.2 : Normal ad tagetial compoets of F. 3 : Normal ad tagetial compoets of the E ad E o two eds of surface dipole M electric forces o two eds of surface dipole M First, we use boudary coditios to determie the electric field E iside the electro, the we calculate the electric force fe. Fig. 1 shows two arbitrary surface dipoles M ad N : M o the upper hemisphere, N o the lower hemisphere. Boudary coditios applied o dipole M give ( Fig. 2 ) E = E + Et actig o the egative charge q of the surface dipole M, E = E + E t = ( 1/) E + Et actig o the positive charge +q, where E = E cos, 0 /2 ad Et = E si. Sice Et = E t, two taget forces ft ad f t produced o two eds q ad +q of dipole M are equal ad opposite ( Fig. 3 ) : ft = f t = q Et = q E si. Two ormal forces f ad f have magitudes f = q E = q E cos, f is cetripetal, f = q E = (1/) q E = (1/) q E cos, f is cetrifugal. Sice the dipole is extremely small, these forces ca be cosidered as if they apply o the same poit o the dipole M. Their resultat force fe actig o the dipole M is thus fe = f + f + ft + f t = f + f sice ft ad f t are equal ad opposite. Sice f ad f are i opposite directios, the magitude ad directio of fe deped o : * whe 1 ( Fig. 4), f f, hece : fe = f - f = ( 1/ - 1 ) q E cos, fe is cetrifugal, (1-a) * whe 1 ( Fig. 5), f f, hece : fe = f - f = (1 1/ ) q E cos, fe is cetripetal, (1-b)

17 17 F.6 : Normal ad tagetial compoets of F. 7 : Normal ad tagetial compoets of the E ad E o two eds of surface dipole N electric forces o two eds of surface dipole N Now we calculate the resultat force fe actig o the dipole N, o the lower hemisphere of the electro : /2, ( hece cos 0 ), Fig. 6 & Fig.7. Sice + =, E = E cos = - E cos ad Et = E si = E si. Boudary coditios give Et = E t ; so, two taget forces ft ad f t produced o two eds q ad +q of the dipole N are equal ad opposite : ft = f t = q E si. Two ormal forces f ad f have magitudes f = q E = q E ( -cos ), f is cetrifugal, f = q E = ( 1/) q E ( - cos ), f is cetripetal. Sice cos 0, (-cos ) is a positive umber ; the resultat force fe actig o the dipole N is * whe 1 ( Fig. 8 ) : fe = f - f = ( 1/ - 1 ) q E ( -cos ), fe is cetripetal, (1-c) * whe 1 ( Fig. 9 ) : fe = f - f = ( 1 1/ ) q E ( -cos ), fe is cetrifugal. (1-d) We have used four expressios ( 1-a ), (1-b ), (1-c ) ad (1-d ) i the determiatio of fe i the mai text. Appedix B : Determiatio of the force F produced o the core of the electro The core is a poit charge ( -q 0 ) subject to the field E. Because of the spherical symmetry of the structure of the of the electro, E at the core must be parallel to the exteral field E. Ad hece we ca write E = k E (5) where k is a positive umber differet from 1 : 0 k 1. k is positive because E is parallel to E ; k 1 because if k = 1, Eq.(5) becomes E = E ; ad this meas that the electric field iside the electro is idepedet of the material of the electro. But sice the exteded electro has a permittivity ( ) differet from that of the surroudig free space ( 0 ), it is reasoable to thik that E depeds o the medium of the electro ; i.e., E differs from E i magitude or k 1. Now let s apply boudary coditios to two poits o both sides of the iterface : M o the surface of the electro ad the ceter O At M : E = E + Et (6) At O : E = ( 1/ ) E + Et (7) Equatios ( 5) ad (6) give E = ke = k E + k Et (8) Comparig (7) ad (8) we get (9) ad (10) ( 1/ ) E = k E k = 1/ 1 ad (9) Et = k Et ( k 1 ) Et = 0 Et = 0 sice k 1 0 (10) Sice Et = 0, Eq.(6) becomes E = E (11) ad Eq.(7) becomes E = (1/) E = ( 1/) E (12)

18 18 So, the electric force F produced o the core (-q 0 ) is F = - q 0 E = - (1/) q 0 E (13) F is thus always egative ; i.e., it always poits i the opposite directio to E ( Fig.17). Fig.18 shows the variatio of F vs accordig to Eq.(13). Note : The above results Et = 0 (i Eq.10) ad E = E (i Eq.11) imply that the poit M must be located at either the orth pole ( = 0 ) or the south pole ( = ) of the electro because Et = E si = 0, hece si = 0. This result comes from the fact that the electro is spherically symmetric : E ad E are colliear (12) ad ormal (11) to the spherical surface oly at the orth ad south poles of the electro. If M takes a arbitrary positio o the surface of the electro, we caot solve the equatios of boudary coditios for the field E at the core O. We have used two Eqs. (12) ad (13) i the determiatio of F' i the mai text. Appedix C : Radiatio ad absorptio of light ( or photos ) I part 2 we maitaied that light is composed of particles called photos which are idetified as electric dipoles that form the outer part of the electro 1(a). I the determiatio of the et electric force Fe, we defied the parameter 'a' by the expressio (16) a (q / q 0 ) cos 2 i, (16) i which represets the structure of the exteded electro, i which is the umber of surface dipoles. The parameter 'a' is thus a positive dimesioless umber which icreases mootoically with the umber of surface dipoles because i cos 2 i is the sum of positive terms cos 2 i. This expressio suggests the idea that the radiatio ad the absorptio of light of the exteded electro are liked to ad 'a' i the followig maer : * whe the electro radiates, it emits its surface dipoles ito space ; this meas that the umber (of surface dipoles ) decreases, ad thus the parameter 'a' decreases accordigly, * whe the electro absorbs photos ( e.g., by irradiatio ), it collects dipoles oto its surface ; this meas that icreases, ad thus the parameter 'a' icreases accordigly. As metioed o page 11 ( sectio 6, part 1 ) that the equatio (20 ) a 1 - a = - ( 1 v 2 / c 2 ) N/ 2 (20) ca suggest a way to relate the radiatio ad absorptio of light of the exteded electro to its velocity ; that is, whe the electro radiates, it slows dow ; ad whe it absorbs light, it speeds up. To prove this statemet, let us extract v 2 / c 2 from Eq.(20) v 2 / c 2 = 1 - [ a - ( a - 1 ) / ] 2/N

19 19 As metioed o page 10 that sice a 1, 1- ( ifiitesimally less tha 1 ) ; so if is cosidered approximately costat, we ca take the derivative of v 2 / c 2 with respect to 'a' : d (v 2 / c 2 ) / da = (2/N) [ ( 1/ ) - 1 ] [ a - ( a - 1 ) / ] ( 2/N ) -1 For N 0 ; a 1 ad 1- or (1-1/a ) 1, we have (2/N) 0 ; [ ( 1/ ) - 1 ] 0 ; [ a - ( a - 1 ) / ] ( 2/N ) -1 0 for all values of the expoet ( 2/N ) - 1, therefore we get d (v 2 / c 2 ) / da 0 This meas that v 2 / c 2 ad 'a' icrease mootoically, that is * whe the electro radiates : 'a' decreases, ad v 2 / c 2 decreases as well ; this meas that the electro slows dow ; i.e., it loses its ( kietic ) eergy. ( For example, i the sychrotro accelerator, electros radiate ad lose their eergy ). * whe the electro absorbs photos : 'a' icreases, ad v 2 / c 2 icreases as well ; this meas that the electro speeds up ; i.e., it gais eergy. ( I the photoelectric effect, electros absorb photos via irradiatio ad gai eergy ). These results are ot ew fidigs, they are already kow i the classical theory of radiatio. What is ew here is that they ca be deduced from a few mathematical expressios which are derived from the exteded model of the electro 1(a). Refereces 1. Nguye H. V. (a) " A New Exteded Model for the Electro ", vixra.org Classical Physics vixra: (b) " A Foudatioal Problem i Physics : Mass versus Electric Charge ", vixra.org Classical Physics vixra: Dirac P.A.M, " Theory of Electro ad Positro ", Nobel lecture, December 12, Sadiku M, " Elemets of Electromagetics", 1994, 2d Editio, p, Hudso J. J. et al, "Improved measuremet of the shape of the electro" Nature 473, p , May 26, 2011, Published olie May 25, Johsto H, "New techique arrows electro dipole momet", Physicsworld. com 46085, May 26, Rohrlich F, " Dyamics of a charged particle", Phys. Rev. E 77, 2008, Pearle P, " Whe ca a classical electro accelerate without radiatig? " Foudatio of physics, Vol. 8, No. 11/12, 1978, p Jackso J.D., " Classical Electrodyamics ", 2 d Editio, Chap.15, p.702