Analytical Calculation of Stored Electrostatic Energy per …inspirehep.net/record/1365921/files/180185.pdfy z L 𝜆 𝜌 Figure 1: The Gaussian surface for an infinite charged line

Research ArticleAnalytical Calculation of Stored Electrostatic Energy perUnit Length for an Infinite Charged Line and an Infinitely LongCylinder in the Framework of Born-Infeld Electrostatics

S K Moayedi M Shafabakhsh and F Fathi

Department of Physics Faculty of Sciences Arak University Arak 38156-8-8349 Iran

Correspondence should be addressed to S K Moayedi s-moayediarakuacir

Received 28 December 2014 Accepted 7 March 2015

Academic Editor George Siopsis

Copyright copy 2015 S K Moayedi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

More than 80 years ago Born-Infeld electrodynamics was proposed in order to remove the point charge singularity in Maxwellelectrodynamics In this work after a brief introduction to Lagrangian formulation of Abelian Born-Infeld model in the presenceof an external source we obtain the explicit forms of Gaussrsquos law and the energy density of an electrostatic field for Born-Infeldelectrostatics The electric field and the stored electrostatic energy per unit length for an infinite charged line and an infinitely longcylinder in Born-Infeld electrostatics are calculated Numerical estimations in this paper show that the nonlinear corrections toMaxwell electrodynamics are considerable only for strong electric fields We present an action functional for Abelian Born-Infeldmodel with an auxiliary scalar field in the presence of an external source This action functional is a generalization of the actionfunctional which was presented by Tseytlin in his studies on low energy dynamics of D-branes (Nucl Phys B469 51 (1996) Int JMod Phys A 19 3427 (2004)) Finally we derive the symmetric energy-momentum tensor for Abelian Born-Infeld model with anauxiliary scalar field

1 Introduction

Maxwell electrodynamics is a very successful theory whichdescribes a wide range of macroscopic phenomena in elec-tricity and magnetism On the other hand in Maxwellelectrodynamics the electric field of a point charge 119902 at theposition of the point charge is singular that is

E (x) =119902

41205871205980 |x|2

x|x||x|rarr 0997888rarr infin (1)

Also inMaxwell electrodynamics the classical self-energy ofa point charge is

119880 =1199022

81205871205980

int

infin

0

119889119903

1199032997888rarr infin (2)

More than 80 years ago Born and Infeld proposed a nonlineargeneralization of Maxwell electrodynamics [1] In their gen-eralization the classical self-energy of a point charge was afinite value [1ndash6] Recent studies in string theory show that

the dynamics of electromagnetic fields on 119863-branes can bedescribed by Born-Infeld theory [7ndash10] In a paper on Born-Infeld theory [8] the concept of a BIon was introduced byGibbons BIon is a finite energy solution of a nonlinear theorywith a distributional source Today many physicists believethat the dark energy in our universe can be described by aBorn-Infeld type scalar field [11]The authors of [12] have pre-sented a non-Abelian generalization of Born-Infeld theory Intheir generalization they have found a one-parameter familyof finite energy solutions in the case of the 119878119880(2) gauge groupIn 2013 Hendi [13] proposed a nonlinear generalizationof Maxwell electrodynamics which is called exponentialelectrodynamics [14 15]The black hole solutions of Einsteinrsquosgravity in the presence of exponential electrodynamics ina 3+1-dimensional spacetime are obtained in [13] In 2014Gaete and Helayel-Neto introduced a new generalization ofMaxwell electrodynamics which is known as logarithmicelectrodynamics [14] They proved that the classical self-energy of a point charge in logarithmic electrodynamics is

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015 Article ID 180185 7 pageshttpdxdoiorg1011552015180185

2 Advances in High Energy Physics

a finite value Recently a novel generalization of Born-Infeldelectrodynamics is presented by Gaete and Helayel-Neto inwhich the authors show that the field energy of a point-like charge is finite only for Born-Infeld like electrodynamics[15] In [16] a nonlinear model for electrodynamics withmanifestly broken gauge symmetry is proposed In the abovementioned model for nonlinear electrodynamics there arenonsingular solitonic solutions which describe charged parti-cles Another interesting theory of nonlinear electrodynamicswas proposed and developed by Heisenberg and his stu-dents Euler and Kockel [17ndash19] They showed that classicalelectrodynamics must be corrected by nonlinear terms dueto the vacuum polarization effects In [20] the chargedblack hole solutions for Einstein-Euler-Heisenberg theory areobtainedThere are three physical motivations in writing thispaper First the exact solutions of nonlinear field equationsare very important in theoretical physics These solutionshelp us to obtain a better understanding of physical realityAccording to the above statements we attempt to obtainparticular cylindrically symmetric solutions in Born-Infeldelectrostatics The search for spherically symmetric solutionsin Born-Infeld electrostaticswill be discussed in futureworksSecond we want to show that the nonlinear corrections inelectrodynamics are considerable only for very strong electricfields and extremely short spatial distances Third we hopeto remove or at least modify the infinities which appear inMaxwell electrostatics This paper is organized as followsIn Section 2 we study Lagrangian formulation of AbelianBorn-Infeld model in the presence of an external source Theexplicit forms of Gaussrsquos law and the energy density of anelectrostatic field for Born-Infeld electrostatics are obtainedIn Section 3 we calculate the electric field together with thestored electrostatic energy per unit length for an infinitecharged line and an infinitely long cylinder in Born-Infeldelectrostatics Summary and conclusions are presented inSection 4 Numerical estimations in Section 4 show that thenonlinear corrections to electric field of infinite charged lineat large radial distances are negligible for weak electric fieldsThere are two appendices in this paper In Appendix A ageneralized action functional for Abelian Born-Infeld modelwith an auxiliary scalar field in the presence of an externalsource is proposed In Appendix B we obtain the symmetricenergy-momentum tensor for Abelian Born-Infeld modelwith an auxiliary scalar field We use SI units throughout thispaper The metric of spacetime has the signature (+ minus minus minus)

2 Lagrangian Formulation of Abelian Born-Infeld Model with an External Source

The Lagrangian density for Abelian Born-Infeld model in a3 + 1-dimensional spacetime is [1ndash6]

LBI = 12059801205732

1 minus radic1 +1198882

21205732119865120583]119865120583]

minus 119869120583119860120583 (3)

where 119865120583] = 120597

120583119860] minus 120597]119860120583 is the electromagnetic field tensor

and 119869120583= (119888120588 J) is an external source for the Abelian field

119860120583

= ((1119888)120601A) The parameter 120573 in (3) is called the

nonlinearity parameter of the model In the limit 120573 rarr infin(3) reduces to the Lagrangian density of the Maxwell fieldthat is

LBI1003816100381610038161003816large120573 = L

119872+ O (120573

minus2) (4)

where L119872

= minus(141205830)119865120583]119865120583]

minus 119869120583119860120583is the Maxwell

Lagrangian density The Euler-Lagrange equation for theBorn-Infeld field 119860120583 is

120597LBI120597119860120582

minus 120597120588(

120597LBI

120597 (120597120588119860120582)) = 0 (5)

If we substitute Lagrangian density (3) in the Euler-Lagrangeequation (5) we will obtain the inhomogeneous Born-Infeldequations as follows

120597120588(

119865120588120582

radic1 + (119888221205732) 119865120583]119865120583]) = 120583

0119869120582 (6)

The electromagnetic field tensor 119865120583] satisfies the Bianchi

identity

120597120583119865]120582 + 120597]119865120582120583 + 120597120582119865120583] = 0 (7)

Equation (7) leads to the homogeneous Maxwell equationsIn 3 + 1-dimensional spacetime the components of 119865

120583] canbe written as follows

119865120583] =

((((((((

(

0119864119909

119888

119864119910

119888

119864119911

119888

minus119864119909

1198880 minus119861

119911119861119910

minus119864119910

119888119861119911

0 minus119861119909

minus119864119911

119888minus119861119910

119861119909

0

))))))))

)

(8)

Using (8) (6) and (7) can be written in the vector form asfollows

nabla sdot (E (x 119905)

radic1 minus (E2 (x 119905) minus 1198882B2 (x 119905)) 1205732) =

120588 (x 119905)1205980

nabla times E (x 119905) = minus120597B (x 119905)

120597119905

nabla times(B (x 119905)

radic1 minus (E2 (x 119905) minus 1198882B2 (x 119905)) 1205732)

= 1205830J (x 119905)

+1

1198882

120597

120597119905(

E (x 119905)radic1 minus (E2 (x 119905) minus 1198882B2 (x 119905)) 1205732

)

nabla sdot B (x 119905) = 0

(9)

Advances in High Energy Physics 3

The symmetric energy-momentum tensor for the AbelianBorn-Infeld model in (3) has been obtained by Accioly [21]as follows

119879120583

120582=

1

1205830

[119865120583]119865]120582

Ω+1205732

1198882(Ω minus 1) 120575

120583

120582] (10)

where Ω = radic1 + (119888221205732)119865120572120574119865120572120574 The classical Born-Infeld

equations (9) for an electrostatic field E(x) are

nabla sdot (E (x)

radic1 minus E2 (x) 1205732) =

120588 (x)1205980

(11)

nabla times E (x) = 0 (12)

Equations (11) and (12) are fundamental equations of Born-Infeld electrostatics [2] Using divergence theorem the inte-gral form of (11) can be written in the form

∮119878

1

radic1 minus E2 (x) 1205732E (x) sdot n 119889119886 = 1

1205980

int119881

120588 (x) 1198893119909 (13)

where 119881 is the three-dimensional volume enclosed by a two-dimensional surface 119878 Equation (13) is Gaussrsquos law in Born-Infeld electrostatics Using (8) and (10) the energy density ofan electrostatic field in Born-Infeld theory is given by

119906 (x) = 12059801205732(

1

radic1 minus E2 (x) 1205732minus 1) (14)

In the limit 120573 rarr infin the modified electrostatic energydensity in (14) smoothly becomes the usual electrostaticenergy density in Maxwell theory that is

119906(x)|large120573 =1

21205980E2 (x) + O (120573

minus2) (15)

3 Calculation of Stored Electrostatic Energyper Unit Length for an InfiniteCharged Line and an Infinitely LongCylinder in Born-Infeld Electrostatics

31 Infinite Charged Line Let us consider an infinite chargedline with a uniform positive linear charge density 120582 whichis located on the 119911-axis Now we find an expression for theelectric field E(x) at a radial distance 120588 from the 119911-axisBecause of the cylindrical symmetry of the problem thesuitable Gaussian surface is a circular cylinder of radius 120588 andlength 119871 coaxial with the 119911-axis (see Figure 1)

x

y

z

L

120582

120588

Figure 1 The Gaussian surface for an infinite charged line Thecylindrical symmetry of the problem implies that E(x) = 119864

120588(120588)e120588

where e120588is the radial unit vector in cylindrical coordinates (120588 120593 119911)

Using the cylindrical symmetry of the problem togetherwith the modified Gaussrsquos law in (13) the electric field for theGaussian surface in Figure 1 becomes

E (x) = 120582

21205871205980120588

1

radic1 + (12058221205871205980120573120588)2

e120588 (16)

In contrast with Maxwell electrostatics the electric field E(x)in (16) has a finite value on the 119911-axis that is

lim120588rarr0

E (x) = 120573e120588 (17)

At large radial distances from the 119911-axis the asymptoticbehavior of the electric field in (16) is given by

E (x) = 120582

21205871205980120588e120588minus

1205823

1612058731205983

012057321205883

e120588+ O (120588

minus5) (18)

The first term on the right-hand side of (18) shows the electricfield of an infinite charged line in Maxwell electrostaticswhile the second and higher order terms in (18) show theeffect of nonlinear corrections By putting (16) in (14) theelectrostatic energy density for an infinite charged line inBorn-Infeld electrostatics can be written as follows

119906 (x) = 12059801205732(radic1 + (

120582

21205871205980120573120588

)

2

minus 1) (19)


x

y

z

L

120588

R

(a)

x

y

z

L

120588

R

(b)Figure 2 The Gaussian surface for an infinitely long cylinder of radius 119877 (a) Inside the cylinder (120588 lt 119877) (b) Outside the cylinder (120588 gt 119877)

Using (19) the stored electrostatic energy per unit length foran infinite charged line in the radial interval 0 le 120588 le Λ isgiven by

119880

119871= int

Λ

0

int

2120587

0

119906 (x) 120588119889 120588119889120593

= 212058712059801205732

Λ

2radicΛ2 + (

120582

21205871205980120573)

2

minusΛ2

2+1

2(

120582

21205871205980120573)

2

sdot ln(Λ + radicΛ2 + (1205822120587120598

0120573)2

(12058221205871205980120573)

)

(20)

It is necessary to note that the above value for 119880119871 has aninfinite value in Maxwell theory In the limit of large 120573 theexpression for 119880119871 in (20) diverges logarithmically as ln120573Hence it seems that the finite regularization parameter 120573removes the logarithmic divergence in (20)

32 Infinitely Long Cylinder In this subsection we determinethe electric field E(x) and stored electrostatic energy per unitlength for an infinitely long cylinder of radius 119877 and uniformpositive volume charge density 120591 As in the previous sub-section we assume that the Gaussian surface is a cylindricalclosed surface of radius 120588 and length 119871 with a common axiswith the infinitely long cylinder (see Figure 2)

According to modified Gaussrsquos law in (13) the electricfield for the Gaussian surfaces in Figure 2 is given by

E (x) =

120591120588

21205980

1

radic1 + (12059112058821205980120573)2

e120588 120588 lt 119877

1205911198772

21205980120588

1

radic1 + (120591119877221205980120573120588)2

e120588 120588 gt 119877

(21)

For the large values of the nonlinearity parameter 120573 thebehavior of the electric field E(x) in (21) is as follows

E (x) =

120591120588

21205980

e120588minus

12059131205883

161205983

01205732

e120588+ O (120573

minus4) 120588 lt 119877

1205911198772

21205980120588e120588minus

12059131198776

161205983

012057321205883

e120588+ O (120573

minus4) 120588 gt 119877

(22)

Hence for the large values of 120573 the electric field E(x) in (22)becomes the electric field of an infinitely long cylinder inMaxwell electrostatics If we substitute (21) into (14) we willobtain the electrostatic energy density for an infinitely longcylinder in Born-Infeld electrostatics as follows

119906 (x) =

12059801205732(radic1 + (

120591120588

21205980120573)

2

minus 1) 120588 lt 119877

12059801205732(radic1 + (

1205911198772

21205980120573120588

)

2

minus 1) 120588 gt 119877

(23)

Using (23) the stored electrostatic energy per unit lengthfor an infinitely long cylinder in the radial interval 0 le 120588 le

Λ (Λ gt 119877) is given by

119880

119871= 2120587120598

01205732

int

119877

0

(radic1 + (120591120588

21205980120573)

2

minus 1)120588119889120588

+int

Λ

119877

(radic1 + (1205911198772

21205980120573120588

)

2

minus 1)120588119889120588


= 212058712059801205732

minusΛ2

2+ (

21205980120573

120591radic3)

2

[(1 + (120591119877

21205980120573)

2

)

32

minus 1]

+Λ2

2

radic1 + (1205911198772

21205980120573Λ

)

2

minus1198772

2radic1 + (

120591119877

21205980120573)

2

+1

2(1205911198772

21205980120573)

2

sdot ln(Λ + Λradic1 + (12059111987722120598

0120573Λ)2

119877 + 119877radic1 + (12059111987721205980120573)2

)

(24)

In the limit of large 120573 the expression for 119880119871 in (24) can beexpanded in powers of 11205732 as follows

119880

119871

1003816100381610038161003816100381610038161003816large120573=12058712059121198774

41205980

(1

4+ ln Λ

119877) + O (120573

minus2) (25)

The first term on the right-hand side of (25) shows the storedelectrostatic energy per unit length for an infinitely longcylinder in the radial interval 0 le 120588 le Λ (Λ gt 119877) in Maxwellelectrostatics

4 Summary and Conclusions

In 1934 Born and Infeld introduced a nonlinear generaliza-tion of Maxwell electrodynamics in which the classical self-energy of a point charge like electron became a finite value [1]We showed that in the limit of large 120573 the modified Gaussrsquoslaw in Born-Infeld electrostatics is

∮119878

[

[

1 +1

2

E2 (x)1205732

+3

8

(E2 (x))2

1205734+ O (120573

minus6)]

]

E (x) sdot n 119889119886

=1

1205980

int119881

120588 (x) 1198893119909

(26)

By using the modified Gaussrsquos law in (13) we calculated theelectric field of an infinite charged line and an infinitely longcylinder in Born-Infeld electrostaticsThe stored electrostaticenergy per unit length for the above configurations of chargedensity has been calculated in the framework of Born-Infeldelectrostatics Born and Infeld attempted to determine 120573 byequating the classical self-energy of the electron in theirtheory with its rest mass energy They obtained the followingnumerical value for the nonlinearity parameter 120573 [1]

120573 = 12 times 1020 Vm (27)

In 1973 Soff et al [22] have estimated a lower bound on 120573This lower bound on 120573 is

120573 ge 17 times 1022 Vm (28)

Recent studies on photonic processes in Born-Infeld theoryshow that the numerical value of 120573 is close to 12 times 1020 Vmin (27) [23] In order to obtain a better understanding ofnonlinear effects in Born-Infeld electrostatics let us estimatethe numerical value of the second termon the right-hand sideof (18) For this purpose we rewrite (18) as follows

E (x) = E0(x) + ΔE (x) + O (120588

minus5) (29)

where

E0 (x) =

120582

21205871205980120588e120588

ΔE (x) = minus1205823

1612058731205983

012057321205883

e120588

(30)

Using (30) the ratio of ΔE(x) to E0(x) is given by

|ΔE (x)|1003816100381610038161003816E0 (x)

1003816100381610038161003816

=1

2

E20(x)1205732

(31)

Let us assume the following approximate but realistic values[24]

119871 = 180m 120588 = 010m 119876 = +24 120583119862

120582 = 133 times 10minus5 Cm

(32)

By putting (27) and (32) into (31) we get

|ΔE (x)| asymp 2 times 10minus28 1003816100381610038161003816E0 (x)

1003816100381610038161003816 (33)

Finally if we put (28) and (32) in (31) we obtain

|ΔE (x)| ≲ 10minus32 1003816100381610038161003816E0 (x)

1003816100381610038161003816 (34)

In fact as is clear from (33) and (34) the nonlinearcorrections to electric field in (18) are very small for weakelectric fields The authors of [25] have suggested a non-linear generalization of Maxwell electrodynamics In theirgeneralization the electric field of a point charge is singularat the position of the point charge but the classical self-energy of the point charge has a finite value Recently Kruglov[26 27] has proposed two different models for nonlinearelectrodynamics In these models both the electric field ofa point charge at the position of the point charge and theclassical self-energy of the point charge have finite values Infuture works we hope to study the problems discussed in thisresearch from the viewpoint of [25ndash27]


Appendices

A A Generalized Action Functional forAbelian Born-Infeld Model with anAuxiliary Scalar Field

Let us consider the following action functional

119878 (119860 120595)

=1

119888int

119905

1199050

intR3[12059801205732(1 minus 120596

11205951205821 (1 +

1198882

21205732119865120583]119865120583])

minus 12059621205951205822) minus 119869

120583119860120583]1198894119909

(A1)

where 120595 is an auxiliary scalar field and 1205961 1205821 1205962 and 120582

2are

four nonzero constantsThe variation of (A1) with respect to120595 and 119860

120583leads to the following classical field equations

120595 = [minus12059621205822

12059611205821

(1 +1198882

21205732119865120583]119865120583])

minus1

]

1(1205821minus1205822)

(A2)

120597120583(1205951205821119865120583]) =

1205830

21205961

119869] (A3)

Substituting (A2) into (A3) we obtain the following classicalfield equation

120597120583

[minus12059621205822

12059611205821

(1 +1198882

21205732119865120572120574119865120572120574)

minus1

]

1205821(1205821minus1205822)

119865120583]

=1205830

21205961

119869]

(A4)

By choosing 1205821= 120582 120582

2= minus120582 and 120596

1= 120596 120596

2= (14120596) (120596 gt

0) (A1) and (A4) can be written as follows

119878 (119860 120595)

=1

119888int

119905

1199050

intR3[12059801205732(1 minus 120596120595

120582(1 +

1198882

21205732119865120583]119865120583])

minus1

4120596120595minus120582) minus 119869120583119860120583]1198894119909

(A5)

120597120583(

119865120583]

radic1 + (119888221205732) 119865120572120574119865120572120574

) = 1205830119869] (A6)

Equation (A5) is the generalized action functional forAbelian Born-Infeld model with an auxiliary scalar field120595 Also (A6) is the inhomogeneous Born-Infeld equation

(see (6)) If we choose 120596 = 12 and 120582 = 1 in (A5) we willobtain the following action functional

119878 (119860 120595)

=1

119888int

119905

1199050

intR3[12059801205732(1 minus

120595

2(1 +

1198882

21205732119865120583]119865120583])

minus1

2120595) minus 119869120583119860120583]1198894119909

(A7)

The above action functional was presented by Tseytlin in hisstudies on low energy dynamics of119863-branes [28]

B The Symmetric Energy-MomentumTensor for Abelian Born-Infeld Model withan Auxiliary Scalar Field

In this appendix we want to obtain the symmetric energy-momentum tensor for Abelian Born-Infeld model with anauxiliary scalar field According to (A5) the Lagrangiandensity for Abelian Born-Infeld model with an auxiliaryscalar field 120595 in the absence of external source 119869120583 is

L = 12059801205732(1 minus 120596120595

120582(1 +

1198882

21205732119865120583]119865120583]) minus

1

4120596120595minus120582) (B1)

From (B1) we obtain the following classical field equations

120595120582=

1

2120596(1 +

1198882

21205732119865120583]119865120583])

minus12

120597120583(120595120582119865120583]) = 0

(B2)

The canonical energy-momentum tensor for (B1) is

119879120572

120574

=120597L

120597 (120597120572119860120578)(120597120574119860120578) +

120597L

120597 (120597120572120595)

(120597120574120595) minus 120575

120572

120574L

= minus 212059801198882120596120595120582119865120572120578(120597120574119860120578)

minus 12059801205732120575120572

120574(1 minus 120596120595

120582(1 +

1198882

21205732119865120583]119865120583]) minus

1

4120596120595minus120582)

(B3)

Using (B2) the canonical energy-momentum tensor 119879120572120574in

(B3) can be rewritten as follows

119879120572

120574

= minus 12059801198882(1 +

1198882

21205732119865120583]119865120583])

minus12

119865120572120578119865120574120578

minus 12059801205732120575120572

120574(1 minus (1 +

1198882

21205732119865120583]119865120583])

12

) + 120597120578119872120578120572

120574

(B4)


where

119872120578120572

120574=

1

1205830

119865120578120572

radic1 + (119888221205732) 119865120583]119865120583]119860120574

119872120572120578

120574= minus119872

120578120572

120574

(B5)

After dropping the total divergence term 120597120578119872120578120572

120574in (B4)

we get the following expression for the symmetric energy-momentum tensor

119879120572

120574= minus 12059801198882(1 +

1198882

21205732119865120583]119865120583])

minus12

119865120572120578119865120574120578

minus 12059801205732120575120572

120574(1 minus (1 +

1198882

21205732119865120583]119865120583])

12

)

(B6)

If we use (8) and (B6) we will obtain the electrostatic energydensity for Abelian Born-Infeld model with an auxiliaryscalar field as follows

119906 (x) = 1198790

0(x) = 120598

01205732(

1

radic1 minus E2 (x) 1205732minus 1) (B7)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank the referees for their carefulreading and constructive comments

References

[1] M Born and L Infeld ldquoFoundations of the new field theoryrdquoProceedings of the Royal Society of London A vol 144 p 4251934

[2] D Fortunato L Orsina and L Pisani ldquoBorn-Infeld type equa-tions for electrostatic fieldsrdquo Journal of Mathematical Physicsvol 43 no 11 pp 5698ndash5706 2002

[3] S H Mazharimousavi and M Halilsoy ldquoGround state H-atomin born-infeld teoryrdquo Foundations of Physics vol 42 no 4 pp524ndash530 2012

[4] L P De Assis P Gaete J A Helayel-Neto and S O VellozoldquoAspects of magnetic field configurations in planar nonlinearelectrodynamicsrdquo International Journal of Theoretical Physicsvol 51 no 2 pp 477ndash486 2012

[5] Z K Vad ldquoSome solutions of Einstein-Born-Infeld field equa-tionsrdquo Acta Physica Hungarica vol 63 no 3-4 pp 353ndash3641988

[6] P Gaete and I Schmidt ldquoCoulombrsquos law modification in non-linear and in noncommutative electrodynamicsrdquo InternationalJournal ofModern Physics A vol 19 no 20 pp 3427ndash3437 2004

[7] C G Callan and J M Maldacena ldquoBrane dynamics from theBorn-Infeld actionrdquoNuclear Physics B vol 513 no 1-2 pp 198ndash212 1998

[8] G W Gibbons ldquoBorn-Infeld particles and Dirichlet 119901-branesrdquoNuclear Physics B vol 514 no 3 pp 603ndash639 1998

[9] G W Gibbons ldquoAspects of born-infeld theory and stringM-theoryrdquo Revista Mexicana de Fısica vol 49 supplement 1 pp19ndash29 2003

[10] B Zwiebach A First Course in String Theory CambridgeUniversity Press Cambridge UK 2nd edition 2009

[11] H Q Lu ldquoPhantom cosmology with a nonlinear born-infeldtype scalar fieldrdquo International Journal of Modern Physics D vol14 no 2 p 355 2005

[12] E Serie T Masson and R Kerner ldquoNon-abelian generalizationof Born-Infeld theory inspired by noncommutative geometryrdquoPhysical Review D vol 68 no 12 Article ID 125003 2003

[13] S H Hendi ldquoAsymptotic ReissnerndashNordstrom black holesrdquoAnnals of Physics vol 333 pp 282ndash289 2013

[14] P Gaete and J Helayel-Neto ldquoFinite field-energy and interpar-ticle potential in logarithmic electrodynamicsrdquo The EuropeanPhysical Journal C vol 74 article 2816 2014

[15] P Gaete and J Helayel-Neto ldquoRemarks on nonlinear electrody-namicsrdquoThe European Physical Journal C vol 74 no 11 article3182 2014

[16] N Riazi and M Mohammadi ldquoComposite lane-emden equa-tion as a nonlinear poisson equationrdquo International Journal ofTheoretical Physics vol 51 no 4 pp 1276ndash1283 2012

[17] H Euler and B Kockel ldquoUber die streuung von licht an lichtnach der diracschen theorierdquo Naturwissenschaften vol 23 no15 pp 246ndash247 1935

[18] H Euler ldquoOber die Streuung von Licht an Licht nach derDiracschen Theorierdquo Annalen der Physik (Leipzig) vol 26 pp398ndash448 1936

[19] W Heisenberg and H Euler ldquoFolgerungen aus der diracschentheorie des positronsrdquo Zeitschrift fur Physik vol 98 no 11-12pp 714ndash732 1936

[20] R Ruffini Y-B Wu and S-S Xue ldquoEinstein-Euler-Heisenbergtheory and charged black holesrdquo Physical Review D vol 88Article ID 085004 2013

[21] A Accioly ldquoEnergy and momentum for the electromagneticfield described by three outstanding electrodynamicsrdquo Ameri-can Journal of Physics vol 65 no 9 pp 882ndash887 1997

[22] G Soff J Rafelski and W Greiner ldquoLower bound to limitingfields in nonlinear electrodynamicsrdquo Physical Review A vol 7no 3 pp 903ndash907 1973

[23] J M Davila C Schubert and M A Trejo ldquoPhotonic processesin Born-Infeld theoryrdquo International Journal of Modern PhysicsA Particles and Fields Gravitation Cosmology vol 29 ArticleID 1450174 2014

[24] D Halliday R Resnick and J Walker Fundamentals of PhysicsExtended Wiley 10th edition 2014

[25] C V Costa D M Gitman and A E Shabad ldquoFinite field-energy of a point charge in QEDrdquo httparxivorgabs13120447

[26] S I Kruglov ldquoAmodel of nonlinear electrodynamicsrdquoAnnals ofPhysics vol 353 pp 299ndash306 2015

[27] S I Kruglov ldquoNonlinear arcsin-electrodynamicsrdquo httparxivorgabs14107633

[28] A A Tseytlin ldquoSelf-duality of Born-Infeld action and Dirichlet3-brane of type IIB superstring theoryrdquo Nuclear Physics B vol469 no 1-2 pp 51ndash67 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


a finite value Recently a novel generalization of Born-Infeldelectrodynamics is presented by Gaete and Helayel-Neto inwhich the authors show that the field energy of a point-like charge is finite only for Born-Infeld like electrodynamics[15] In [16] a nonlinear model for electrodynamics withmanifestly broken gauge symmetry is proposed In the abovementioned model for nonlinear electrodynamics there arenonsingular solitonic solutions which describe charged parti-cles Another interesting theory of nonlinear electrodynamicswas proposed and developed by Heisenberg and his stu-dents Euler and Kockel [17ndash19] They showed that classicalelectrodynamics must be corrected by nonlinear terms dueto the vacuum polarization effects In [20] the chargedblack hole solutions for Einstein-Euler-Heisenberg theory areobtainedThere are three physical motivations in writing thispaper First the exact solutions of nonlinear field equationsare very important in theoretical physics These solutionshelp us to obtain a better understanding of physical realityAccording to the above statements we attempt to obtainparticular cylindrically symmetric solutions in Born-Infeldelectrostatics The search for spherically symmetric solutionsin Born-Infeld electrostaticswill be discussed in futureworksSecond we want to show that the nonlinear corrections inelectrodynamics are considerable only for very strong electricfields and extremely short spatial distances Third we hopeto remove or at least modify the infinities which appear inMaxwell electrostatics This paper is organized as followsIn Section 2 we study Lagrangian formulation of AbelianBorn-Infeld model in the presence of an external source Theexplicit forms of Gaussrsquos law and the energy density of anelectrostatic field for Born-Infeld electrostatics are obtainedIn Section 3 we calculate the electric field together with thestored electrostatic energy per unit length for an infinitecharged line and an infinitely long cylinder in Born-Infeldelectrostatics Summary and conclusions are presented inSection 4 Numerical estimations in Section 4 show that thenonlinear corrections to electric field of infinite charged lineat large radial distances are negligible for weak electric fieldsThere are two appendices in this paper In Appendix A ageneralized action functional for Abelian Born-Infeld modelwith an auxiliary scalar field in the presence of an externalsource is proposed In Appendix B we obtain the symmetricenergy-momentum tensor for Abelian Born-Infeld modelwith an auxiliary scalar field We use SI units throughout thispaper The metric of spacetime has the signature (+ minus minus minus)

2 Lagrangian Formulation of Abelian Born-Infeld Model with an External Source

The Lagrangian density for Abelian Born-Infeld model in a3 + 1-dimensional spacetime is [1ndash6]

LBI = 12059801205732

1 minus radic1 +1198882

21205732119865120583]119865120583]

minus 119869120583119860120583 (3)

where 119865120583] = 120597

120583119860] minus 120597]119860120583 is the electromagnetic field tensor

and 119869120583= (119888120588 J) is an external source for the Abelian field

119860120583

= ((1119888)120601A) The parameter 120573 in (3) is called the

nonlinearity parameter of the model In the limit 120573 rarr infin(3) reduces to the Lagrangian density of the Maxwell fieldthat is

LBI1003816100381610038161003816large120573 = L

119872+ O (120573

minus2) (4)

where L119872

= minus(141205830)119865120583]119865120583]

minus 119869120583119860120583is the Maxwell

Lagrangian density The Euler-Lagrange equation for theBorn-Infeld field 119860120583 is

120597LBI120597119860120582

minus 120597120588(

120597LBI

120597 (120597120588119860120582)) = 0 (5)

If we substitute Lagrangian density (3) in the Euler-Lagrangeequation (5) we will obtain the inhomogeneous Born-Infeldequations as follows

120597120588(

119865120588120582

radic1 + (119888221205732) 119865120583]119865120583]) = 120583

0119869120582 (6)

The electromagnetic field tensor 119865120583] satisfies the Bianchi

identity

120597120583119865]120582 + 120597]119865120582120583 + 120597120582119865120583] = 0 (7)

Equation (7) leads to the homogeneous Maxwell equationsIn 3 + 1-dimensional spacetime the components of 119865

120583] canbe written as follows

119865120583] =

((((((((

(

0119864119909

119888

119864119910

119888

119864119911

119888

minus119864119909

1198880 minus119861

119911119861119910

minus119864119910

119888119861119911

0 minus119861119909

minus119864119911

119888minus119861119910

119861119909

0

))))))))

)

(8)

Using (8) (6) and (7) can be written in the vector form asfollows

nabla sdot (E (x 119905)

radic1 minus (E2 (x 119905) minus 1198882B2 (x 119905)) 1205732) =

120588 (x 119905)1205980

nabla times E (x 119905) = minus120597B (x 119905)

120597119905

nabla times(B (x 119905)

radic1 minus (E2 (x 119905) minus 1198882B2 (x 119905)) 1205732)

= 1205830J (x 119905)

+1

1198882

120597

120597119905(

E (x 119905)radic1 minus (E2 (x 119905) minus 1198882B2 (x 119905)) 1205732

)

nabla sdot B (x 119905) = 0

(9)



119879120583

120582=

1

1205830

[119865120583]119865]120582

Ω+1205732

1198882(Ω minus 1) 120575

120583

120582] (10)



nabla sdot (E (x)


120588 (x)1205980

(11)



∮119878

1


1205980

int119881

120588 (x) 1198893119909 (13)


119906 (x) = 12059801205732(

1



119906(x)|large120573 =1

21205980E2 (x) + O (120573

minus2) (15)



x

y

z

L

120582

120588


120588(120588)e120588



E (x) = 120582

21205871205980120588

1

radic1 + (12058221205871205980120573120588)2

e120588 (16)


lim120588rarr0

E (x) = 120573e120588 (17)


E (x) = 120582

21205871205980120588e120588minus

1205823

1612058731205983

012057321205883

e120588+ O (120588

minus5) (18)


119906 (x) = 12059801205732(radic1 + (

120582

21205871205980120573120588

)

2

minus 1) (19)


x

y

z

L

120588

R

(a)

x

y

z

L

120588

R



119880

119871= int

Λ

0

int

2120587

0

119906 (x) 120588119889 120588119889120593

= 212058712059801205732

Λ

2radicΛ2 + (

120582

21205871205980120573)

2

minusΛ2

2+1

2(

120582

21205871205980120573)

2

sdot ln(Λ + radicΛ2 + (1205822120587120598

0120573)2

(12058221205871205980120573)

)

(20)




E (x) =

120591120588

21205980

1

radic1 + (12059112058821205980120573)2

e120588 120588 lt 119877

1205911198772

21205980120588

1

radic1 + (120591119877221205980120573120588)2

e120588 120588 gt 119877

(21)


E (x) =

120591120588

21205980

e120588minus

12059131205883

161205983

01205732

e120588+ O (120573

minus4) 120588 lt 119877

1205911198772

21205980120588e120588minus

12059131198776

161205983

012057321205883

e120588+ O (120573

minus4) 120588 gt 119877

(22)


119906 (x) =

12059801205732(radic1 + (

120591120588

21205980120573)

2

minus 1) 120588 lt 119877

12059801205732(radic1 + (

1205911198772

21205980120573120588

)

2

minus 1) 120588 gt 119877

(23)



119880

119871= 2120587120598

01205732

int

119877

0

(radic1 + (120591120588

21205980120573)

2

minus 1)120588119889120588

+int

Λ

119877

(radic1 + (1205911198772

21205980120573120588

)

2

minus 1)120588119889120588


= 212058712059801205732

minusΛ2

2+ (

21205980120573

120591radic3)

2

[(1 + (120591119877

21205980120573)

2

)

32

minus 1]

+Λ2

2

radic1 + (1205911198772

21205980120573Λ

)

2

minus1198772

2radic1 + (

120591119877

21205980120573)

2

+1

2(1205911198772

21205980120573)

2

sdot ln(Λ + Λradic1 + (12059111987722120598

0120573Λ)2

119877 + 119877radic1 + (12059111987721205980120573)2

)

(24)


119880

119871

1003816100381610038161003816100381610038161003816large120573=12058712059121198774

41205980

(1

4+ ln Λ

119877) + O (120573

minus2) (25)




∮119878

[

[

1 +1

2

E2 (x)1205732

+3

8

(E2 (x))2

1205734+ O (120573

minus6)]

]

E (x) sdot n 119889119886

=1

1205980

int119881

120588 (x) 1198893119909

(26)


120573 = 12 times 1020 Vm (27)


120573 ge 17 times 1022 Vm (28)


E (x) = E0(x) + ΔE (x) + O (120588

minus5) (29)

where

E0 (x) =

120582

21205871205980120588e120588

ΔE (x) = minus1205823

1612058731205983

012057321205883

e120588

(30)


|ΔE (x)|1003816100381610038161003816E0 (x)

1003816100381610038161003816

=1

2

E20(x)1205732

(31)


119871 = 180m 120588 = 010m 119876 = +24 120583119862


(32)



1003816100381610038161003816 (33)


|ΔE (x)| ≲ 10minus32 1003816100381610038161003816E0 (x)

1003816100381610038161003816 (34)



Appendices



119878 (119860 120595)

=1

119888int

119905

1199050

intR3[12059801205732(1 minus 120596

11205951205821 (1 +

1198882

21205732119865120583]119865120583])

minus 12059621205951205822) minus 119869

120583119860120583]1198894119909

(A1)


2are



120595 = [minus12059621205822

12059611205821

(1 +1198882

21205732119865120583]119865120583])

minus1

]

1(1205821minus1205822)

(A2)

120597120583(1205951205821119865120583]) =

1205830

21205961

119869] (A3)


120597120583

[minus12059621205822

12059611205821

(1 +1198882

21205732119865120572120574119865120572120574)

minus1

]

1205821(1205821minus1205822)

119865120583]

=1205830

21205961

119869]

(A4)

By choosing 1205821= 120582 120582

2= minus120582 and 120596

1= 120596 120596

2= (14120596) (120596 gt


119878 (119860 120595)

=1

119888int

119905

1199050

intR3[12059801205732(1 minus 120596120595

120582(1 +

1198882

21205732119865120583]119865120583])

minus1

4120596120595minus120582) minus 119869120583119860120583]1198894119909

(A5)

120597120583(

119865120583]

radic1 + (119888221205732) 119865120572120574119865120572120574

) = 1205830119869] (A6)



119878 (119860 120595)

=1

119888int

119905

1199050

intR3[12059801205732(1 minus

120595

2(1 +

1198882

21205732119865120583]119865120583])

minus1

2120595) minus 119869120583119860120583]1198894119909

(A7)




L = 12059801205732(1 minus 120596120595

120582(1 +

1198882

21205732119865120583]119865120583]) minus

1

4120596120595minus120582) (B1)


120595120582=

1

2120596(1 +

1198882

21205732119865120583]119865120583])

minus12

120597120583(120595120582119865120583]) = 0

(B2)


119879120572

120574

=120597L

120597 (120597120572119860120578)(120597120574119860120578) +

120597L

120597 (120597120572120595)

(120597120574120595) minus 120575

120572

120574L

= minus 212059801198882120596120595120582119865120572120578(120597120574119860120578)

minus 12059801205732120575120572

120574(1 minus 120596120595

120582(1 +

1198882

21205732119865120583]119865120583]) minus

1

4120596120595minus120582)

(B3)



119879120572

120574

= minus 12059801198882(1 +

1198882

21205732119865120583]119865120583])

minus12

119865120572120578119865120574120578

minus 12059801205732120575120572

120574(1 minus (1 +

1198882

21205732119865120583]119865120583])

12

) + 120597120578119872120578120572

120574

(B4)


where

119872120578120572

120574=

1

1205830

119865120578120572

radic1 + (119888221205732) 119865120583]119865120583]119860120574

119872120572120578

120574= minus119872

120578120572

120574

(B5)


120574in (B4)


119879120572

120574= minus 12059801198882(1 +

1198882

21205732119865120583]119865120583])

minus12

119865120572120578119865120574120578

minus 12059801205732120575120572

120574(1 minus (1 +

1198882

21205732119865120583]119865120583])

12

)

(B6)


119906 (x) = 1198790

0(x) = 120598

01205732(

1




Acknowledgment


References


































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119879120583

120582=

1

1205830

[119865120583]119865]120582

Ω+1205732

1198882(Ω minus 1) 120575

120583

120582] (10)



nabla sdot (E (x)


120588 (x)1205980

(11)



∮119878

1


1205980

int119881

120588 (x) 1198893119909 (13)


119906 (x) = 12059801205732(

1



119906(x)|large120573 =1

21205980E2 (x) + O (120573

minus2) (15)



x

y

z

L

120582

120588


120588(120588)e120588



E (x) = 120582

21205871205980120588

1

radic1 + (12058221205871205980120573120588)2

e120588 (16)


lim120588rarr0

E (x) = 120573e120588 (17)


E (x) = 120582

21205871205980120588e120588minus

1205823

1612058731205983

012057321205883

e120588+ O (120588

minus5) (18)


119906 (x) = 12059801205732(radic1 + (

120582

21205871205980120573120588

)

2

minus 1) (19)


x

y

z

L

120588

R

(a)

x

y

z

L

120588

R



119880

119871= int

Λ

0

int

2120587

0

119906 (x) 120588119889 120588119889120593

= 212058712059801205732

Λ

2radicΛ2 + (

120582

21205871205980120573)

2

minusΛ2

2+1

2(

120582

21205871205980120573)

2

sdot ln(Λ + radicΛ2 + (1205822120587120598

0120573)2

(12058221205871205980120573)

)

(20)




E (x) =

120591120588

21205980

1

radic1 + (12059112058821205980120573)2

e120588 120588 lt 119877

1205911198772

21205980120588

1

radic1 + (120591119877221205980120573120588)2

e120588 120588 gt 119877

(21)


E (x) =

120591120588

21205980

e120588minus

12059131205883

161205983

01205732

e120588+ O (120573

minus4) 120588 lt 119877

1205911198772

21205980120588e120588minus

12059131198776

161205983

012057321205883

e120588+ O (120573

minus4) 120588 gt 119877

(22)


119906 (x) =

12059801205732(radic1 + (

120591120588

21205980120573)

2

minus 1) 120588 lt 119877

12059801205732(radic1 + (

1205911198772

21205980120573120588

)

2

minus 1) 120588 gt 119877

(23)



119880

119871= 2120587120598

01205732

int

119877

0

(radic1 + (120591120588

21205980120573)

2

minus 1)120588119889120588

+int

Λ

119877

(radic1 + (1205911198772

21205980120573120588

)

2

minus 1)120588119889120588


= 212058712059801205732

minusΛ2

2+ (

21205980120573

120591radic3)

2

[(1 + (120591119877

21205980120573)

2

)

32

minus 1]

+Λ2

2

radic1 + (1205911198772

21205980120573Λ

)

2

minus1198772

2radic1 + (

120591119877

21205980120573)

2

+1

2(1205911198772

21205980120573)

2

sdot ln(Λ + Λradic1 + (12059111987722120598

0120573Λ)2

119877 + 119877radic1 + (12059111987721205980120573)2

)

(24)


119880

119871

1003816100381610038161003816100381610038161003816large120573=12058712059121198774

41205980

(1

4+ ln Λ

119877) + O (120573

minus2) (25)




∮119878

[

[

1 +1

2

E2 (x)1205732

+3

8

(E2 (x))2

1205734+ O (120573

minus6)]

]

E (x) sdot n 119889119886

=1

1205980

int119881

120588 (x) 1198893119909

(26)


120573 = 12 times 1020 Vm (27)


120573 ge 17 times 1022 Vm (28)


E (x) = E0(x) + ΔE (x) + O (120588

minus5) (29)

where

E0 (x) =

120582

21205871205980120588e120588

ΔE (x) = minus1205823

1612058731205983

012057321205883

e120588

(30)


|ΔE (x)|1003816100381610038161003816E0 (x)

1003816100381610038161003816

=1

2

E20(x)1205732

(31)


119871 = 180m 120588 = 010m 119876 = +24 120583119862


(32)



1003816100381610038161003816 (33)


|ΔE (x)| ≲ 10minus32 1003816100381610038161003816E0 (x)

1003816100381610038161003816 (34)



Appendices



119878 (119860 120595)

=1

119888int

119905

1199050

intR3[12059801205732(1 minus 120596

11205951205821 (1 +

1198882

21205732119865120583]119865120583])

minus 12059621205951205822) minus 119869

120583119860120583]1198894119909

(A1)


2are



120595 = [minus12059621205822

12059611205821

(1 +1198882

21205732119865120583]119865120583])

minus1

]

1(1205821minus1205822)

(A2)

120597120583(1205951205821119865120583]) =

1205830

21205961

119869] (A3)


120597120583

[minus12059621205822

12059611205821

(1 +1198882

21205732119865120572120574119865120572120574)

minus1

]

1205821(1205821minus1205822)

119865120583]

=1205830

21205961

119869]

(A4)

By choosing 1205821= 120582 120582

2= minus120582 and 120596

1= 120596 120596

2= (14120596) (120596 gt


119878 (119860 120595)

=1

119888int

119905

1199050

intR3[12059801205732(1 minus 120596120595

120582(1 +

1198882

21205732119865120583]119865120583])

minus1

4120596120595minus120582) minus 119869120583119860120583]1198894119909

(A5)

120597120583(

119865120583]

radic1 + (119888221205732) 119865120572120574119865120572120574

) = 1205830119869] (A6)



119878 (119860 120595)

=1

119888int

119905

1199050

intR3[12059801205732(1 minus

120595

2(1 +

1198882

21205732119865120583]119865120583])

minus1

2120595) minus 119869120583119860120583]1198894119909

(A7)




L = 12059801205732(1 minus 120596120595

120582(1 +

1198882

21205732119865120583]119865120583]) minus

1

4120596120595minus120582) (B1)


120595120582=

1

2120596(1 +

1198882

21205732119865120583]119865120583])

minus12

120597120583(120595120582119865120583]) = 0

(B2)


119879120572

120574

=120597L

120597 (120597120572119860120578)(120597120574119860120578) +

120597L

120597 (120597120572120595)

(120597120574120595) minus 120575

120572

120574L

= minus 212059801198882120596120595120582119865120572120578(120597120574119860120578)

minus 12059801205732120575120572

120574(1 minus 120596120595

120582(1 +

1198882

21205732119865120583]119865120583]) minus

1

4120596120595minus120582)

(B3)



119879120572

120574

= minus 12059801198882(1 +

1198882

21205732119865120583]119865120583])

minus12

119865120572120578119865120574120578

minus 12059801205732120575120572

120574(1 minus (1 +

1198882

21205732119865120583]119865120583])

12

) + 120597120578119872120578120572

120574

(B4)


where

119872120578120572

120574=

1

1205830

119865120578120572

radic1 + (119888221205732) 119865120583]119865120583]119860120574

119872120572120578

120574= minus119872

120578120572

120574

(B5)


120574in (B4)


119879120572

120574= minus 12059801198882(1 +

1198882

21205732119865120583]119865120583])

minus12

119865120572120578119865120574120578

minus 12059801205732120575120572

120574(1 minus (1 +

1198882

21205732119865120583]119865120583])

12

)

(B6)


119906 (x) = 1198790

0(x) = 120598

01205732(

1




Acknowledgment


References


































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




x

y

z

L

120588

R

(a)

x

y

z

L

120588

R



119880

119871= int

Λ

0

int

2120587

0

119906 (x) 120588119889 120588119889120593

= 212058712059801205732

Λ

2radicΛ2 + (

120582

21205871205980120573)

2

minusΛ2

2+1

2(

120582

21205871205980120573)

2

sdot ln(Λ + radicΛ2 + (1205822120587120598

0120573)2

(12058221205871205980120573)

)

(20)




E (x) =

120591120588

21205980

1

radic1 + (12059112058821205980120573)2

e120588 120588 lt 119877

1205911198772

21205980120588

1

radic1 + (120591119877221205980120573120588)2

e120588 120588 gt 119877

(21)


E (x) =

120591120588

21205980

e120588minus

12059131205883

161205983

01205732

e120588+ O (120573

minus4) 120588 lt 119877

1205911198772

21205980120588e120588minus

12059131198776

161205983

012057321205883

e120588+ O (120573

minus4) 120588 gt 119877

(22)


119906 (x) =

12059801205732(radic1 + (

120591120588

21205980120573)

2

minus 1) 120588 lt 119877

12059801205732(radic1 + (

1205911198772

21205980120573120588

)

2

minus 1) 120588 gt 119877

(23)



119880

119871= 2120587120598

01205732

int

119877

0

(radic1 + (120591120588

21205980120573)

2

minus 1)120588119889120588

+int

Λ

119877

(radic1 + (1205911198772

21205980120573120588

)

2

minus 1)120588119889120588


= 212058712059801205732

minusΛ2

2+ (

21205980120573

120591radic3)

2

[(1 + (120591119877

21205980120573)

2

)

32

minus 1]

+Λ2

2

radic1 + (1205911198772

21205980120573Λ

)

2

minus1198772

2radic1 + (

120591119877

21205980120573)

2

+1

2(1205911198772

21205980120573)

2

sdot ln(Λ + Λradic1 + (12059111987722120598

0120573Λ)2

119877 + 119877radic1 + (12059111987721205980120573)2

)

(24)


119880

119871

1003816100381610038161003816100381610038161003816large120573=12058712059121198774

41205980

(1

4+ ln Λ

119877) + O (120573

minus2) (25)




∮119878

[

[

1 +1

2

E2 (x)1205732

+3

8

(E2 (x))2

1205734+ O (120573

minus6)]

]

E (x) sdot n 119889119886

=1

1205980

int119881

120588 (x) 1198893119909

(26)


120573 = 12 times 1020 Vm (27)


120573 ge 17 times 1022 Vm (28)


E (x) = E0(x) + ΔE (x) + O (120588

minus5) (29)

where

E0 (x) =

120582

21205871205980120588e120588

ΔE (x) = minus1205823

1612058731205983

012057321205883

e120588

(30)


|ΔE (x)|1003816100381610038161003816E0 (x)

1003816100381610038161003816

=1

2

E20(x)1205732

(31)


119871 = 180m 120588 = 010m 119876 = +24 120583119862


(32)



1003816100381610038161003816 (33)


|ΔE (x)| ≲ 10minus32 1003816100381610038161003816E0 (x)

1003816100381610038161003816 (34)



Appendices



119878 (119860 120595)

=1

119888int

119905

1199050

intR3[12059801205732(1 minus 120596

11205951205821 (1 +

1198882

21205732119865120583]119865120583])

minus 12059621205951205822) minus 119869

120583119860120583]1198894119909

(A1)


2are



120595 = [minus12059621205822

12059611205821

(1 +1198882

21205732119865120583]119865120583])

minus1

]

1(1205821minus1205822)

(A2)

120597120583(1205951205821119865120583]) =

1205830

21205961

119869] (A3)


120597120583

[minus12059621205822

12059611205821

(1 +1198882

21205732119865120572120574119865120572120574)

minus1

]

1205821(1205821minus1205822)

119865120583]

=1205830

21205961

119869]

(A4)

By choosing 1205821= 120582 120582

2= minus120582 and 120596

1= 120596 120596

2= (14120596) (120596 gt


119878 (119860 120595)

=1

119888int

119905

1199050

intR3[12059801205732(1 minus 120596120595

120582(1 +

1198882

21205732119865120583]119865120583])

minus1

4120596120595minus120582) minus 119869120583119860120583]1198894119909

(A5)

120597120583(

119865120583]

radic1 + (119888221205732) 119865120572120574119865120572120574

) = 1205830119869] (A6)



119878 (119860 120595)

=1

119888int

119905

1199050

intR3[12059801205732(1 minus

120595

2(1 +

1198882

21205732119865120583]119865120583])

minus1

2120595) minus 119869120583119860120583]1198894119909

(A7)




L = 12059801205732(1 minus 120596120595

120582(1 +

1198882

21205732119865120583]119865120583]) minus

1

4120596120595minus120582) (B1)


120595120582=

1

2120596(1 +

1198882

21205732119865120583]119865120583])

minus12

120597120583(120595120582119865120583]) = 0

(B2)


119879120572

120574

=120597L

120597 (120597120572119860120578)(120597120574119860120578) +

120597L

120597 (120597120572120595)

(120597120574120595) minus 120575

120572

120574L

= minus 212059801198882120596120595120582119865120572120578(120597120574119860120578)

minus 12059801205732120575120572

120574(1 minus 120596120595

120582(1 +

1198882

21205732119865120583]119865120583]) minus

1

4120596120595minus120582)

(B3)



119879120572

120574

= minus 12059801198882(1 +

1198882

21205732119865120583]119865120583])

minus12

119865120572120578119865120574120578

minus 12059801205732120575120572

120574(1 minus (1 +

1198882

21205732119865120583]119865120583])

12

) + 120597120578119872120578120572

120574

(B4)


where

119872120578120572

120574=

1

1205830

119865120578120572

radic1 + (119888221205732) 119865120583]119865120583]119860120574

119872120572120578

120574= minus119872

120578120572

120574

(B5)


120574in (B4)


119879120572

120574= minus 12059801198882(1 +

1198882

21205732119865120583]119865120583])

minus12

119865120572120578119865120574120578

minus 12059801205732120575120572

120574(1 minus (1 +

1198882

21205732119865120583]119865120583])

12

)

(B6)


119906 (x) = 1198790

0(x) = 120598

01205732(

1




Acknowledgment


References


































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




= 212058712059801205732

minusΛ2

2+ (

21205980120573

120591radic3)

2

[(1 + (120591119877

21205980120573)

2

)

32

minus 1]

+Λ2

2

radic1 + (1205911198772

21205980120573Λ

)

2

minus1198772

2radic1 + (

120591119877

21205980120573)

2

+1

2(1205911198772

21205980120573)

2

sdot ln(Λ + Λradic1 + (12059111987722120598

0120573Λ)2

119877 + 119877radic1 + (12059111987721205980120573)2

)

(24)


119880

119871

1003816100381610038161003816100381610038161003816large120573=12058712059121198774

41205980

(1

4+ ln Λ

119877) + O (120573

minus2) (25)




∮119878

[

[

1 +1

2

E2 (x)1205732

+3

8

(E2 (x))2

1205734+ O (120573

minus6)]

]

E (x) sdot n 119889119886

=1

1205980

int119881

120588 (x) 1198893119909

(26)


120573 = 12 times 1020 Vm (27)


120573 ge 17 times 1022 Vm (28)


E (x) = E0(x) + ΔE (x) + O (120588

minus5) (29)

where

E0 (x) =

120582

21205871205980120588e120588

ΔE (x) = minus1205823

1612058731205983

012057321205883

e120588

(30)


|ΔE (x)|1003816100381610038161003816E0 (x)

1003816100381610038161003816

=1

2

E20(x)1205732

(31)


119871 = 180m 120588 = 010m 119876 = +24 120583119862


(32)



1003816100381610038161003816 (33)


|ΔE (x)| ≲ 10minus32 1003816100381610038161003816E0 (x)

1003816100381610038161003816 (34)



Appendices



119878 (119860 120595)

=1

119888int

119905

1199050

intR3[12059801205732(1 minus 120596

11205951205821 (1 +

1198882

21205732119865120583]119865120583])

minus 12059621205951205822) minus 119869

120583119860120583]1198894119909

(A1)


2are



120595 = [minus12059621205822

12059611205821

(1 +1198882

21205732119865120583]119865120583])

minus1

]

1(1205821minus1205822)

(A2)

120597120583(1205951205821119865120583]) =

1205830

21205961

119869] (A3)


120597120583

[minus12059621205822

12059611205821

(1 +1198882

21205732119865120572120574119865120572120574)

minus1

]

1205821(1205821minus1205822)

119865120583]

=1205830

21205961

119869]

(A4)

By choosing 1205821= 120582 120582

2= minus120582 and 120596

1= 120596 120596

2= (14120596) (120596 gt


119878 (119860 120595)

=1

119888int

119905

1199050

intR3[12059801205732(1 minus 120596120595

120582(1 +

1198882

21205732119865120583]119865120583])

minus1

4120596120595minus120582) minus 119869120583119860120583]1198894119909

(A5)

120597120583(

119865120583]

radic1 + (119888221205732) 119865120572120574119865120572120574

) = 1205830119869] (A6)



119878 (119860 120595)

=1

119888int

119905

1199050

intR3[12059801205732(1 minus

120595

2(1 +

1198882

21205732119865120583]119865120583])

minus1

2120595) minus 119869120583119860120583]1198894119909

(A7)




L = 12059801205732(1 minus 120596120595

120582(1 +

1198882

21205732119865120583]119865120583]) minus

1

4120596120595minus120582) (B1)


120595120582=

1

2120596(1 +

1198882

21205732119865120583]119865120583])

minus12

120597120583(120595120582119865120583]) = 0

(B2)


119879120572

120574

=120597L

120597 (120597120572119860120578)(120597120574119860120578) +

120597L

120597 (120597120572120595)

(120597120574120595) minus 120575

120572

120574L

= minus 212059801198882120596120595120582119865120572120578(120597120574119860120578)

minus 12059801205732120575120572

120574(1 minus 120596120595

120582(1 +

1198882

21205732119865120583]119865120583]) minus

1

4120596120595minus120582)

(B3)



119879120572

120574

= minus 12059801198882(1 +

1198882

21205732119865120583]119865120583])

minus12

119865120572120578119865120574120578

minus 12059801205732120575120572

120574(1 minus (1 +

1198882

21205732119865120583]119865120583])

12

) + 120597120578119872120578120572

120574

(B4)


where

119872120578120572

120574=

1

1205830

119865120578120572

radic1 + (119888221205732) 119865120583]119865120583]119860120574

119872120572120578

120574= minus119872

120578120572

120574

(B5)


120574in (B4)


119879120572

120574= minus 12059801198882(1 +

1198882

21205732119865120583]119865120583])

minus12

119865120572120578119865120574120578

minus 12059801205732120575120572

120574(1 minus (1 +

1198882

21205732119865120583]119865120583])

12

)

(B6)


119906 (x) = 1198790

0(x) = 120598

01205732(

1




Acknowledgment


References


































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Appendices



119878 (119860 120595)

=1

119888int

119905

1199050

intR3[12059801205732(1 minus 120596

11205951205821 (1 +

1198882

21205732119865120583]119865120583])

minus 12059621205951205822) minus 119869

120583119860120583]1198894119909

(A1)


2are



120595 = [minus12059621205822

12059611205821

(1 +1198882

21205732119865120583]119865120583])

minus1

]

1(1205821minus1205822)

(A2)

120597120583(1205951205821119865120583]) =

1205830

21205961

119869] (A3)


120597120583

[minus12059621205822

12059611205821

(1 +1198882

21205732119865120572120574119865120572120574)

minus1

]

1205821(1205821minus1205822)

119865120583]

=1205830

21205961

119869]

(A4)

By choosing 1205821= 120582 120582

2= minus120582 and 120596

1= 120596 120596

2= (14120596) (120596 gt


119878 (119860 120595)

=1

119888int

119905

1199050

intR3[12059801205732(1 minus 120596120595

120582(1 +

1198882

21205732119865120583]119865120583])

minus1

4120596120595minus120582) minus 119869120583119860120583]1198894119909

(A5)

120597120583(

119865120583]

radic1 + (119888221205732) 119865120572120574119865120572120574

) = 1205830119869] (A6)



119878 (119860 120595)

=1

119888int

119905

1199050

intR3[12059801205732(1 minus

120595

2(1 +

1198882

21205732119865120583]119865120583])

minus1

2120595) minus 119869120583119860120583]1198894119909

(A7)




L = 12059801205732(1 minus 120596120595

120582(1 +

1198882

21205732119865120583]119865120583]) minus

1

4120596120595minus120582) (B1)


120595120582=

1

2120596(1 +

1198882

21205732119865120583]119865120583])

minus12

120597120583(120595120582119865120583]) = 0

(B2)


119879120572

120574

=120597L

120597 (120597120572119860120578)(120597120574119860120578) +

120597L

120597 (120597120572120595)

(120597120574120595) minus 120575

120572

120574L

= minus 212059801198882120596120595120582119865120572120578(120597120574119860120578)

minus 12059801205732120575120572

120574(1 minus 120596120595

120582(1 +

1198882

21205732119865120583]119865120583]) minus

1

4120596120595minus120582)

(B3)



119879120572

120574

= minus 12059801198882(1 +

1198882

21205732119865120583]119865120583])

minus12

119865120572120578119865120574120578

minus 12059801205732120575120572

120574(1 minus (1 +

1198882

21205732119865120583]119865120583])

12

) + 120597120578119872120578120572

120574

(B4)


where

119872120578120572

120574=

1

1205830

119865120578120572

radic1 + (119888221205732) 119865120583]119865120583]119860120574

119872120572120578

120574= minus119872

120578120572

120574

(B5)


120574in (B4)


119879120572

120574= minus 12059801198882(1 +

1198882

21205732119865120583]119865120583])

minus12

119865120572120578119865120574120578

minus 12059801205732120575120572

120574(1 minus (1 +

1198882

21205732119865120583]119865120583])

12

)

(B6)


119906 (x) = 1198790

0(x) = 120598

01205732(

1




Acknowledgment


References


































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




where

119872120578120572

120574=

1

1205830

119865120578120572

radic1 + (119888221205732) 119865120583]119865120583]119860120574

119872120572120578

120574= minus119872

120578120572

120574

(B5)


120574in (B4)


119879120572

120574= minus 12059801198882(1 +

1198882

21205732119865120583]119865120583])

minus12

119865120572120578119865120574120578

minus 12059801205732120575120572

120574(1 minus (1 +

1198882

21205732119865120583]119865120583])

12

)

(B6)


119906 (x) = 1198790

0(x) = 120598

01205732(

1




Acknowledgment


References


































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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