Analytical Calculation of Stored Electrostatic Energy Per Unit Length

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Maxwell electrodynamics is a very successful theory which describes a wide range of macroscopic phenomena in electricity and magnetism. On the other hand, in Maxwell electrodynamics, the electric field of a point charge q at the position of the point charge is singular, i.e., q x |x|→0 E(x)= . 4πǫ x 2 x −−−→ ∞ 0| | | | Also, in Maxwell electrodynamics, the classical self-energy of a point charge is q2 ∞ dr U = . 8πǫ r2 →∞ 0 Z0 More than 80 years ago, Born and Infeld proposed a nonlinear generalization of Maxwell electrodynamics [1]. In their generalization, the classical self-energy of a point charge was a finite value [1-6]. Recent studies in string theory show that the dynamics of electromagnetic fields on D-branes can be described by Born-Infeld theory [7-10]. In a paper on Born-Infeld theory [8], the concept of a BIon was introduced by Gibbons. BIon is a finite energy solution of a nonlinear theory with a distributional source. Today, many physicists believe that the dark energy in our universe can be described by a Born-Infeld type scalar field [11]. The authors of Ref. [12] have presented a non-Abelian generalization of Born-Infeld theory. In their generalization, they have found a one-parameter family of finite energy solutions in the case of the SU(2) gauge group. In 2013, Hendi [13] proposed a nonlinear generalization of Maxwell electrodynamics which is called exponential electrodynamics [14,15]. The black hole solutions of Einstein’s gravity in the presence of exponential electrodynamics in a 3 + 1-dimensional spacetime are obtained in Ref. [13]. In 2014, Gaete and Helayel-Neto introduced a new generalization of Maxwell electrodynamics which is known as logarithmic electrodynamics [14]. They proved that the classical self-energy of a point charge in logarithmic electrodynamics is a finite value. Recently, a novel generalization of Born-Infeld electrodynamics is presented by Gaete and Helayel-Neto in which the authors show that the field energy of a point-like charge is finite only for Born-Infeld like electrodynamics [15]. In Ref. [16], a nonlinear model for electrodynamics with manifestly broken gauge symmetry is proposed. In the above mentioned model for nonlinear electrodynamics, there are non-singular solitonic solutions which de- scribe charged particles. Another interesting theory of nonlinear electrodynamics was proposed and developed by Heisenberg and his students Euler and Kockel [17-19]. They showed that classical electrodynamics must be corrected by nonlinear terms due to the vacuum polarization effects. In Ref. [20], the charged black hole solutions for Einstein- Euler-Heisenberg theory are obtained. There are three physical motivations in writing this paper. First, the exact solutions of nonlinear field equations are very important in

2 theoretical physics. These solutions help us to obtain a better understanding of physical reality. According to above statements, we attempt to obtain particular cylindrically symmetric solutions in Born-Infeld electrostatics. The search for spherically symmetric solutions in Born-Infeld electrostatics will be discussed in future works. Second, we want to show that the nonlinear corrections in electrodynamics are considerable only for very strong electric fields and extremely short spatial distances. Third, we hope to remove or at least modify the infinities which appear in Maxwell electrostatics. This paper is organized as follows. In Section 2, we study Lagrangian formulation of Abelian Born-Infeld model in the presence of an external source. The explicit forms of Gauss’s law and the energy density of an electrostatic field for Born-Infeld electrostatics are obtained. In Section 3, we calculate the electric field together with the stored electrostatic energy per unit length for an infinite charged line and an infinitely long cylinder in Born-Infeld electrostatics. Summary and conclusions are presented in Section 4. Numerical estimations in Section 4 show that the nonlinear corrections to electric field of an infinite charged line at large radial distances are negligible for weak electric fields. There are two appendices in this paper. In Appendix A, a generalized action functional for Abelian Born-Infeld model with an auxiliary scalar field in the presence of an external source is proposed. In Appendix B, we obtain the symmetric energy-momentum tensor for Abelian Born-Infeld model with an auxiliary scalar field. We use SI units throughout this paper. The metric of spacetime has the signature (+, , , ). − − − 2 Lagrangian Formulation of Abelian Born-Infeld Model with an External Source

The Lagrangian density for Abelian Born-Infeld model in a 3 + 1-dimensional spacetime is [1-6] c2 = ǫ β2 1 1+ F F µν J µA , (1) LBI 0 − 2β2 µν − µ s µ where Fµν = ∂µAν ∂ν Aµ is the electromagnetic field tensor and J = (cρ, J) is an − µ 1 external source for the Abelian field A =( c φ, A). The parameter β in Eq. (1) is called the nonlinearity parameter of the model. In the limit β , Eq. (1) reduces to the Lagrangian density of the Maxwell field, i.e., → ∞

= + (β−2), (2) LBI |large β LM O

3 1 µν µ where M = 4µ0 FµνF J Aµ is the Maxwell Lagrangian density. The Euler-Lagrange equationL for the− Born-Infeld− ﬁeld Aµ is

∂ ∂ LBI ∂ LBI =0. (3) ∂A − ρ ∂(∂ A ) λ ρ λ If we substitute Lagrangian density (1) in the Euler-Lagrange equation (3), we will obtain the inhomogeneous Born-Infeld equations as follows:

F ρλ ∂ = µ J λ. (4) ρ 2 0 c µν 1+ 2β2 Fµν F q The electromagnetic ﬁeld tensor Fµν satisﬁes the Bianchi identity:

∂µFνλ + ∂ν Fλµ + ∂λFµν =0. (5)

Equation (5) leads to the homogeneous Maxwell equations. In 3+1-dimensional spacetime, the components of Fµν can be written as follows:

0 Ex/c Ey/c Ez/c E /c 0 B B F = x z y . (6) µν  −E /c B −0 B  − y z − x  Ez/c By Bx 0   − −    Using Eq. (6), Eqs. (4) and (5) can be written in the vector form as follows:

E(x, t) ρ(x, t) ∇ = , (7) 2 2 2 · E (x,t)−c B (x,t) ǫ0 1 2 − β q ∂B(x, t) ∇ E(x, t) = , (8) × − ∂t B(x, t) 1 ∂ E(x, t) ∇ = µ0 J(x, t)+ , (9) × E2(x,t)−c2B2(x,t) c2 ∂t E2(x,t)−c2B2(x,t) 1 2 1 2 − β − β q ∇ B(x, t) = 0. q (10) · The symmetric energy-momentum tensor for the Abelian Born-Infeld model in Eq. (1) has been obtained by Accioly [21] as follows:

1 F µν F β2 T µ = νλ + (Ω 1)δµ , (11) λ µ Ω c2 − λ 0 4 2 c αγ where Ω := 1+ 2β2 Fαγ F . The classical Born-Infeld equations (7)-(10) for an electrostatic ﬁeld Eq(x) are

∇ E(x) ρ(x) 2 = , (12) · E (x) ǫ0 1 2 − β q ∇ E(x) = 0. (13) × Equations (12) and (13) are fundamental equations of Born-Infeld electrostatics [2]. Using divergence theorem, the integral form of Eq. (12) can be written in the form 1 1 E(x).n da = ρ(x)d3x, (14) E2 x S ( ) ǫ0 V I 1 2 Z − β q where V is the three-dimensional volume enclosed by a two-dimensional surface S. Equa- tion (14) is Gauss’s law in Born-Infeld electrostatics. Using Eqs. (6) and (11), the energy density of an electrostatic ﬁeld in Born-Infeld theory is given by

2 1 u(x)= ǫ0β 1 . (15) E2(x) − 1 2 − β q In the limit β , the modified electrostatic energy density in Eq. (15) smoothly → ∞ becomes the usual electrostatic energy density in Maxwell theory, i.e., 1 u(x) = ǫ E2(x)+ (β−2). (16) |large β 2 0 O 3 Calculation of Stored Electrostatic Energy per Unit Length for an Infinite Charged Line and an In- finitely Long Cylinder in Born-Infeld Electrostat- ics

3.1 Infinite Charged Line Let us consider an infinite charged line with a uniform positive linear charge density λ which is located on the z-axis. Now, we find an expression for the electric field E(x) at a radial distance ρ from the z-axis. Because of the cylindrical symmetry of the problem, the suitable Gaussian surface is a circular cylinder of radius ρ and length L, coaxial with the z-axis (see Fig. 1).

5 Figure 1: The Gaussian surface for an inﬁnite charged line. The cylindrical symmetry of the problem implies that E(x) = Eρ(ρ) eˆρ, where eˆρ is the radial unit vector in cylindrical coordinates(ρ, ϕ, z).

Using the cylindrical symmetry of the problem together with the modiﬁed Gauss’s law in Eq. (14), the electric ﬁeld for the Gaussian surface in Fig. 1 becomes

λ 1 E(x)= eˆρ. (17) 2πǫ0ρ λ 2 1+ 2πǫ0βρ q In contrast with Maxwell electrostatics, the electric field E(x) in Eq. (17) has a finite value on the z-axis, i.e., lim E(x)= β eˆρ. (18) ρ →0 At large radial distances from the z-axis, the asymptotic behavior of the electric field in Eq. (17) is given by

λ λ3 E x e e −5 ( )= ˆρ 3 3 2 3 ˆρ + (ρ ). (19) 2πǫ0ρ − 16π ǫ0β ρ O The first term on the right-hand side of Eq. (19) shows the electric field of an infinite charged line in Maxwell electrostatics, while the second and higher order terms in Eq. (19) show the effect of nonlinear corrections. By putting Eq. (17) in Eq. (15), the electrostatic

6 energy density for an inﬁnite charged line in Born-Infeld electrostatics can be written as follows:

2 λ 2 u(x)= ǫ0β 1+ 1 . (20) s 2πǫ0βρ − Using Eq. (20), the stored electrostatic energy per unit length for an inﬁnite charged line in the radial interval 0 ρ Λ is given by ≤ ≤ U Λ 2π = u(x)ρdρdϕ L Z0 Z0 Λ λ Λ2 = 2πǫ β2 Λ2 + 2 0 2 2πǫ β − 2 s 0 λ 2 Λ+ Λ2 + 1 λ 2 2πǫ β + ln r 0 . (21) 2 2πǫ β λ 0 2πǫ0β

U It is necessary to note that the above value for L has an inﬁnite value in Maxwell theory. U In the limit of large β, the expression for L in Eq. (21) diverges logarithmically as ln β. Hence, it seems that the ﬁnite regularization parameter β removes the logarithmic divergence in Eq. (21).

3.2 Infinitely Long Cylinder In this subsection, we determine the electric field E(x) and stored electrostatic energy per unit length for an infinitely long cylinder of radius R and uniform positive volume charge density τ. As in the previous subsection, we assume that the Gaussian surface is a cylindrical closed surface of radius ρ and length L with a common axis with the infinitely long cylinder (see Fig. 2).

According to modified Gauss’s law in Eq. (14), the electric field for the Gaussian surfaces in Fig. 2 is given by τρ 1 eˆρ ; ρ < R, 2ǫ0 τρ 2  1+  2ǫ0β E(x)=  2r (22)  τR 1  eˆρ ; ρ > R. 2ǫ0ρ τR2 1+ 2  2ǫ βρ  s 0    7 Figure 2: The Gaussian surface for an infinitely long cylinder of radius R. Left) inside the cylinder (ρ < R). Right) outside the cylinder (ρ > R).

For the large values of the nonlinearity parameter β, the behavior of the electric ﬁeld E(x) in Eq. (22) is as follows:

τρ τ 3ρ3 e e −4 ˆρ 3 2 ˆρ + (β ); ρ < R, 2ǫ0 − 16ǫ0β O E(x)=  (23)  τR2 τ 3R6  e e −4  ˆρ 3 2 3 ˆρ + (β ); ρ > R. 2ǫ0ρ − 16ǫ0β ρ O  Hence, for the large values of β, the electric field E(x) in Eq. (23) becomes the electric field of an infinitely long cylinder in Maxwell electrostatics. If we substitute Eq. (22) into Eq. (15), we will obtain the electrostatic energy density for an infinitely long cylinder in Born-Infeld electrostatics as follows:

τρ ǫ β2 1+ 2 1 ; ρ < R, 0 2ǫ β −  r 0 u(x)= (24)  2  2 τR 2 ǫ0β 1+ 1 ; ρ > R. s 2ǫ0βρ −   Using Eq. (24), the stored electrostatic energy per unit length for an inﬁnitely long cylinder in the radial interval 0 ρ Λ (Λ > R) is given by ≤ ≤

8 U R τρ Λ τR2 = 2πǫ β2 1+ 2 1 ρdρ + 1+ 2 1 ρdρ L 0 2ǫ β − 2ǫ βρ − ( Z0 r 0 ZR s 0 ) 2 3 2 2 2 Λ 2ǫ0β 2 τR 2 2 Λ τR 2 = 2πǫ0β + 1+ 1 + 1+ − 2 τ√3 2ǫ0β − 2 2ǫ0βΛ ( s τR2 Λ+Λ 1+ 2 2 2 R τR 2 1 τR 2 s 2ǫ0βΛ 1+ + ln . (25) − 2 2ǫ β 2 2ǫ β τR s 0 0 R + R 1+ 2 !) 2ǫ β r 0 U 1 In the limit of large β, the expression for in Eq. (25) can be expanded in powers of L β2 as follows: 2 4 U πτ R 1 Λ −2 large β = ( + ln )+ (β ). (26) L | 4ǫ0 4 R O The ﬁrst term on the right-hand side of Eq. (26) shows the stored electrostatic energy per unit length for an inﬁnitely long cylinder in the radial interval 0 ρ Λ (Λ > R) in Maxwell electrostatics. ≤ ≤

4 Summary and Conclusions

In 1934, Born and Infeld introduced a nonlinear generalization 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 β, the modified Gauss’s law in Born-Infeld electrostatics is 2 1 E2(x) 3 E2(x) 1 1+ + + (β−6) E(x).n da = ρ(x)d3x. (27) 2 β2 8 β4 O ǫ IS 0 ZV By using the modified Gauss’s law in Eq. (14), we calculated the electric field of an infinite charged line and an infinitely long cylinder in Born-Infeld electrostatics. The stored electrostatic energy per unit length for the above configurations of charge density has been calculated in the framework of Born-Infeld electrostatics. Born and Infeld attempted to determine β by equating the classical self-energy of the electron in their theory with its rest mass energy. They obtained the following numerical value for the nonlinearity parameter β [1]: V β =1.2 1020 . (28) × m 9 In 1973, Soff, Rafelski and Greiner [22] have estimated a lower bound on β. This lower bound on β is V β 1.7 1022 . (29) ≥ × m Resent studies on photonic processes in Born-Infeld theory show that the numerical value of β is close to 1.2 1020 V/m in Eq. (28) [23]. In order to obtain a better understanding of nonlinear effects× in Born-Infeld electrostatics, let us estimate the numerical value of the second term on the right-hand side of Eq. (19). For this purpose, we rewrite Eq. (19) as follows: E(x)= E (x) + ∆E(x)+ (ρ−5), (30) 0 O where λ E0(x) := eˆρ, (31) 2πǫ0ρ λ3 E x e ∆ ( ) := 3 3 2 3 ˆρ. (32) −16π ǫ0β ρ

Using Eqs. (31) and (32), the ratio of ∆E(x) to E0(x) is given by ∆E(x) 1 E2(x) | | = 0 . (33) E (x) 2 β2 | 0 | Let us assume the following approximate but realistic values [24]: C L =1.80 m, ρ =0.10 m, Q = +24 µC, λ =1.33 10−5 . (34) × m By putting Eqs. (28) and (34) into Eq. (33), we get ∆E(x) 2 10−28 E (x) . (35) | | ≈ × | 0 | Finally, if we put Eqs. (29) and (34) in Eq. (33), we obtain ∆E(x) . 10−32 E (x) . (36) | | | 0 | In fact, as is clear from Eqs. (35) and (36), the nonlinear corrections to electric field in Eq. (19) are very small for weak electric fields. The authors of Ref. [25] have suggested a nonlinear generalization of Maxwell electrodynamics. In their generalization, the electric field of a point charge is singular at 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 nonlinear electrodynamics. In these models, both the electric field of a point charge at the position of the point charge and the classical self-energy of the point charge have finite values. In future works, we hope to study the problems discussed in this research from the viewpoint of Refs. [25-27].

10 Acknowledgments

We would like to thank the referees for their careful reading and constructive comments.

Appendix A: A Generalized Action Functional for Abelian Born-Infeld Model with an Auxiliary Scalar Field

Let us consider the following action functional:

t 2 1 2 λ1 c µν λ2 µ 4 S(A, ψ)= ǫ0β 1 ω1 ψ (1 + 2 Fµν F ) ω2 ψ J Aµ d x, (A.1) c R3 − 2β − − Zt0 Z

where ψ is an auxiliary scalar field and ω1, λ1, ω2 and λ2 are four non-zero constants. The variation of Eq. (A.1) with respect to ψ and Aµ leads to the following classical field equations: 1 ω λ c2 −1 ψ = 2 2 1+ F F µν λ1 λ2 , (A.2) − ω λ 2β2 µν − 1 1 λ1 µν µ0 ν ∂µ(ψ F )= J . (A.3) 2ω1 Substituting Eq. (A.2) into Eq. (A.3), we obtain the following classical field equation:

λ1 ω λ c2 −1 µ ∂ 2 2 1+ F F αγ λ1 λ2 F µν = 0 J ν. (A.4) µ − ω λ 2β2 αγ − 2ω 1 1 1 1 By choosing λ = λ, λ = λ and ω = ω, ω = (ω > 0), Eqs.(A.1) and (A.4) can be 1 2 − 1 2 4ω written as follows:

t 2 1 2 λ c µν 1 −λ µ 4 S(A, ψ)= ǫ0β 1 ω ψ (1 + 2 Fµν F ) ψ J Aµ d x, (A.5) c R3 − 2β − 4ω − Zt0 Z F µν ∂ = µ J ν . (A.6) µ 2 0 c αγ 1+ 2β2 Fαγ F Equation (A.5) is the generalizedq action functional for Abelian Born-Infeld model with an auxiliary scalar ﬁeld ψ. Also, Eq. (A.6) is the inhomogeneous Born-Infeld equation

11 1 (see Eq. (4)). If we choose ω = and λ = 1 in Eq. (A.5), we will obtain the following 2 action functional: t 2 1 2 ψ c µν 1 µ 4 S(A, ψ)= ǫ0β 1 (1 + 2 Fµν F ) J Aµ d x. (A.7) c R3 − 2 2β − 2ψ − Zt0 Z The above action functional was presented by Tseytlin in his studies on low energy dynamics of D-branes [28].

Appendix B: The Symmetric Energy-Momentum Ten- sor for Abelian Born-Infeld Model with an Auxiliary Scalar Field

In this appendix, we want to obtain the symmetric energy-momentum tensor for Abelian Born-Infeld model with an auxiliary scalar field. According to Eq. (A.5), the Lagrangian density for Abelian Born-Infeld model with an auxiliary scalar field ψ in the absence of external source J µ is c2 1 = ǫ β2 1 ωψλ 1+ F F µν ψ−λ . (B.1) L 0 − 2β2 µν − 4ω From Eq. (B.1), we obtain the following classical field equations:

2 1 1 c − 2 ψλ = 1+ F F µν , (B.2) 2ω 2β2 µν λ µν ∂µ ψ F =0. (B.3) The canonical energy-momentum tensor for Eq. (B.1) is

α ∂ ∂ α T γ = L (∂γ Aη)+ L (∂γ ψ) δ γ ∂(∂αAη) ∂(∂αψ) − L c2 1 = 2ǫ c2ωψλF αη(∂ A ) ǫ β2δα 1 ωψλ 1+ F F µν ψ−λ .(B.4) − 0 γ η − 0 γ − 2β2 µν − 4ω α Using Eqs. (B.2) and (B.3), the canonical energy-momentum tensor T γ in Eq. (B.4) can be rewritten as follows: 2 1 α c − 2 T = ǫ c2 1+ F F µν F αηF γ − 0 2β2 µν γη c2 1 ǫ β2δα 1 1+ F F µν 2 + ∂ M ηα , (B.5) − 0 γ − 2β2 µν η γ 12 where 1 F ηα M ηα := A , M αη = M ηα . (B.6) γ 2 γ γ γ µ0 c µν − 1+ 2β2 Fµν F q ηα After dropping the total divergence term ∂ηM γ in Eq. (B.5), we get the following expression for the symmetric energy-momentum tensor:

1 2 2 1 α c − 2 c T = ǫ c2 1+ F F µν F αηF ǫ β2δα 1 1+ F F µν 2 . (B.7) γ − 0 2β2 µν γη − 0 γ − 2β2 µν If we use Eqs. (6) and (B.7), we will obtain the electrostatic energy density for Abelian Born-Infeld model with an auxiliary scalar ﬁeld as follows:

0 2 1 u(x)= T 0(x)= ǫ0β 1 . (B.8) E2(x) − 1 2 − β q References

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