# Analytical Calculation of Stored Electrostatic Energy per Unit Length for an Infinite Charged Line and an Infinitely Long Cylinder in the Framework of Born-Infeld Electrostatics

Authors: S. K. Moayedi; M. Shafabakhsh; F. Fathi
Journal: Advances in High Energy Physics
 (2015)

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## Abstract

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


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## Body

## 1. Introduction

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 chargeq at the position of the point charge is singular; that is, (1)E(x)=q4πϵ0x2xx⟶x→0∞.Also, in Maxwell electrodynamics, the classical self-energy of a point charge is (2)U=q28πϵ0∫0∞‍drr2⟶∞.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 [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 [13]. In 2014, Gaete and Helayël-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 Helayël-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 [16], a nonlinear model for electrodynamics with manifestly broken gauge symmetry is proposed. In the above mentioned model, for nonlinear electrodynamics, there are nonsingular solitonic solutions which describe 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 [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 theoretical physics. These solutions help us to obtain a better understanding of physical reality. According to the 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 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 a3+1-dimensional spacetime is [1–6](3)LBI=ϵ0β21-1+c22β2FμνFμν-JμAμ,where Fμν=∂μAν-∂νAμ is the electromagnetic field tensor and Jμ=(cρ,J) is an external source for the Abelian field Aμ=(1/cϕ,A). The parameter β in (3) is called the nonlinearity parameter of the model. In the limit β→∞, (3) reduces to the Lagrangian density of the Maxwell field; that is,(4)LBIlargeβ=LM+Oβ-2,where LM=-1/4μ0FμνFμν-JμAμ is the Maxwell Lagrangian density. The Euler-Lagrange equation for the Born-Infeld field Aμ is(5)∂LBI∂Aλ-∂ρ∂LBI∂∂ρAλ=0.If we substitute Lagrangian density (3) in the Euler-Lagrange equation (5), we will obtain the inhomogeneous Born-Infeld equations as follows:(6)∂ρFρλ1+c2/2β2FμνFμν=μ0Jλ.The electromagnetic field tensor Fμν satisfies the Bianchi identity:(7)∂μFνλ+∂νFλμ+∂λFμν=0.Equation (7) leads to the homogeneous Maxwell equations. In 3+1-dimensional spacetime, the components of Fμν can be written as follows:(8)Fμν=0ExcEycEzc-Exc0-BzBy-EycBz0-Bx-Ezc-ByBx0.Using (8), (6) and (7) can be written in the vector form as follows:(9)∇·Ex,t1-E2x,t-c2B2x,t/β2=ρ(x,t)ϵ0,∇×E(x,t)=-∂B(x,t)∂t,∇×Bx,t1-E2x,t-c2B2x,t/β2=μ0J(x,t)+1c2∂∂tEx,t1-E2x,t-c2B2x,t/β2,∇·B(x,t)=0.The symmetric energy-momentum tensor for the Abelian Born-Infeld model in (3) has been obtained by Accioly [21] as follows:(10)Tλμ=1μ0FμνFνλΩ+β2c2Ω-1δλμ,where Ω:=1+c2/2β2FαγFαγ. The classical Born-Infeld equations (9) for an electrostatic field E(x) are(11)∇·Ex1-E2x/β2=ρ(x)ϵ0,(12)∇×E(x)=0.Equations (11) and (12) are fundamental equations of Born-Infeld electrostatics [2]. Using divergence theorem, the integral form of (11) can be written in the form(13)∮S‍11-E2x/β2Ex·nda=1ϵ0∫V‍ρxd3x,where V is the three-dimensional volume enclosed by a two-dimensional surface S. Equation (13) is Gauss’s law in Born-Infeld electrostatics. Using (8) and (10), the energy density of an electrostatic field in Born-Infeld theory is given by(14)u(x)=ϵ0β211-E2x/β2-1.In the limit β→∞, the modified electrostatic energy density in (14) smoothly becomes the usual electrostatic energy density in Maxwell theory; that is,(15)u(x)largeβ=12ϵ0E2x+Oβ-2.


## 3. Calculation of Stored Electrostatic Energy per Unit Length for an Infinite Charged Line and an Infinitely Long Cylinder in Born-Infeld Electrostatics

### 3.1. Infinite Charged Line

Let us consider an infinite charged line with a uniform positive linear charge densityλ 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 Figure 1).Figure 1
The Gaussian surface for an infinite charged line. The cylindrical symmetry of the problem implies thatE(x)=Eρ(ρ)e^ρ, where e^ρ 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 (13), the electric field for the Gaussian surface in Figure 1 becomes(16)E(x)=λ2πϵ0ρ11+λ/2πϵ0βρ2e^ρ.In contrast with Maxwell electrostatics, the electric field E(x) in (16) has a finite value on the z-axis; that is,(17)limρ→0Ex=βe^ρ.At large radial distances from the z-axis, the asymptotic behavior of the electric field in (16) is given by(18)Ex=λ2πϵ0ρe^ρ-λ316π3ϵ03β2ρ3e^ρ+Oρ-5.The first term on the right-hand side of (18) shows the electric field of an infinite charged line in Maxwell electrostatics, while the second and higher order terms in (18) show the effect of nonlinear corrections. By putting (16) in (14), the electrostatic energy density for an infinite charged line in Born-Infeld electrostatics can be written as follows:(19)ux=ϵ0β21+λ2πϵ0βρ2-1.Using (19), the stored electrostatic energy per unit length for an infinite charged line in the radial interval 0≤ρ≤Λ is given by(20)UL=∫0Λ‍∫02πuxρdρdφ=2πϵ0β2Λ2Λ2+λ2πϵ0β2-Λ22+12λ2πϵ0β2Λ+Λ2+λ/2πϵ0β2λ/2πϵ0βhhhhhhhhh·ln⁡Λ+Λ2+λ/2πϵ0β2λ/2πϵ0β.It is necessary to note that the above value for U/L has an infinite value in Maxwell theory. In the limit of large β, the expression for U/L in (20) diverges logarithmically as ln⁡β. Hence, it seems that the finite regularization parameter β removes the logarithmic divergence in (20).


### 3.2. Infinitely Long Cylinder

In this subsection, we determine the electric fieldE(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 Figure 2).Figure 2
The Gaussian surface for an infinitely long cylinder of radiusR. (a) Inside the cylinder (ρ<R). (b) Outside the cylinder (ρ>R).
(a)
(b)According to modified Gauss’s law in (13), the electric field for the Gaussian surfaces in Figure 2 is given by(21)E(x)=τρ2ϵ011+τρ/2ϵ0β2e^ρ;ρ<R,τR22ϵ0ρ11+τR2/2ϵ0βρ2e^ρ;ρ>R.For the large values of the nonlinearity parameterβ, the behavior of the electric field E(x) in (21) is as follows:(22)E(x)=τρ2ϵ0e^ρ-τ3ρ316ϵ03β2e^ρ+O(β-4);ρ<R,τR22ϵ0ρe^ρ-τ3R616ϵ03β2ρ3e^ρ+O(β-4);ρ>R.Hence, for the large values of β, the electric field E(x) in (22) becomes the electric field of an infinitely long cylinder in Maxwell electrostatics. If we substitute (21) into (14), we will obtain the electrostatic energy density for an infinitely long cylinder in Born-Infeld electrostatics as follows:(23)u(x)=ϵ0β21+τρ2ϵ0β2-1;ρ<R,ϵ0β21+τR22ϵ0βρ2-1;ρ>R.Using (23), the stored electrostatic energy per unit length for an infinitely long cylinder in the radial interval 0≤ρ≤Λ(Λ>R) is given by(24)UL=2πϵ0β21+τR22ϵ0βρ2∫0R‍1+τρ2ϵ0β2-1ρdρhhhhhhhhh+∫RΛ‍1+τR22ϵ0βρ2-1ρdρ=2πϵ0β2-Λ22+2ϵ0βτ321+τR2ϵ0β23/2-1Λ+Λ1+τR2/2ϵ0βΛ2R+R1+τR/2ϵ0β2hihhhhhhhh+Λ221+τR22ϵ0βΛ2hihhhhhhhh-R221+τR2ϵ0β2+12τR22ϵ0β2hhihhhhhhh·ln⁡Λ+Λ1+τR2/2ϵ0βΛ2R+R1+τR/2ϵ0β2.In the limit of large β, the expression for U/L in (24) can be expanded in powers of 1/β2 as follows:(25)ULlargeβ=πτ2R44ϵ014+ln⁡ΛR+Oβ-2.The first term on the right-hand side of (25) 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.


## 3.1. Infinite Charged Line

Let us consider an infinite charged line with a uniform positive linear charge densityλ 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 Figure 1).Figure 1
The Gaussian surface for an infinite charged line. The cylindrical symmetry of the problem implies thatE(x)=Eρ(ρ)e^ρ, where e^ρ 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 (13), the electric field for the Gaussian surface in Figure 1 becomes(16)E(x)=λ2πϵ0ρ11+λ/2πϵ0βρ2e^ρ.In contrast with Maxwell electrostatics, the electric field E(x) in (16) has a finite value on the z-axis; that is,(17)limρ→0Ex=βe^ρ.At large radial distances from the z-axis, the asymptotic behavior of the electric field in (16) is given by(18)Ex=λ2πϵ0ρe^ρ-λ316π3ϵ03β2ρ3e^ρ+Oρ-5.The first term on the right-hand side of (18) shows the electric field of an infinite charged line in Maxwell electrostatics, while the second and higher order terms in (18) show the effect of nonlinear corrections. By putting (16) in (14), the electrostatic energy density for an infinite charged line in Born-Infeld electrostatics can be written as follows:(19)ux=ϵ0β21+λ2πϵ0βρ2-1.Using (19), the stored electrostatic energy per unit length for an infinite charged line in the radial interval 0≤ρ≤Λ is given by(20)UL=∫0Λ‍∫02πuxρdρdφ=2πϵ0β2Λ2Λ2+λ2πϵ0β2-Λ22+12λ2πϵ0β2Λ+Λ2+λ/2πϵ0β2λ/2πϵ0βhhhhhhhhh·ln⁡Λ+Λ2+λ/2πϵ0β2λ/2πϵ0β.It is necessary to note that the above value for U/L has an infinite value in Maxwell theory. In the limit of large β, the expression for U/L in (20) diverges logarithmically as ln⁡β. Hence, it seems that the finite regularization parameter β removes the logarithmic divergence in (20).


## 3.2. Infinitely Long Cylinder

In this subsection, we determine the electric fieldE(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 Figure 2).Figure 2
The Gaussian surface for an infinitely long cylinder of radiusR. (a) Inside the cylinder (ρ<R). (b) Outside the cylinder (ρ>R).
(a)
(b)According to modified Gauss’s law in (13), the electric field for the Gaussian surfaces in Figure 2 is given by(21)E(x)=τρ2ϵ011+τρ/2ϵ0β2e^ρ;ρ<R,τR22ϵ0ρ11+τR2/2ϵ0βρ2e^ρ;ρ>R.For the large values of the nonlinearity parameterβ, the behavior of the electric field E(x) in (21) is as follows:(22)E(x)=τρ2ϵ0e^ρ-τ3ρ316ϵ03β2e^ρ+O(β-4);ρ<R,τR22ϵ0ρe^ρ-τ3R616ϵ03β2ρ3e^ρ+O(β-4);ρ>R.Hence, for the large values of β, the electric field E(x) in (22) becomes the electric field of an infinitely long cylinder in Maxwell electrostatics. If we substitute (21) into (14), we will obtain the electrostatic energy density for an infinitely long cylinder in Born-Infeld electrostatics as follows:(23)u(x)=ϵ0β21+τρ2ϵ0β2-1;ρ<R,ϵ0β21+τR22ϵ0βρ2-1;ρ>R.Using (23), the stored electrostatic energy per unit length for an infinitely long cylinder in the radial interval 0≤ρ≤Λ(Λ>R) is given by(24)UL=2πϵ0β21+τR22ϵ0βρ2∫0R‍1+τρ2ϵ0β2-1ρdρhhhhhhhhh+∫RΛ‍1+τR22ϵ0βρ2-1ρdρ=2πϵ0β2-Λ22+2ϵ0βτ321+τR2ϵ0β23/2-1Λ+Λ1+τR2/2ϵ0βΛ2R+R1+τR/2ϵ0β2hihhhhhhhh+Λ221+τR22ϵ0βΛ2hihhhhhhhh-R221+τR2ϵ0β2+12τR22ϵ0β2hhihhhhhhh·ln⁡Λ+Λ1+τR2/2ϵ0βΛ2R+R1+τR/2ϵ0β2.In the limit of large β, the expression for U/L in (24) can be expanded in powers of 1/β2 as follows:(25)ULlargeβ=πτ2R44ϵ014+ln⁡ΛR+Oβ-2.The first term on the right-hand side of (25) 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(26)∮S‍1+12E2xβ2+38E2x2β4+Oβ-6Ex·nda=1ϵ0∫V‍ρxd3x.By using the modified Gauss’s law in (13), 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]:(27)β=1.2×1020V/m.In 1973, Soff et al. [22] have estimated a lower bound on β. This lower bound on β is(28)β≥1.7×1022V/m.Recent studies on photonic processes in Born-Infeld theory show that the numerical value of β is close to 1.2×1020 V/m in (27) [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 (18). For this purpose, we rewrite (18) as follows:(29)E(x)=E0(x)+ΔE(x)+Oρ-5,where(30)E0(x):=λ2πϵ0ρe^ρ,ΔEx:=-λ316π3ϵ03β2ρ3e^ρ.Using (30), the ratio of ΔE(x) to E0(x) is given by(31)ΔExE0x=12E02xβ2.Let us assume the following approximate but realistic values [24]:(32)L=1.80m,ρ=0.10m,Q=+24μC,λ=1.33×10-5C/m.By putting (27) and (32) into (31), we get(33)|ΔE(x)|≈2×10-28E0x.Finally, if we put (28) and (32) in (31), we obtain(34)ΔEx≲10-32E0x.In fact, as is clear from (33) and (34), the nonlinear corrections to electric field in (18) are very small for weak electric fields. The authors of [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 [25–27].



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