Exact Calculation of the Capacitance and the Electrostatic Potential Energy for a Nonlinear Parallel-Plate Capacitor in a Two-Parameter Modification of Born-Infeld Electrodynamics
aa r X i v : . [ phy s i c s . c l a ss - ph ] A p r Exact Calculation of the Capacitance and theElectrostatic Potential Energy for a NonlinearParallel-Plate Capacitor in a Two-ParameterModification of Born-Infeld Electrodynamics
S. K. Moayedi ∗ , F. Fathi † Department of Physics, Faculty of Sciences, Arak University, Arak 38156-8-8349, Iran
Abstract
The nonlinear capacitors are important devices in modern technologies and applied physics. The aim of thispaper is to calculate exactly the capacitance and the electrostatic potential energy of a nonlinear parallel-plate capacitor by using a two-parameter modification of Born-Infeld electrodynamics. Our calculationsshow that the capacitance and the electrostatic potential energy of a nonlinear parallel-plate capacitor inmodified Born-Infeld theory have the weak field expansions C = ǫ Ad + O ( q ) and U = q ǫ Ad ) + O ( q ),where q is the amount of electric charge on each plate of the capacitor. It is demonstrated that the results ofthis paper are in agreement with the results of Maxwell electrodynamics for weak electric fields. Numericalevaluations show that the nonlinear electrodynamical effects in modified Born-Infeld theory are negligible inthe weak field regime. Keywords:
Classical field theories; Applied classical electromagnetism; Other special classical fieldtheories; Nonlinear or nonlocal theories and models; Nonlinear Capacitor
PACS:
The nonlinear capacitors have found a wide range of applications in circuit theory in electrical engineering anddifferent branches of applied physics [1-7]. In a nonlinear capacitor, in contrast with the ordinary capacitors,the capacitance is a function of voltage, i.e., C = f ( △ φ ) where △ φ is the potential difference between the platesof the capacitor and the function f ( △ φ ) can be determined empirically or from a fundamental electromagnetictheory like Maxwell electrodynamics or Born-Infeld theory [1,2,8-10]. In Ref. [2], it has been shown that for anonlinear capacitor which its capacitance depends linearly on the voltage, the usual relation U = C ( △ φ ) isnot satisfied. The authors of Ref. [6] have shown that the electrostatic potential energy of a nonlinear capacitorcan be expanded as follows: U = αq + βq + γq + ..., (1)where α , β , γ , and ... are material dependent constants. Today we know that the interaction between the chargedbodies can be described by Maxwell equations classically [8]. On the other hand, Maxwell electrodynamics suffers ∗ Corresponding author, E-mail: [email protected] † E-mail: [email protected] [9]: L BI = ǫ β (cid:26) − s c β F µν F µν − c β ( F µν ⋆ F µν ) (cid:27) , (2)where F µν = ∂ µ A ν − ∂ ν A µ is the Faraday tensor, A µ = ( c φ, A ) is the gauge potential, ⋆F µν = ǫ µναβ F αβ isthe dual field tensor, and β is the maximum value of the electric field in Born-Infeld theory. In string theorythe dynamics of electromagnetic fields on D -branes can be represented by a Born-Infeld type theory [10]. Thesolutions of Born-Infeld equations for an infinite charged line and an infinitely long cylinder have been obtainedin Ref. [11]. In 1930s, Heisenberg, Euler, and Kockel studied the scattering of light by light according to Dirac’shole theory [12-14]. They showed that Maxwell electrodynamics should be corrected by adding nonlinear termsdue to the quantum electrodynamical effects [12-14]. It must be emphasized that, for weak electromagneticfields, the Lagrangian density and the energy density of nonlinear electrodynamics have the following explicitexpressions: L = ∞ X i =0 ∞ X j =0 c i,j F i G j , (3) u = ∞ X i =0 ∞ X j =0 c i,j (cid:0) ǫ iF ( i − G j E + ( j − F i G j (cid:1) , (4)where c i,j are field-independent parameters, F := (cid:0) ǫ E − B µ (cid:1) , and G := q ǫ µ ( E . B ) [15,16]. Note that inthe weak field regime (2) is a particular example of (3). The effect of nonlinear corrections on the electricfield between the plates of a parallel-plate capacitor has been studied in the framework of Heisenberg-Euler-Kockel electrostatics [17]. In a recent paper, the capacitance and the electrostatic potential energy for aparallel-plate and spherical capacitors have been computed in ordinary Born-Infeld theory [18]. Iacopini andZavattini suggested and developed a ( p, τ )-two-parameter modification of Born-Ifeld electrodynamics, in whichthe electrostatic self-energy of a point charge becomes a finite value for p < p takes the values { , , , } . It isverified that the results of Section 4 for p = are compatible with those obtained previously in [18]. Numericalevaluations in summary and conclusions indicate that the nonlinear corrections to the electrostatic potentialenergy of a parallel-plate capacitor in modified Born-Infeld electrostatics are not important in the weak fieldregime. We use SI units in this paper. The space-time metric has the signature (+ , − , − , − ). A Brief Review of Modified Born-Infeld Electrodynamics
The modified Born-Infeld electrodynamics in a (3+1)-dimensional Minkowski space-time is described by thefollowing Lagrangian density (see Eq. (B.10) in Ref. [19]): L p,τ = 12 p ǫ β (cid:26) − (cid:20) c β F µν F µν − τ c β (cid:0) F µν ⋆ F µν (cid:1) (cid:21) p (cid:27) − J µ A µ , (5)where p < τ is another dimensionless constant, and J µ = ( cρ, J ) is an externalcurrent for the U (1) gauge field A µ . The parameter β in Eq. (5) is called the Born-Infeld parameter and showsthe upper limit of the electric field in modified Born-Infeld electrodynamics. It is necessary to note that for p = , τ = 1 the Lagrangian density in Eq. (5) becomes the standard Born-Infeld Lagrangian density [9], whilefor p = 1, τ = 0 we obtain the Maxwell Lagrangian density. The equation of motion for the vector field A λ is ∂ L p,τ ∂A λ − ∂ σ (cid:18) ∂ L p,τ ∂ ( ∂ σ A λ ) (cid:19) = 0 . (6)If we put Eq. (5) into Eq. (6), we will get the inhomogeneous modified Born-Infeld equations as follows: ∂ σ (cid:18) F σλ − τ c β (cid:0) F µν ⋆ F µν (cid:1) ⋆ F σλ (cid:2) c β F µν F µν − τ c β (cid:0) F µν ⋆ F µν (cid:1) (cid:3) − p (cid:19) = µ J λ . (7)The dual field tensor ⋆F µν satisfies the following Bianchi identity: ∂ µ ⋆ F µν = 0 . (8)In (3+1)-dimensional space-time, the Faraday 2-form F and its dual ⋆F have the following expressions [20]: F = F i dx ∧ dx i + 12 F ij dx i ∧ dx j = E i dt ∧ dx i − ǫ ijk B i dx j ∧ dx k , (9) ⋆F = − B i dt ∧ dx i − ǫ ijk E i dx j ∧ dx k , (10)where i, j, k = 1 , ,
3, and { E i } = { E x , E y , E z } , { B i } = { B x , B y , B z } . Using Eqs. (9) and (10), Eqs. (7) and (8) take the following vector forms: ∇ · D ( x , t ) = ρ ( x , t ) , (11) ∇ × H ( x , t ) = J ( x , t ) + ∂ D ( x , t ) ∂t , (12) ∇ · B ( x , t ) = 0 , (13) ∇ × E ( x , t ) = − ∂ B ( x , t ) ∂t , (14)where D ( x , t ) and H ( x , t ) are given by D ( x , t ) = ǫ E ( x , t ) + τ c β (cid:0) E ( x , t ) . B ( x , t ) (cid:1) B ( x , t )Ω p,τ p − , (15) H ( x , t ) = 1 µ B ( x , t ) − τ β (cid:0) E ( x , t ) . B ( x , t ) (cid:1) E ( x , t )Ω p,τ p − , (16)3nd Ω p,τ is defined as follows: Ω p,τ := (cid:20) c β F µν F µν − τ c β (cid:0) F µν ⋆ F µν (cid:1) (cid:21) p . (17)Now, let us study the electrostatic case where B = J = 0 and all other physical quantities are time independent.In this case the modified Born-Infeld equations (11)-(14) are ∇ · (cid:18) E ( x )[1 − E ( x ) β ] − p (cid:19) = ρ ( x ) ǫ , (18) ∇ × E ( x ) = 0 . (19)The above equations are basic equations of modified Born-Infeld electrostatics [19]. By using the divergencetheorem in vector calculus, we get the integral form of Eq. (18) as follows: I C − E ( x ) β ] − p E ( x ) . ˆ n da = 1 ǫ Z C ρ ( x ) d x, (20)where C is a 2-chain which is the boundary of a 3-chain C , i.e., C = ∂C [20]. Equation (20) is Gauss’s lawin modified Born-Infeld electrostatics. In this section, we obtain the symmetric energy-momentum tensor for modified Born-Infeld electrodynamics.According to Eq. (5), the Lagrangian density for modified Born-Infeld electrodynamics in the absence of externalcurrent J µ is L p,τ = 12 p ǫ β (cid:26) − (cid:20) c β F µν F µν − τ c β (cid:0) F µν ⋆ F µν (cid:1) (cid:21) p (cid:27) . (21)From (21), we derive the following classical field equation: ∂ σ (cid:18) F σλ − τ c β (cid:0) F µν ⋆ F µν (cid:1) ⋆ F σλ Ω p,τ p − (cid:19) = 0 . (22)The canonical energy-momentum tensor for Eq. (21) is [21-23]Θ σ η = ∂ L p,τ ∂ ( ∂ σ A λ ) ( ∂ η A λ ) − δ σ η L p,τ . (23)If we substitute (21) into (23) and use (22), we will obtain the following expression for the canonical energy-momentum tensor Θ σ η :Θ σ η = 1 µ F σλ − τ c β (cid:0) F µν ⋆ F µν (cid:1) ⋆ F σλ Ω p,τ p − F λη + 12 p ǫ β (Ω p,τ − δ σ η + ∂ λ R λσ η , (24)4here R λσ η := 1 µ F λσ − τ c β (cid:0) F µν ⋆ F µν (cid:1) ⋆ F λσ Ω p,τ p − A η , (25) R σλ η = −R λσ η . (26)It is well known that the canonical energy-momentum tensor Θ σ η in (23) is generally not symmetric [21-24].Belinfante showed that the canonical energy-momentum tensor Θ σ η in Eq. (23) can be written as follows [21]:Θ σ η = T σ η + ∂ λ R λσ η , (27)where the second- and third-order tensors T σ η and R λσ η must satisfy the following conditions: T ση = T ησ , (28a) R λσ η = −R σλ η . (28b)The second-order tensor T ση in the above equations is called the symmetric energy-momentum tensor [22]. Acomparison between Eqs. (24) and (27) clearly shows that the symmetric energy-momentum tensor for modifiedBorn-Infeld electrodynamics is T σ η = 1 µ F σλ − τ c β (cid:0) F µν ⋆ F µν (cid:1) ⋆ F σλ Ω p,τ p − F λη + 12 p ǫ β (Ω p,τ − δ σ η . (29)After straightforward but tedious calculations, one finds that in the presence of an external current the symmetricenergy-momentum tensor T σ η in (29) satisfies the following equation: ∂ σ T σ η = J σ F ση . (30)Using Eqs. (9) and (10) together with Eq. (29), the energy density of modified Born-Infeld electrodynamics isgiven by u ( x , t ) = T ( x , t )= 12 p ǫ β (cid:26) (2 p − (cid:18) E ( x ,t ) β + τ (cid:0) E ( x ,t ) .c B ( x ,t ) (cid:1) β (cid:19) + c B ( x ,t ) β + 1 (cid:20) − (cid:0) E ( x ,t ) − c B ( x ,t ) (cid:1) β − τ (cid:0) E ( x ,t ) .c B ( x ,t ) (cid:1) β (cid:21) − p − (cid:27) . (31)According to Eq. (31), the energy density of an electrostatic field in modified Born-Infeld electrodynamicsbecomes u ( x ) = T ( x ) = 12 p ǫ β (cid:20) (2 p − E ( x ) β + 1 (cid:0) − E ( x ) β (cid:1) − p − (cid:21) . (32)For p = , the modified electrostatic energy density in Eq. (32) becomes the energy density of an electrostaticfield in Born-Infeld electrodynamics, i.e., u ( x ) = ǫ β (cid:18) q − E ( x ) β − (cid:19) . (33) It is obvious that for source-free modified Born-Infeld theory the right-hand side of (30) vanishes, i.e., ∂ σ T σ η = 0. Calculation of the Capacitance and the Electrostatic PotentialEnergy of a Nonlinear Parallel-Plate Capacitor in Modified Born-Infeld Theory
In order to calculate the capacitance of a nonlinear parallel-plate capacitor in modified Born-Infeld electrostatics,we assume a capacitor composed of two large parallel conducting plates with area A and separation d (see Figure1).Figure 1: A parallel-plate capacitor. The Gaussian surface is represented by dashed lines. The symmetry of the problemimplies that E ( x ) = E z ( − ˆ e z ), where ˆ e z is the unit vector in the z -direction. By applying modified Gauss’s law in (20) to the Gaussian surface in Figure 1, we obtain the following equationfor E z E z Γ + (cid:18) qβ ǫ A (cid:19) Γ E z − (cid:18) qǫ A (cid:19) Γ = 0 , (34)where Γ := − p . Now, let us obtain the exact solutions of Eq. (34) for p ∈ { , , , } .For p = (Γ = 2), Eq. (34) becomes the quadratic equation (cid:20) (cid:18) qβǫ A (cid:19) (cid:21) E z − (cid:18) qǫ A (cid:19) = 0 . For p = (Γ = 3), Eq. (34) becomes the cubic equation E z + (cid:18) qβ ǫ A (cid:19) E z − (cid:18) qǫ A (cid:19) = 0 . For p = (Γ = 4), Eq. (34) becomes the following quartic equation E z + (cid:18) qβ ǫ A (cid:19) E z − (cid:18) qǫ A (cid:19) = 0 . Finally, for p = (Γ = 6), (34) becomes the following sextic equation E z + (cid:18) qβ ǫ A (cid:19) E z − (cid:18) qǫ A (cid:19) = 0 . Note that the above sextic equation by a suitable change of variable reduces to a cubic equation. In Galoistheory, the Abel-Ruffini theorem or Abel’s impossibility theorem says that: for all n ≥ , there is a polynomial n Q [ x ] of degree n that is not solvable by irreducible radicals over Q [25,26]. According to Abel’s impossibilitytheorem, equation (34) has the following exact solutions for p ∈ { , , , } E z p = 12 = qǫ A Π p = 12 ( q ) , (35a) E z p = 23 = qǫ A Π p = 23 ( q ) , (35b) E z p = 34 = qǫ A Π p = 34 ( q ) , (35c) E z p = 56 = qǫ A Π p = 56 ( q ) , (35d)where Π p = 12 ( q ) := 1 q qβǫ A ) , (36a)Π p = 23 ( q ) := vuut − β (cid:18) qǫ A (cid:19) + s − β (cid:18) qǫ A (cid:19) + vuut − β (cid:18) qǫ A (cid:19) − s − β (cid:18) qǫ A (cid:19) − β (cid:18) qǫ A (cid:19) , (36b)Π p = 34 ( q ) := vuuts β (cid:18) qǫ A (cid:19) − β (cid:18) qǫ A (cid:19) , (36c)Π p = 56 ( q ) := vuuut vuut
12 + s
14 + 127 β (cid:18) qǫ A (cid:19) + vuut − s
14 + 127 β (cid:18) qǫ A (cid:19) . (36d)When the Born-Infeld parameter β takes the large values, the behavior of the electric fields in (35a)-(35d) aregiven by E z p = 12 = (cid:18) qǫ A (cid:19)(cid:20) − β − (cid:18) qǫ A (cid:19) + 38 β − (cid:18) qǫ A (cid:19) + O (cid:0) β − (cid:1)(cid:21) , (37a) E z p = 23 = (cid:18) qǫ A (cid:19)(cid:20) − β − (cid:18) qǫ A (cid:19) + 19 β − (cid:18) qǫ A (cid:19) + O (cid:0) β − (cid:1)(cid:21) , (37b) E z p = 34 = (cid:18) qǫ A (cid:19)(cid:20) − β − (cid:18) qǫ A (cid:19) + 132 β − (cid:18) qǫ A (cid:19) + O (cid:0) β − (cid:1)(cid:21) , (37c) E z p = 56 = (cid:18) qǫ A (cid:19)(cid:20) − β − (cid:18) qǫ A (cid:19) − β − (cid:18) qǫ A (cid:19) + O (cid:0) β − (cid:1)(cid:21) . (37d)It must be emphasized that in obtaining the above results, the Mathematica software has been used [27]. All ofthe electric fields (37a)-(37d), have the Maxwellian limit in the weak field regime. The first term on the right-hand side of (37a)-(37d) represent the electric field between the plates of a parallel-plate capacitor in Maxwell Q is the field of rational numbers (see page 116 in Ref. [26]). E ( x ) in the following way: E ( x ) = − ∇ φ ( x ) , (38)where φ ( x ) is the electrostatic potential. From (38) we obtain the following relation: △ φ = − Z fi E ( x ) .d l , (39)where △ φ = φ f − φ i is the potential difference between the initial and final points, and d l is an infinitesimaldisplacement vector. If we use Eqs. (35) and (39), we will get the following expressions for the potentialdifference between the plates of a nonlinear parallel-plate capacitor for p ∈ { , , , }△ φ p = 12 = − Z + − qǫ A Π p = 12 ( q )( − ˆ e z ) . (ˆ e z dz )= qdǫ A Π p = 12 ( q ) , (40a) △ φ p = 23 = − Z + − qǫ A Π p = 23 ( q )( − ˆ e z ) . (ˆ e z dz )= qdǫ A Π p = 23 ( q ) , (40b) △ φ p = 34 = − Z + − qǫ A Π p = 34 ( q )( − ˆ e z ) . (ˆ e z dz )= qdǫ A Π p = 34 ( q ) , (40c) △ φ p = 56 = − Z + − qǫ A Π p = 56 ( q )( − ˆ e z ) . (ˆ e z dz )= qdǫ A Π p = 56 ( q ) . (40d)As is well known, in electrostatics the capacitance C of a capacitor is the ratio of the amount of charge on eachplate of a capacitor to the potential difference between the plates of the capacitor, i.e., C = q △ φ . (41)It is necessary to note that Eq. (41) is also applicable for determination of the capacitance of nonlinearcapacitors [2,17,18]. After inserting (40) into (41), the capacitance of a nonlinear parallel-plate capacitor inmodified Born-Infeld electrostatics for p ∈ { , , , } becomes C p = 12 = ǫ Ad p = 12 ( q ) , (42a) C p = 23 = ǫ Ad p = 23 ( q ) , (42b) C p = 34 = ǫ Ad p = 34 ( q ) , (42c) C p = 56 = ǫ Ad p = 56 ( q ) . (42d)8he above equations have the following weak field expansions: C p = 12 = (cid:18) ǫ Ad (cid:19)(cid:20) β − (cid:18) qǫ A (cid:19) − β − (cid:18) qǫ A (cid:19) + O (cid:0) β − (cid:1)(cid:21) , (42e) C p = 23 = (cid:18) ǫ Ad (cid:19)(cid:20) β − (cid:18) qǫ A (cid:19) − β − (cid:18) qǫ A (cid:19) + O (cid:0) β − (cid:1)(cid:21) , (42f) C p = 34 = (cid:18) ǫ Ad (cid:19)(cid:20) β − (cid:18) qǫ A (cid:19) + 132 β − (cid:18) qǫ A (cid:19) + O (cid:0) β − (cid:1)(cid:21) , (42g) C p = 56 = (cid:18) ǫ Ad (cid:19)(cid:20) β − (cid:18) qǫ A (cid:19) + 124 β − (cid:18) qǫ A (cid:19) + O (cid:0) β − (cid:1)(cid:21) . (42h)Equations (42e)-(42h) show that the capacitance of a nonlinear parallel-plate capacitor in modified Born-Infeldelectrostatics is a function of the amount of charge on each plate of the capacitor. Now, let us compute theenergy density between the plates of a nonlinear parallel-plate capacitor in modified Born-Infeld electrostatics.If we put (35) into (32), we will get the following results: u ( x ) p = 12 = ǫ β (cid:20) p = 12 ( q ) − (cid:21) , (43a) u ( x ) p = 23 = 34 ǫ β (cid:20) β (cid:18) qǫ A Π p = 23 ( q ) (cid:19) + 1Π p = 23 ( q ) − (cid:21) , (43b) u ( x ) p = 34 = 23 ǫ β (cid:20) β (cid:18) qǫ A Π p = 34 ( q ) (cid:19) + 1Π p = 34 ( q ) − (cid:21) , (43c) u ( x ) p = 56 = 35 ǫ β (cid:20) β (cid:18) qǫ A Π p = 56 ( q ) (cid:19) + 1Π p = 56 ( q ) − (cid:21) . (43d)9sing Eq. (43), the electrostatic potential energy of a nonlinear parallel-plate capacitor in modified Born-Infeldelectrostatics according to Figure 1 becomes U p = 12 = Z area of a plate da Z d dz u ( x ) p = 12 = ǫ β (cid:20)r qβǫ A ) − (cid:21) Ad, (44a) U p = 23 = Z area of a plate da Z d dz u ( x ) p = 23 = 34 ǫ β (cid:20) β (cid:18) qǫ A Π p = 23 ( q ) (cid:19) + 1Π p = 23 ( q ) − (cid:21) Ad, (44b) U p = 34 = Z area of a plate da Z d dz u ( x ) p = 34 = 23 ǫ β (cid:20) β (cid:18) qǫ A Π p = 34 ( q ) (cid:19) + 1Π p = 34 ( q ) − (cid:21) Ad, (44c) U p = 56 = Z area of a plate da Z d dz u ( x ) p = 56 = 35 ǫ β (cid:20) β (cid:18) qǫ A Π p = 56 ( q ) (cid:19) + 1Π p = 56 ( q ) − (cid:21) Ad. (44d)For the large values of the Born-Infeld parameter β , the behavior of the electrostatic potential energies in(44a)-(44d) are given by U p = 12 | large β = U M (cid:20) − β − (cid:18) qǫ A (cid:19) + 18 β − (cid:18) qǫ A (cid:19) + O (cid:0) β − (cid:1)(cid:21) , (45a) U p = 23 | large β = U M (cid:20) − β − (cid:18) qǫ A (cid:19) + 127 β − (cid:18) qǫ A (cid:19) + O (cid:0) β − (cid:1)(cid:21) , (45b) U p = 34 | large β = U M (cid:20) − β − (cid:18) qǫ A (cid:19) + 196 β − (cid:18) qǫ A (cid:19) + O (cid:0) β − (cid:1)(cid:21) , (45c) U p = 56 | large β = U M (cid:20) − β − (cid:18) qǫ A (cid:19) − β − (cid:18) qǫ A (cid:19) + O (cid:0) β − (cid:1)(cid:21) , (45d)where U M = q C M and C M = ǫ Ad are the electrostatic potential energy and the capacitance of a parallel-platecapacitor in Maxwell electrostatics respectively. Equation (45) shows that the relation U M = q C M is not true fora parallel-plate capacitor in modified Born-Infeld electrostatics. In the limit of β → ∞ , equations (45a)-(45d)reduce to the following equation: U p = 12 | β = ∞ = U p = 23 | β = ∞ = U p = 34 | β = ∞ = U p = 56 | β = ∞ = U M . (46)10 Summary and Conclusions
In 1930s Born-Infeld theory was introduced in order to remove the infinite self-energy of the electron in Maxwellelectrodynamics [9]. In Born-Infeld electrodynamics the absolute value of the electric field has an upper limit β , i.e., | E | ≤ β . In Born-Infeld paper the numerical value of β was [9,28]: β Born − Infeld = 1 . × Vm . (47)Soff, Rafelski, and Greiner obtained the following lower bound on β [29]: β Soff ≥ . × Vm . (48)In a paper about photon-photon scattering and photon splitting in a magnetic field in Born-Infeld theory, Davilaand his coworkers obtained the following new lower bound on β [30]: β Davila ≥ . × Vm . (49)In 1983, E. Iacopini and E. Zavattini introduced a ( p, τ )-two-parameter modification of Born-Infeld electro-dynamics, in which the self-energy of a point-like charge becomes finite for p < p ∈ { , , , } . In order to have a deeper understanding of nonlinear effects in modified Born-Infeldelectrostatics, let us rewrite (45a) as follows: U p = 12 | large β = U M + U first − order nonlinear correctionM + U second − order nonlinear correctionM + O (cid:0) β − (cid:1) , (50)where U first − order nonlinear correctionM := − β ǫ Ad U M , (51a) U second − order nonlinear correctionM := 12 β ǫ A d U M . (51b)Now, let us estimate the numerical values of U M , U first − order nonlinear correctionM , and U second − order nonlinear correctionM in (50). For this aim, we use the following typical values for a parallel-plate capacitor (see page 804 in Ref.[31]): A = 100 cm , d = 1 . mm, q = 1 . × − C. (52)If we put Eqs. (47), (48), (49), and (52) into Eq. (50), we get U M = 6 . × − J, (53a) U first − order nonlinear correctionM ( Born − Infeld ) = − . × − J, (53b) U second − order nonlinear correctionM ( Born − Infeld ) = 7 . × − J, (53c) U first − order nonlinear correctionM ( Soff ) = − . × − J, (53d) U second − order nonlinear correctionM ( Soff ) = 1 . × − J, (53e) U first − order nonlinear correctionM ( Davila ) = − . × − J, (53f) U second − order nonlinear correctionM ( Davila ) = 1 . × − J. (53g)11t must be noted that in (53d)-(53g) the minimum value of β in Eqs. (48) and (49) has been used. Equations(53a)-(53g) tell us that the nonlinear corrections to electrostatic potential energy in a parallel-plate capacitorare not important in the weak field limit. For p = (Γ = 8), Eq. (34) becomes E z + (cid:18) qβ ǫ A (cid:19) E z − (cid:18) qǫ A (cid:19) = 0 . 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