The body under consideration is a uniformly charged three-dimensional cube with length L. The cube con- tains a uniformly distributed positive charge, Q, thus,.
Coulomb self-energy of a uniformly charged three-dimensional cube Orion Ciftja Department of Physics, Prairie View A&M University, Prairie View, TX 77446, USA (Dated: October 1, 2010) In a previous work [Physics Letters A 374 (2010) 981-983] we reported the exact calculation of the Coulomb self-energy of a uniformly charged two-dimensional square. The purpose of this brief extension of the earlier work is to report the corresponding exact result for the Coulomb selfenergy of a uniformly charged three-dimensional cube. The calculations were carried out by using a straightforward generalization of the same mathematical method adopted in the earlier study. PACS numbers: 73.43.Cd, 73.20.Dx.
In a previous work [1], we reported the exact calculation of the Coulomb self-energy of a uniformly charged square. In that study, we introduced a mathematical transformation which ultimately led us to the exact calculation of the Coulomb self-energy. This brief extension of the earlier work reports the corresponding exact result for the Coulomb self-energy of a uniformly charged threedimensional cube. The result was obtained by using a straightforward generalization of the same mathematical method adopted in the earlier study [1]. The body under consideration is a uniformly charged three-dimensional cube with length L. The cube contains a uniformly distributed positive charge, Q, thus, the uniform charge density is: ρ0 =
Q . L3
(1)
We assume a Coulomb interaction between elementary charges dq1 = ρ0 d3 r1 and dq2 = ρ0 d3 r2 and write the Coulomb self-energy of the three-dimensional cube as: Z Z ke ρ20 1 3 Vbb = d r1 d 3 r2 , (2) 2 |~r1 − ~r2 | D D where Vbb is the background-background (bb) electrostatic energy, namely, the Coulomb self-energy term, ke is Coulomb’s electric constant, ~ri = (xi , yi , zi ) (i=1,2) are three-dimensional position vectors and D is the cubic domain of integration. We choose a Cartesian system of coordinates with origin at the center of the cube. Thus, the domain of integration is D : − L2 ≤ xi ≤ + L2 ; − L2 ≤ yi ≤ + L2 ; − L2 ≤ zi ≤ + L2 . In three dimensions we have: Z ∞ 2 2 2 2 1 2 =√ du e−u [(x1 −x2 ) +(y1 −y2 ) +(z1 −z2 ) ] . |~r1 − ~r2 | π 0 (3) One then proceeds to substitute Eq.(3) into Eq.(2) and to introduce the dimensionless variables, Xi = xi /L, Yi = yi /L and t = u L. At this stage, one can carry out the calculations by using the same approach as in the earlier study [1] to eventually obtain: Z ∞ ke Q2 1 3 √ dt [g(t)] , (4) Vbb = L π 0
where the auxiliary function, g(t) is given by: √ 2 −1 + e−t + π t erf (t) , (5) t2 √ Rz 2 and erf (z) = 2/ π 0 dx e−x is an error function [2]. The important result so far is to have succeeded on expressing the six-dimensional integral arising from the expression for the Coulomb self-energy of a uniformly charged three-dimensional cube as a simple onedimensional integral. Despite the mathematical difficulties faced, were able to carry out exactly the integration in Eq.(4). Here we report the exact value for the Coulomb self-energy of a uniformly charged threedimensional cube with charge Q and length L that reads: ( ) √ √ h √ √ i ke Q2 1 + 2 − 2 3 π − + ln 1 + 2 2 + 3 . Vbb = L 5 3 (6) If one approximates it numerically, this gives: g(t) =
Vbb ≈ 0.941156
ke Q2 . L
(7)
We note that the present result for the exact Coulomb self-energy of a uniformly charged three-dimensional cube is an extension of our previous work [1] on the exact Coulomb self-energy for a uniformly charged twodimensional square. Both calculations rely on a mathematical tranformation that can be straightforwardly generalizated to any dimension and this observation was the key ingredient that allowed us to obtain the exact Coulomb self-energies in both cases. Approximating the Coulomb self-energy of a charge distribution within a three-dimensional domain and the mutual Coulomb energy of two charge distributions often constitutes a computational bottleneck in the simulation of physical systems [3]. The numerical evaluation of the six-dimensional integrals arising from Coulomb interactions requires sophisticated computational techniques and is very time-consuming. For instance, a recently developed computational technique from integral geometry was implemented in numerical calculations using NAG standard software packages [4] to obtain a very precise
2 numerical value (see Eq.(15) in Ref. [3]) for the Coulomb self-energy of a uniformly charged cube. With an estimated error of the order of 10−14 , the reported value 2 was: ≈ 0.94115632219486 keLQ . It is very refreshing to discover that the above numerical value is nothing else but the approximated value of our exact analytic result given in Eq.( 6). Similarly to the case of two-dimensional electronic systems [5, 6], we remark that the present result can be used in studies of electrostatic problems involving cubic geometries or in systematic studies of properties of finite systems of electrons embedded in a three-dimensional cubic background of uniform positive charge.
Acknowledgements
This research was supported in part by the NSF Grant No. DMR-0804568.
[1] O. Ciftja, Phys. Lett. A 374 (2010) 981. [2] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Ninth printing, Dover, 1970, pg. 297. [3] D. Finocchiaro, M. Pellegrini, and P. Bientinesi, J. Comput. Phys. 146, 707 (1998). [4] Numerical Algorithms Group (http://www.nag.com/). [5] O. Ciftja, Physica B 404, 227 (2009). [6] O. Ciftja, Physica B 404, 2244 (2009).
