Electrostatics_of_a_Conducting_Toroidal_Surface_and_a_Point_Charge

Electrostatics of a conducting toroidal surface and a point charge J. A. Hernandes Departamento de Fisica, ICE, Universidade Federal de Juiz de Fora, 36036-900 Juiz de Fora, MG, Brazil Abstract We find the exact expression for the electrostatic potential due to a grounded toroidal conducting surface and a point charge outside the toroidal surface. We use the approach provided by Green’s method. We also find the induced surface charge on the toroidal conductor and the net force between the toroid and the point charge. We show how the potential and induced charges behave on some special cases. Keywords: Electric potential, induced charges, electrostatics, toroidal conductor 1. Introduction Recently, there’s been a renewed interest in the toroidal geometry and toroidal functions, particularly on the fields of astrophysics [CT99, CTRS00], nanoparticles [MKH+ 07] and toroidal function calculations [And06, GS97]. The toroidal coordinates system is well known, and Laplace’s equation is separable in this system. This is suitable for solving problems involving the gravitational or electric potential in the toroidal geometry, like the electrostatics of a charged toroid, [Smy89, p. 239], [MF53, p. 1304], or even the magnetostatics of steady currents along the toroid, [HA03, HMLA08]. The aim of this work is to calculate the electrostatic potential that arises due to the proximity of a point charge outside a conducting toroid. We present the solution of this problem for the electric potential, along with the force between them and the surface charges induced on the toroid surface. To our knowledge this has never been done before. To this end we consider the Green’s function method, [Jac99, Chap. 1-3]. Email address: jahernandes@fisica.ufjf.br (J. A. Hernandes) Preprint submitted to Elsevier September 14, 2010 2. Toroidal coordinates, scale factors and differential operators We use toroidal coordinates, given by (η, χ, ϕ) [MS88, p. 112], defined by: x=a sinh η cos ϕ, cosh η − cos χ z=a y=a sinh η sin ϕ, cosh η − cos χ (1) and the coordinates are constrained by: 0 ≤ η < ∞, −π ≤ χ ≤ π and −π ≤ ϕ ≤ π. The inverse relations are given by: η = arctanh x2 2a x2 + y 2 , + y 2 + z 2 + a2 χ = arctan 2za , x 2 + y 2 + z 2 − a2 (2) sin χ , cosh η − cos χ y ϕ = arctan . x It is convenient to present the following relations: cosh η = x 2 + y 2 + z 2 + a2 (x2 + y 2 + z 2 − a2 )2 + 4a2 z 2 (x2 + y 2 + z 2 − a2 )2 + 4a2 z 2 x 2 + y 2 + z 2 − a2 , cos χ = . (3) The scale factors hi , for coordinate ξi relative to the cartesian coordinates [AW05, sec. 2.1] are given by: hi = ∂x ∂ξi 2 + ∂y ∂ξi 2 + ∂x ∂ξi 2 , which result for toroidal coordinates as the following scale factors: hη = hχ = a , cosh η − cos χ hϕ = a sinh η . cosh η − cos χ (4) 3. Solution of Poisson’s equation We want to obtain the electric potential φ due to a point charge q near a conducting toroidal shell. In toroidal coordinates, the surface can be described by η = η0 . The point charge is at position r = (η , χ , ϕ ), and can be located inside (η > η0 ) or outside (η < η0 ) the toroid. We suppose the medium to be vacuum, and we use SI units. We intend to solve Poisson’s equation: 2 φ=− ρ . ε0 (5) 2 By standard Green’s function method, the solution of Poisson’s equation for this case with Dirichlet boundary condition (in which the potential on the toroidal surface is maintained at a constant value φ0 ) is given by: φ(r) = 1 4πε0 ρ(r )G(r, r )dv − 1 4π φ(r ) S V ∂G da , ∂n (6) where V is the volume which contains the point charge (the volume of the toroid if the charge is inside, or all of the surrounding space if the charge is outside), S is the toroidal surface, ∂/∂n is the normal derivative at the surface directed outwards from V , and G(r, r ) is Green’s function satisfying the equation: 2 r G(r, r ) = −4πδ(r − r ). (7) As we set the toroidal conducting surface to be in electrostatic equilibrium at a constant potential φ0 we impose that G(r, r ) = 0 at this surface. We expand the Dirac delta function in toroidal coordinates as given by: δ(r − r ) = = δ(η − η ) δ(χ − χ ) δ(ϕ − ϕ ) hη hχ hϕ (8) (cosh η − cos χ)3 δ(η − η )δ(χ − χ )δ(ϕ − ϕ ). a3 sinh η The delta functions for χ and ϕ can be expanded by: δ(χ − χ ) = δ(ϕ − ϕ ) = 1 2π 1 2π ∞ eip(χ−χ ) , p=−∞ ∞ (9) eiq(ϕ−ϕ ) . q=−∞ (10) Laplace’s equation in toroidal coordinates is R-separable: the solution of 2 φ = √ 0 is given by a combination of functions φpq = cosh η − cos χHpq (η)Xp (χ)Φq (φ). Therefore, we can expand Green’s function in a similar manner [CTRS00, Eq. (48)]: G(r, r ) = ∞ 1 4π 2 e cosh η − cos χ ∞ cosh η − cos χ (11) × ip(χ−χ ) q=−∞ eiq(ϕ−ϕ ) gpq (η, η ). p=−∞ By substituting the delta function, Eq. (8), and Green’s function expansion, Eq. (11), on Green’s equation, Eq. (7), we obtain: gpq + 1 cosh η q2 + p2 − gpq gpq − 2 sinh η 4 sinh η √ cosh η − cos χ 4π √ δ(η − η ), =− a sinh η cosh η − cos χ 3 (12) where the derivatives ( ) on the function gpq are relative to the argument cosh η. q For η = η , the solutions for gpq are toroidal Legendre functions Pp− 1 (cosh η) 2 1 and Qp− 2 (cosh η). Suppose that ψ1,pq (η< ) satisfies the boundary conditions for η < η and ψ2,pq (η> ) satisfies the boundary conditions for η > η . Generally, the functions ψ1,pq and ψ2,pq can be written as a linear combination of the toroidal Legendre functions: q ψ1,pq (η< ) = Apq Pp− 1 (cosh η< ) + Bpq Qq 1 (cosh η< ), p− 2 2 (13) (14) q ψ2,pq (η> ) = Cpq Pp− 1 (cosh η> ) + Dpq Qq 1 (cosh η> ). p− 2 2 Symmetry on the arguments of Green’s function G(r, r ) = G(r , r) requires that our solution must be gpq (η, η ) = ψ1,pq (η< )ψ2,pq (η> ), where η< or η> is the smaller or the larger between η and η , respectively. Since we have the asymptotic behavior limη→0 Qp− 1 (cosh η) → ∞, we must have Bpq = 0. 2 Green’s function must vanish at the toroidal conducting shell, at η = η0 , yielding q Dpq = −Cpq Pp− 1 (cosh η0 )/Qq 1 (cosh η0 ). The function gpq can then be written p− 2 2 as: q q gpq (η, η ) = Hpq Pp− 1 (cosh η< ) Pp− 1 (cosh η> ) 2 2 −Qq 1 (cosh η> ) p− 2 q Pp− 1 (cosh η0 ) 2 Qq 1 (cosh η0 ) p− 2 , (15) where Hpq = Apq Cpq can be determined by the discontinuity implied by the delta function. Integrating Eq. (15) from η − to η + (here η0 and we take the limit → 0), we obtain: dgpq dη η + η − = −Hpq sinh η q Pp− 1 (cosh η0 ) 2 2 Qq 1 (cosh η0 ) p− 4π . a sinh η (16) q ×W Pp− 1 (cosh η ), Qq 1 (cosh η ) = − p− 2 2 m In the previous equation, we used the Wronskian W {Pν (z), Qm (z)} = Γ(ν + ν m 2 m + 1)(−1) /[Γ(ν − m + 1)(1 − z )], [SG99, Eq. (2.4)]. We then obtain the coefficient Hpq : Hpq = − Qq 1 (cosh η0 ) Γ(p − q + 1/2) 4π p− (−1)q q 2 . a Pp− 1 (cosh η0 ) Γ(p + q + 1/2) 2 (17) Substituting on Green’s function, we finally obtain: G(r, r ) = ∞ 1 πa ∞ cosh η − cos χ ) Γ(p cosh η − cos χ eip(χ−χ p=−∞ ) × (−1)q eiq(ϕ−ϕ q=−∞ − q + 1/2) q P 1 (cosh η< ) Qq 1 (cosh η> ) p− 2 Γ(p + q + 1/2) p− 2 4 Qq 1 (cosh η0 ) p− q −Pp− 1 (cosh η> ) q 2 2 Pp− 1 (cosh η0 ) 2 . (18) We can use the following relations of toroidal Legendre functions, [AS65, Sec. 8.2], and Gamma functions, [AS65, Eq. 6.1.17]: q q P−p− 1 (z) = Pp− 1 (z), 2 2 (19) (20) (21) (22) (23) Qq 1 (z) = Qq 1 (z), p− −p− 2 2 −q Pp− 1 (z) = 2 Γ(p − q + 1/2) q P 1 (z), Γ(p + q + 1/2) p− 2 Γ(p − q + 1/2) q Q 1 (z), Γ(p + q + 1/2) p− 2 Q−q 1 (z) = p− 2 Γ(p − q + 1/2) Γ(−p − q + 1/2) = , Γ(−p + q + 1/2) Γ(p + q + 1/2) to rewrite Green’s function as: G(r, r ) = ∞ ∞ 1 πa cosh η − cos χ cosh η − cos χ p=0 (2 − δ0p ) cos p(χ − χ ) × q=0 (−1)q (2 − δ0q ) cos q(ϕ − ϕ ) Qq 1 (cosh η> ) p− 2 − Γ(p − q + 1/2) q P 1 (cosh η< ) Γ(p + q + 1/2) p− 2 . (24) × Qq 1 (cosh η0 ) p− q Pp− 1 (cosh η> ) q 2 2 Pp− 1 (cosh η0 ) 2 4. Conductor held at zero potential with external point charge Suppose that the conducting toroidal surface η = η0 is held at a constant potential φ0 = 0. We have a point charge q at r = (η , χ , ϕ ) outside the toroid, η < η0 . The potential can be obtained from Eq. (6) as: φ(r) = q 2ε a 4π 0 ∞ ∞ cosh η − cos χ cosh η − cos χ p=0 (2 − δ0p ) cos p(χ − χ ) × q=0 (−1)q (2 − δ0q ) cos q(ϕ − ϕ ) Qq 1 (cosh η> ) p− 2 − Γ(p − q + 1/2) q P 1 (cosh η< ) Γ(p + q + 1/2) p− 2 , (25) × Qq 1 (cosh η0 ) p− q Pp− 1 (cosh η> ) q 2 2 Pp− 1 (cosh η0 ) 2 where η< (η> ) is the smaller (greater) between η and η . Notice that η → 0 corresponds to infinite distances from the origin, or to the z-axis. η → ∞ 5 corresponds to the circle of radius a on xy-plane, which is inside the toroidal surface η0 . We can further simplify the expression for the potential by using the following formula [CTRS00, Eq. (48)]: 1 1 = |r − r | aπ ∞ ∞ cosh η − cos χ cosh η − cos χ eip(χ−χ ) p=−∞ × (−1)q eiq(ϕ−ϕ ) q=−∞ Γ(p − q + 1/2) q P 1 (cosh η< )Qq 1 (cosh η> ). p− 2 Γ(p + q + 1/2) p− 2 (26) The distance |r − r |, in toroidal coordinates can be expressed as: √ a 2ζ √ |r − r | = √ , cosh η − cos χ cosh η − cos χ where: ζ = cosh η cosh η − cos(χ − χ ) − sinh η sinh η cos(ϕ − ϕ ). The potential can then be written as: φ(r) = ∞ (27) (28) q 4πε0 a cosh η − cos χ ∞ cosh η − cos χ (29) 1 q × √ − Apq cos q(ϕ − ϕ )Pp− 1 (cosh η) , cos p(χ − χ ) 2 2ζ p=0 q=0 where the coefficient Apq is given by: q 1 Γ(p − q + 1/2) Qp− 1 (cosh η0 ) q q 2 P 1 (cosh η ). Apq = (2 − δ0p )(−1) (2 − δ0q ) q π Γ(p + q + 1/2) Pp− 1 (cosh η0 ) p− 2 2 (30) Figure 1 shows the equipotential lines of Eq. (29) at xy-plane. 4.1. Surface charges The induced charge distribution σi on the toroidal conducting surface can be obtained through Gauss’ law: σi (χ, ϕ) = −ε0 (−ˆ) · E(η0 ) = ε0 η =− q (cosh η0 − cos χ)3/2 4π 2 a2 sinh η0 ∞ cosh η − cos χ ∂φ a ∂η ∞ η=η0 cosh η − cos χ p=0 (2 − δ0p ) cos p(χ − χ ) . (31) × q=0 (−1)q (2 − δ0q ) cos q(ϕ − ϕ ) 6 q Pp− 1 (cosh η ) 2 q Pp− 1 (cosh η0 ) 2 The total induced charge qi can be found by integrating over the toroidal surface: π π qi = −π hχ dχ −π ∞ σ(χ, ϕ)hϕ dϕ 1 Qp− 2 (cosh η0 ) 1 Pp− 2 (cosh η0 ) √ q 2 =− π cosh η − cos χ p=0 (2 − δ0p ) cos pχ Pp− 1 (cosh η ) 2 ∞ √ = −q 2 cosh η − cos χ Ap cos pχ , p=0 (32) where we used the following integral representation of toroidal Legendre function, [Bat53, p. 156]: 1 1 Qp− 2 (z) = √ 2 2 π −π cos pχdχ √ z − cos χ (33) and the coefficient Ap = Ap0 from Eq. (30) with q = 0: Ap = 1 Qp− 2 (cosh η0 ) 1 P 1 (cosh η ). (2 − δ0p ) π Pp− 1 (cosh η0 ) p− 2 2 (34) Notice that for the point charge approaching the toroidal surface we have: η →η0 lim qi = −q, (35) where we used the following relation, [HA03, Eq. (16)]: √ ∞ 2 1 √ = (2 − δ0p ) cos pχ Qp− 1 (cosh η ). 2 π p=0 cosh η − cos χ (36) 4.2. Force between point charge and conductor The force on the point charge q due to the induced charges on the toroid can be calculated by the electric field on r without the contribution of the point charge itself. From E = − φ we have: Fq = q E(r ) = −q = cosh η − cos χ a ∞ ∂φ ∂φ 1 ∂φ η+ ˆ χ+ ˆ ϕ ˆ ∂η ∂χ sinh η ∂ϕ ∞ q Pp− 1 (cosh η ) 2 r q2 Apq (cosh η − cos χ ) 8πε0 a2 p=0 q=0 2 q q +2(cosh η − cos χ )Pp− 1 (cosh η ) sinh η η + sin χ Pp− 1 (cosh η )χ , ˆ ˆ 2 (37) q q where Pp− 1 (z) = dPp− 1 (z)/dz is the derivative of the toroidal Legendre func2 2 tion over the total argument. ˜ Figure 2 shows the normalized force Fq = Fq /(q 2 /8πε0 a2 ) of Eq. (37) along the x-axis. 7 4.3. Point charge on z-axis Suppose that the external point charge is located on the z-axis. In toroidal coordinates, we have η = 0 and cosh η = 1 (we must also have χ = 0). The q toroidal Legendre functions have the special value Pp− 1 (1) = δ0q . The potential 2 simplifies to: φ= q 4πε0 a 1 π ∞ cosh η − cos χ 1 − cos χ √ 2 1 cosh η − cos(χ − χ ) Pp− 1 (cosh η) . 2 (38) − p=0 (2 − δ0p ) cos p(χ − χ ) Qp− 1 (cosh η0 ) 2 1 Pp− 2 (cosh η0 ) The surface charge density can be written as: σ=− q (cosh η0 − cos χ)3/2 4π 2 a2 sinh η0 ∞ 1 − cos χ p=0 (2 − δ0p ) cos p(χ − χ ) . (39) 1 Pp− 2 (cosh η0 ) The total charge qi is given by: √ ∞ 1 Qp− 2 (cosh η0 ) q 2 qi = − . 1 − cos χ (2 − δ0p ) cos pχ π Pp− 1 (cosh η0 ) p=0 2 and the force on the point charge q is given by: Fq = q2 8πε0 a2 √ 1 − cos χ π ∞ (40) p=0 (2 − δ0p ) Qp− 1 (cosh η0 ) 2 1 Pp− 2 (cosh η0 ) × q q Pp− 1 (cosh η ) + 2(cosh η − cos χ )Pp− 1 (cosh η ) 2 2 × sinh η η + sin χ ˆ q Pp− 1 (cosh η 2 )χ . ˆ (41) 4.4. Potential far from the system Here we consider the potential due to the system (conducting toroid with a, where point charge) from a great distance from the origin, r = ρ2 + z 2 ρ = x2 + y 2 . In this case we have: χ = arctan 2za 2za ≈ 2 r 2 − a2 r r2 2aρ 2aρ ≈ 2 2 +a r ≈1+2 1, 1, aρ r2 2 η = arctanh cosh η = r 2 + a2 (r2 − a2 )2 + 4a2 z 2 8 =1+ , cos χ = (r2 − a2 )2 + 4a2 z 2 (r2 − r 2 − a2 ≈1−2 2 az r2 ≈ 2 , cosh η − cos χ = a a2 )2 + 4a2 z 2 a√ 2, r Far from the origin we have η< = η and η> = η . The toroidal Legendre functions can be approximated as, [Bat53, Sec. 3.9.2]: q Pp− 1 (1 + ) = δ0q + 2 ζ = cosh η − cos χ . √ δ1q p2 − 1/4 √ . 2 (42) The first order approximation will then take into consideration only the first term q = 0 of the summation on index q, which yields: φ(r a) = 1 (q + qi ), 4πε0 r (43) which is the potential of two point charges q and qi , where qi is given by Eq. (32). 4.5. Thin toroid A thin conducting toroidal surface can be caracterized by η0 1. We can use the following approximations for the toroidal Legendre functions with great 1), [Bat53, Sec. 3.9.2]: arguments (z = cosh η q P− 1 (z) = 2 2/π ln(2z) − ψ(1/2 − q) − γ √ , Γ(1/2 − q) z (p > 0) (p ≥ 0) (44) (45) (46) Γ(p) (2z)p−1/2 √ , 2 Γ(p − q + 1/2) π √ (−1)q π Γ(p + q + 1/2) q Qp− 1 (z) = , 2 Γ(p + 1) (2z)p+1/2 q Pp− 1 (z) = where ψ(k) = Γ (k)/Γ(k) is the digamma function, and γ = 0.577216 is the Euler gamma, [AS65, Sec. 6.3]. Using the relation Γ(q + 1/2)Γ(1/2 − q) = π(−1)q , [AS65, Eq. 6.1.17], we can also write: Qq 1 (z) − 2 q P− 1 (z) 2 = π 2 /2 , ln(2z) − ψ(1/2 − q) − γ (p > 0) (47) Qq 1 (z) p− 2 q Pp− 1 (z) 2 = (−1)q π Γ(p + q + 1/2)Γ(p − q + 1/2) , (2z)2p Γ(p + 1)Γ(p) (48) so that the potential will be determined by the terms with index p = 0 on Eq. (25): φ(η0 1) = q 4πε0 a cosh η − cos χ 9 cosh η − cos χ 1 π2 × √ − 2 2ζ ∞ q=0 (49) The induced surface charge density and the induced total charge are given by, respectively: σi (η0 1) = − q √ cosh η0 4π 2πa2 ∞ q q P− 1 (cosh η)P− 1 (cosh η ) (2 − δ0q ) 2 2 . cos q(ϕ − ϕ ) Γ2 (q + 1/2) ln(2 cosh η0 ) − ψ(1/2 − q) − γ cosh η − cos χ q=0 (−1)q (2 − δ0q ) × cos q(ϕ − ϕ ) qi (η0 2 , ln(2 cosh η0 ) − ψ(1/2 − q) − γ √ π cosh η − cos χ P− 1 (cosh η ) 2 √ , 1) = −q 2 ln(8 cosh η0 ) q Γ(1/2 − q)P− 1 (cosh η ) (50) (51) where we used that ψ(1/2) = − ln 4 − γ, [AS65, Eq. 6.3.3]. For the special case where the point charge is close to the surface we have: qi (η0 > η 1) = −q ln(8 cosh η ) η ≈ −q . ln(8 cosh η0 ) η0 (52) 5. Discussion The conducting spherical surface and a point charge nearby is very wellknown problem in electrostatics, see for instance [Jac99, Sec. 2.2]. In the case of a sphere, the only degree of freedom is the distance from the point charge to the center of the sphere. The case presented here goes in the same direction, presenting a complete electrostatics problem that can be treated analytically by using the toroidal Legendre functions. We have solved the problem in the most general case, when the point charge is located anywhere in the space outside the toroid. Because of the obvious symmetry of rotation around the z-axis, we have two degrees of freedom for the location of the point charge: in toroidal coordinates, we have η < η0 , and −π ≤ χ ≤ π. Here we used that η0 is the toroidal conducting surface, η → ∞ corresponds to the circle of radius a in the xy-plane, χ = 0 corresponds to the xy-plane with x2 + y 2 ≥ a, and χ = ±π corresponds to the xy-plane with x2 + y 2 ≤ a. A closed-loop circuit can be idealised as a toroidal ring. With or without current, the presence of a point charge nearby creates an induced surface charge distribution on the toroid, and an electric field in the space surrounding the charges. This electrostatic electric field is of zeroth-order, as it doesn’t depend on the current in the toroid. To our knowledge, this problem has never been solved exactly before, and our solution fills this gap. 10 References [And06] M. Andrews. Alternative separation of laplace’s equation in toroidal coordinates and its application to electrostatics. Journal of Electrostatics, 64:664–672, 2006. [AS65] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1965. [AW05] G. B. Arfken and H. J. Weber. Mathematical Methods for Physicists. Elsevier, San Diego, sixth edition, 2005. [Bat53] H. Bateman. Higher Transcendental Functions, volume 1. McGraw-Hill, New York, 1953. [CT99] H. S. Cohl and Tohline. A compact cylindrical green’s function expansion for the solution of potential problems. Astrophysical Journal, 527:86–101, 1999. [CTRS00] H. S. Cohl, J. E. Tohline, A. R. P. Ray, and H. M. Srivastava. Developments in determining the gravitational potential using toroidal functions. Astron. Nachr., 321:363–372, 2000. [GS97] A. Gil and J. Segura. Evaluation of legendre functions of argument greater than one. Computer Physics Communications, 105:273–283, 1997. [HA03] J. A. Hernandes and A. K. T. Assis. Electric potential for a resistive toroidal conductor carrying a steady azimuthal current. Physical Review E, 68:046611, 2003. [HMLA08] J. A. Hernandes, A. J. Mania, F. R. T Luna, and A. K. T. Assis. The internal and external electric fields for a resistive toroidal conductor carrying a steady poloidal current. Physica Scripta, 78:015403, 2008. [Jac99] J. D. Jackson. Classical Electrodynamics. John Wiley & Sons, New York, third edition, 1999. [MF53] P. M. Morse and H. Feshbach. Methods of Theoretical Physics, volume 2. McGraw-Hill, New York, 1953. [MKH+ 07] A. Mary, D. M. Koller, A. Hohenau, J. R. Krenn, A. Bouhelier, and A. Dereux. Optical absorption of torus-shaped metal nanoparticles in the visible range. Physical Review B, 76:245422, 2007. [MS88] P. Moon and D. E. Spencer. Field Theory Handbook. Springer, Berlin, 2nd edition, 1988. 11 [SG99] J. Segura and A. Gil. Evaluation of associated legendre functions off the cut and prabolic cylinder functions. Electronic Transactions on Numerical Analysis, 9:137–146, 1999. [Smy89] W. R. Smythe. Static and Dynamic Electricity. Hemisphere, New York, 1989. 12 Figure 1: Equipotential lines at xy-plane. We used (η , χ , ϕ ) = (1, 0, 0) and η0 = 2. The two dashed circles centered at the origin indicate the limits of the toroidal surface η0 (between the circles, i.e. inside the toroid, the potential is constant at φ0 ). The dot at the right indicate the position of the point charge q. 13 Figure 2: Equipotential lines at xy-plane. We used η0 = 2. The two points indicate the limits of the toroidal surface η0 (between these points, i.e. inside the toroid, the force is not shown). 14