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Electromagnetic response of two interacting topological insulator spheres in external fields

T0 review · 1 major / 0 minor · reviewed 2026-06-30 · grok-4.3

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Pith's one-line read Two interacting topological insulator spheres generate induced magnetic fields through magnetoelectric coupling when placed in an external electric field.

desk verdict This paper extends single-sphere TI calculations to two spheres with perturbative series for the induced magnetostatic response, but leaves the convergence of its recurrence solution unquantified. read the letter →

arxiv 2606.28982 v1 pith:XBPNH7AP submitted 2026-06-27 cond-mat.other math-phmath.MP

J. Cornejo Gómez , M. Ibarra-Meneses , L. Medel Onofre , A. Martín-Ruiz This is my paper · ORCID
classification cond-mat.othermath-phmath.MP
keywords topologicalinsulatorsmagnetoelectriceffectaxionfieldbisphericalcoordinateselectromagneticresponseinducedmagneticfieldssphereinteractions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper solves the electrostatic problem for two spheres exactly in bispherical coordinates for both parallel and perpendicular field orientations. From the zeroth-order electrostatic coefficients, it derives the leading-order magnetoelectric response, which is interfacial and produces distinct magnetostatic configurations. This provides closed-form series for the induced vector potential and magnetic field. A sympathetic reader would care because it gives an analytically controlled way to understand electromagnetic interactions between topological insulators.

What carries the argument

Piecewise constant axion field at the spherical interfaces that induces topological magnetoelectric coupling, with the electrostatic solution obtained via bispherical coordinates leading to three-term recurrence relations.

What would settle it

An experiment measuring the induced magnetic field around two topological insulator spheres in a uniform electric field and comparing it to the predicted series expansions.

Watch

Extended reading notes

Core claim

The magnetoelectric-induced response is computed to leading order in the fine-structure constant, yielding closed-form series representations for the induced vector potential and magnetic field in terms of the zeroth-order electrostatic coefficients. The induced sources are purely interfacial and generate distinct magnetostatic field configurations in the parallel and perpendicular geometries.

Load-bearing premise

The gapped surface states of the topological insulators are modeled by a piecewise constant axion field localized at the interfaces.

Editorial extensions

If this is right

  • The formalism applies to isotropic magnetoelectric media with effective scalar response.
  • Interaction-induced magnetostatics can be described analytically for coupled spherical systems.
  • The response is computed perturbatively for nonoverlapping spheres.
  • Distinct field patterns arise depending on field orientation relative to the line joining the spheres.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Similar methods could extend to more than two spheres or different shapes.
  • The results suggest experimental tests of topological magnetoelectric effects in paired systems.
  • The general applicability to magnetoelectric media opens paths to engineered responses in composite materials.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The paper examines the static electromagnetic response of two spherical topological insulators in a uniform external electric field, modeling gapped surface states via a piecewise constant axion field that induces localized magnetoelectric coupling at the interfaces. The electrostatic problem is solved exactly in bispherical coordinates for both parallel and perpendicular field orientations, yielding three-term recurrence relations for the mode coefficients that are solved perturbatively for nonoverlapping spheres; the magnetoelectric-induced response is then computed to leading order in the fine-structure constant, producing closed-form series for the induced vector potential and magnetic field in terms of the zeroth-order electrostatic coefficients.

Significance. If the perturbative treatment of the recurrences is rigorously controlled, the work supplies an analytically tractable framework for interaction-induced magnetostatics in coupled spherical magnetoelectric systems, extending single-sphere results to interacting geometries while remaining parameter-free at leading order. The closed-form series representations constitute a concrete strength that could facilitate comparisons with numerics or experiments on topological-insulator composites.

major comments (1)
  1. [Abstract] Abstract: the central claim of an 'analytically controlled description of interaction-induced magnetostatics' rests on the perturbative solution of the three-term recurrence relations for nonoverlapping spheres, yet the manuscript provides neither a radius of convergence for the expansion nor explicit error bounds on the truncation; this omission directly affects the validity range of the closed-form series for the induced fields.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comment on the abstract. We respond to the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim of an 'analytically controlled description of interaction-induced magnetostatics' rests on the perturbative solution of the three-term recurrence relations for nonoverlapping spheres, yet the manuscript provides neither a radius of convergence for the expansion nor explicit error bounds on the truncation; this omission directly affects the validity range of the closed-form series for the induced fields.

    Authors: We agree that the manuscript does not supply an explicit radius of convergence or truncation-error bounds for the perturbative treatment of the three-term recurrences, and that this information would strengthen the claim of an analytically controlled description. The expansion is performed in the small parameter set by the ratio of the sphere radii to their center-to-center separation (strictly less than unity for non-overlapping spheres), with the magnetoelectric response retained only to leading order in the fine-structure constant. In the revised manuscript we will insert a short paragraph (most naturally in the section describing the bispherical solution) that recalls the standard convergence criterion for such recurrences and supplies a leading-order estimate of the truncation error in terms of that separation parameter. This addition will make the validity range of the closed-form series explicit without altering the existing derivations. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

Direct perturbative solution of axion-modified Maxwell equations; no circular reductions

full rationale

The derivation proceeds by solving the zeroth-order electrostatic problem in bispherical coordinates for two spheres, obtaining three-term recurrences for the mode coefficients that are treated perturbatively for non-overlapping spheres, then inserting those coefficients into closed-form series for the induced vector potential and magnetic field at leading order in the fine-structure constant (or effective magnetoelectric coupling). This is a standard boundary-value expansion of the modified Maxwell equations with piecewise axion term; the electrostatic coefficients are computed from the external field and geometry rather than fitted, and the magnetoelectric response is generated by the interfacial sources without redefinition or self-citation chains. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The claim relies on the standard axion electrodynamics model for TIs and the applicability of bispherical coordinates to the two-sphere geometry, with no new entities introduced.

assumptions (3)
  • domain assumption Gapped surface states of topological insulators are described by a piecewise constant axion field inducing magnetoelectric coupling at interfaces.
    Stated in the abstract as the model for the surface states.
  • standard math The electromagnetic response can be solved using Maxwell's equations modified by the axion term in the presence of external fields.
    Standard framework for axion electrodynamics in topological insulators.
  • domain assumption For nonoverlapping spheres, the mode expansions can be solved perturbatively using three-term recurrence relations.
    Assumed in the solution method described.

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Pith. "Pith review of Electromagnetic response of two interacting topological insulator spheres in external fields." pith.science (2026). https://pith.science/paper/XBPNH7AP

@misc{pith2026260628982,
  author       = {Pith},
  title        = {Pith review of: Electromagnetic response of two interacting topological insulator spheres in external fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XBPNH7AP}},
  note         = {Machine review of arXiv:2606.28982}
}
read the original abstract

We study the static electromagnetic response of two spherical topological insulators embedded in a dielectric medium and subjected to a uniform external electric field. The gapped surface states are described by a piecewise constant axion field, which induces a topological magnetoelectric coupling localized at the spherical interfaces. {More generally, the same formalism applies to isotropic magnetoelectric media characterized by an effective scalar magnetoelectric response.} The electrostatic problem is solved at zeroth order using bispherical coordinates, allowing for an exact treatment of both parallel and perpendicular orientations of the external field relative to the center-to-center axis. The resulting mode expansions are determined by three-term recurrence relations, which are solved perturbatively for nonoverlapping spheres. The { magnetoelectric}-induced response is then computed to leading order in the fine-structure constant {(or, more generally, in the effective coupling strength)}. The induced sources are purely interfacial and generate distinct magnetostatic field configurations in the parallel and perpendicular geometries. Closed-form series representations for the induced vector potential and magnetic field are obtained in terms of the zeroth-order electrostatic coefficients. These results provide an analytically controlled description of {interaction-induced magnetostatics in coupled spherical magnetoelectric systems}.

Figures

Figures reproduced from arXiv: 2606.28982 by the authors.

Figure 1
Figure 1. FIG. 1: Schematic configuration of the system. Two spherical TIs of radius [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Zeroth-order electrostatic configuration for two identical dielectric spheres embedded in a homogeneous dielectric [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Zeroth-order electrostatic configuration for two identical dielectric spheres embedded in a homogeneous dielectric [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Magnetostatic field lines for two identical permeable spheres embedded in a homogeneous medium and subjected to a [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Axion-induced magnetic response for two topological insulator spheres subjected to an external electric field parallel to [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Axion-induced magnetic field and surface Hall current for the perpendicular configuration. The upper panels display [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

65 extracted references · 65 canonical work pages

  1. [1]

    The second term inside the brackets represents the particular solution associated with the applied field and guarantees the correct far-field behavior, while the coefficientsCn encode the electrostatic response of the two-sphere system. Inside the upper sphere, regularity atη→+∞restricts the solution to decaying modes, and the potential can be written as ...

    work page

  2. [2]

    Explicit solutions up to second order were constructed, providing a quantitatively accurate seed for the subsequent axion- induced analysis. 16 FIG. 6: Axion-induced magnetic field and surface Hall current for the perpendicular configuration. The upper panels display the magnetic field lines generated by the axion-induced surface currents: the upper-left ...

    work page 2025

  3. [3]

    From boundary conditions to recurrence relations The electrostatic potentials inside and outside the spheres are expanded as ϕe(η, ξ) = (coshη−cosξ) 1/2 ∞X n=0 h An sinh(¯nη)−23/2E0a¯ne−¯nη i Pn(cosξ),(A1) ϕ+(η, ξ) = (coshη−cosξ) 1/2 ∞X n=0 Bne−¯nηPn(cosξ),(A2) with¯n=n+ 1/2. Imposing continuity of the potential at the spherical surfaceη=η 0 leads to the ...

    work page

  4. [4]

    Inspection of Eq

    Perturbative structure of the recurrence For nonoverlapping spheres one hase −η0 <1, while for physical dielectrics|∆|<1. Inspection of Eq. (A5) shows that the coupling between neighboring coefficientsA n±1 is suppressed by factors ofe −2nη0, which motivates a perturbative expansion in the small parameter∆e −2nη0. We therefore write An =A (0) n +A (1) n +...

    work page

  5. [5]

    Zeroth-order solution At leading order the recurrence relation simplifies considerably and reduces to a linear difference equation with polynomial coefficients inn. Guided by its structure, we seek a solution of the form A(0) n =α n+β.(A8) Substitution into the zeroth-order recurrence equation and matching the coefficients of equal powers ofnyields a syst...

    work page

  6. [6]

    First-order correction At first order, the recurrence relation becomes inhomogeneous, with a source term entirely determined byA (0) n . Since the inhomogeneity is proportional toe −2nη0, we adopt the ansatz A(1) n = ∆e−2nη0 α(1)n+β (1) .(A10) Substitution into the first-order recurrence relation and comparison of the terms proportional ton 2 immediately ...

    work page

  7. [7]

    Second-order correction The second-order recurrence relation is driven by the first-order coefficientsA(1) n . The same structural considerations motivate the ansatz A(2) n = ∆e−2nη0 α(2)n+β (2) .(A14) Matching the highest-order terms innyields α(2) =e −η0 α(1),(A15) while the remaining constantβ (2) is obtained by collecting the lower-order terms. Expand...

    work page

  8. [8]

    As a consequence, only bispherical harmonics with azimuthal indexm= 1contribute, and the angular structure is entirely captured by the combinationP 1 n(cosξ) sinϕ

    Mode selection and exterior potential In the perpendicular configuration the far-field electrostatic potential is ϕ0 =−E 0y,(B1) which, in bispherical coordinates, exhibits an explicitsinϕangular dependence. As a consequence, only bispherical harmonics with azimuthal indexm= 1contribute, and the angular structure is entirely captured by the combinationP 1...

    work page

Show all 65 references
  1. [9]

    Interior potential and continuity of the potential Inside the upper sphere (η > η 0), regularity asη→+∞requires exponentially decaying modes. The interior potential is therefore expanded as ϕ+(η, ξ, ϕ) = (coshη−cosξ) 1/2 ∞X n=1 Dne−¯nηP 1 n(cosξ) sinϕ.(B5) Continuity of the el...

  2. [10]

    The latter are reduced to theP 1 n±1 basis using the standard recurrence relations for associated Legendre functions

    Continuity of the normal displacement and recurrence relation The second boundary condition enforces continuity of the normal component of the electric displacement field, ϵi ∂ηϕ+(η0, ξ, ϕ) =ϵ e ∂ηϕe(η0, ξ, ϕ).(B7) Upon differentiation, the resulting expressions contain terms ...

  3. [11]

    Inspection of Eq

    Perturbative structure of the recurrence For nonoverlapping spheres one hase −η0 <1, while for physical dielectrics|∆|<1. Inspection of Eq. (B9) shows that terms coupling different multipoles are suppressed by factors of∆e −2nη0. This motivates a perturbative expansion of the ...

  4. [12]

    Zeroth-order solution At leading order all∆e −2nη0 terms are neglected, and the recurrence reduces to (n−1)e η0 C(0) n−1 + h ∆ sinhη0 −(2n+ 1) coshη 0 i C(0) n + (n+ 2)e −η0 C(0) n+1 =e −2η0 −1.(B11) Since the right-hand side is independent ofn, a constant solutionC (0) n =C (...

  5. [13]

    First-order correction At first order the recurrence becomes inhomogeneous, with a source determined entirely byC (0) n . Guided by the structure of the suppressed terms, we adopt the ansatz C(1) n = ∆e−2nη0 α(1) + β(1) n +O(e −6η0).(B13) Matching the leading contributions inn...

  6. [14]

    Second-order correction The second-order recurrence is driven by the first-order solution. Using the same structural ansatz, C(2) n = ∆e−2nη0 α(2) + β(2) n ,(B16) one finds from the leading-order balance α(2) =e −η0 α(1),(B17) and from the subleading terms β(2) = 1−∆ 3−∆ e−5η0...

  7. [15]

    Using the bispherical expression for∇θ in Eq

    Reduction of the source to the interfaces Since the zeroth-order solution is axisymmetric,ϕ (0) =ϕ (0)(η, ξ)and∂ ϕϕ(0) = 0. Using the bispherical expression for∇θ in Eq. (28) (withη 1 =η 0 andη 2 =−η 0) one finds that the source term in Amp`ere’s law is purely azimuthal, ∇θ× ∇...

  8. [16]

    Derivative of the zeroth-order potential at the interfaces For the parallel configuration the exterior potential is expanded as in the main text, ϕ(0) e (η, ξ) = (coshη−cosξ) 1/2 ∞X n=0 h An sinh(¯nη)−23/2E0a¯ne−¯nη i Pn(cosξ),¯n=n+ 1 2 .(C4) Since∂ ξPn(cosξ) =−sinξ P ′ n(cosξ...

  9. [17]

    Substituting Eq

    Bispherical expansion of the Green function and angular projection We use the standard bispherical-harmonic expansion of the free-space Green functionG 0(r,r ′), G0(r,r ′) = (coshη−cosξ) 1/2(coshη ′ −cosξ ′)1/2 ∞X l=0 lX m=−l gl(η, η′)Y m l (ξ, ϕ)Y m l (ξ′, ϕ′)∗,(C7) whereg l(...

  10. [18]

    Equation (C11) is the expression quoted in the main text, with the kernelg n inherited from the Green-function expansion (C7)

    Final series forA (1) ϕ Collecting the survivingm=±1contributions and using the relation betweenY ±1 l andP 1 l (cosξ), the angular projection yields a compact bispherical series of the form A(1) ϕ (η, ξ) =− αθ π (coshη−cosξ) 1/2 ∞X n=0 h An sinh(¯nη0)−2 3/2E0a¯ne−¯nη0 ih gn(η...

  11. [19]

    Induced magnetic field and streamline invariant GivenA (1) ϕ (η, ξ), the first-order magnetic field follows fromB (1) =∇ ×A (1). For an axisymmetric purely azimuthal potential, the curl in bispherical coordinates reduces to B(1) = (coshη−cosξ) 2 a2 sinξ h ˆη∂ ξ − ˆξ∂ η i sinξ ...

  12. [20]

    In this case, Eqs

    Generalization to two distinct spheres If the spheres are not identical, their interfaces are located atη=η 1 >0andη=η 2 <0, and the axion jumps are ∆θi =θ m −θ i. In this case, Eqs. (C1) and (C3) generalize by the replacement θ δ(η−η 0) +δ(η+η 0) − →∆θ 1 δ(η−η 1) + ∆θ2 δ(η−η ...

  13. [21]

    Structure of the axion-induced source At first order in the axion coupling, the vector potential satisfies ∇2A(1) =− α π ∇θ× ∇ϕ (0).(D1) Using the bispherical-coordinate expression for the gradient operator and the fact that∇θis purely normal to the interfaces, the source term...

  14. [22]

    Cartesian decomposition of the source To facilitate the angular integrations, it is convenient to rewrite the operator acting on the angular functions in terms of Cartesian components. Using the explicit expressions of the bispherical basis vectors and identifying the angular-...

  15. [23]

    (D3) into the Green-function representation (24), theη′ integration collapses onto the two spherical interfaces

    Green-function expansion and angular projections Substituting Eq. (D3) into the Green-function representation (24), theη′ integration collapses onto the two spherical interfaces. Expanding the Green’s function in spherical harmonics and using iLx = i 2(L+ +L −), iL y = 1 2(L+ ...

  16. [24]

    D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett.49, 405 (1982)

    1982

  17. [25]

    M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys.82, 3045 (2010)

    2010

  18. [26]

    Qi and S.-C

    X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys.83, 1057 (2011)

    2011

  19. [27]

    L. Fu, C. L. Kane, and E. J. Mele, Topological insulators in three dimensions, Phys. Rev. Lett.98, 106803 (2007)

    2007

  20. [28]

    J. E. Moore and L. Balents, Topological invariants of time-reversal-invariant band structures, Phys. Rev. B75, 121306 (2007)

    2007

  21. [29]

    Roy, Topological phases and the quantum spin Hall effect in three dimensions, Phys

    R. Roy, Topological phases and the quantum spin Hall effect in three dimensions, Phys. Rev. B79, 195322 (2009)

    2009

  22. [30]

    Okamoto, Y

    M. Okamoto, Y . Takane, and K.-I. Imura, One-dimensional topological insulator: a model for studying finite-size effects in topological insulator thin films, Phys. Rev. B89, 125425 (2014)

    2014

  23. [31]

    Governale, B

    M. Governale, B. B. Bhandari, F. Taddei, K.-I. Imura, and U. Z ¨ulicke, Finite-size effects in cylindrical topological insulators, New J. Phys.22, 063055 (2020)

    2020

  24. [32]

    A. M. Cook and A. E. B. Nielsen, Finite-size topology, Phys. Rev. B108, 045144 (2023)

    2023

  25. [33]

    X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Topological field theory of time-reversal invariant insulators, Phys. Rev. B78, 195424 (2008)

    2008

  26. [34]

    A. M. Essin, J. E. Moore, and D. Vanderbilt, Magnetoelectric polarizability and axion electrodynamics in crystalline insulators, Phys. Rev. Lett.102, 146805 (2009)

    2009

  27. [35]

    X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, Inducing a magnetic monopole with topological surface states, Science323, 1184 (2009)

    2009

  28. [36]

    Karch, Electric-magnetic duality and topological insulators, Phys

    A. Karch, Electric-magnetic duality and topological insulators, Phys. Rev. Lett.103, 171601 (2009)

    2009

  29. [37]

    Mart ´ın-Ruiz, M

    A. Mart ´ın-Ruiz, M. Cambiaso, and L. F. Urrutia, Green’s function approach to Chern-Simons extended electrodynamics: An effective theory describing topological insulators, Phys. Rev. D92, 125015 (2015)

    2015

  30. [38]

    A. G. Grushin and A. Cortijo, Tunable Casimir repulsion with three-dimensional topological insulators, Phys. Rev. Lett.106, 020403 (2011)

    2011

  31. [39]

    A. G. Grushin, P. Rodriguez-Lopez, and A. Cortijo, Effect of finite temperature and uniaxial anisotropy on the Casimir effect with three-dimensional topological insulators, Phys. Rev. B84, 045119 (2011)

    2011

  32. [40]

    Mart ´ın-Ruiz, M

    A. Mart ´ın-Ruiz, M. Cambiaso, and L. F. Urrutia, A Green’s function approach to the Casimir effect on topological insulators with planar symmetry, Europhysics Letters113, 60005 (2016)

    2016

  33. [41]

    N. A. Spaldin, M. Fiebig, and M. Mostovoy, The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect, Journal of Physics: Condensed Matter20, 434203 (2008)

    2008

  34. [42]

    Malashevich, I

    A. Malashevich, I. Souza, S. Coh, and D. Vanderbilt, Theory of orbital magnetoelectric response, New Journal of Physics12, 053032 (2010)

    2010

  35. [43]

    Scaramucci, E

    A. Scaramucci, E. Bousquet, M. Fechner, M. Mostovoy, and N. A. Spaldin, Linear magnetoelectric effect by orbital magnetism, Phys. Rev. Lett.109, 197203 (2012)

    2012

  36. [44]

    J. D. Jackson,Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998)

    1998

  37. [45]

    P. M. Morse and H. Feshbach,Methods of Theoretical Physics(McGraw-Hill, New York, 1953)

    1953

  38. [46]

    Goyette and A

    A. Goyette and A. Navon, Two dielectric spheres in an electric field, Phys. Rev. B13, 4320 (1976)

    1976

  39. [47]

    R. D. Stoy, Solution procedure for the Laplace equation in bispherical coordinates for two spheres in a uniform external field: Perpendic- ular orientation, Journal of Applied Physics66, 5093 (1989)

    1989

  40. [48]

    Lekner, Electrostatics of two charged conducting spheres, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences468, 2829 (2012)

    J. Lekner, Electrostatics of two charged conducting spheres, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences468, 2829 (2012)

    2012

  41. [49]

    Levine and G

    S. Levine and G. Olaofe, Scattering of electromagnetic waves by two equal spherical particles, Journal of Colloid and Interface Science 27, 442 (1968)

    1968

  42. [50]

    Batool and I

    S. Batool and I. A. Qureshi, Scattering of electromagnetic waves by two equal spherical particles, Optik249, 168271 (2022)

    2022

  43. [51]

    Prodan and P

    E. Prodan and P. Nordlander, Plasmon hybridization in spherical nanoparticles, J. Chem. Phys.120, 5444 (2004). 26

    2004

  44. [52]

    Markel, Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure, Journal of Modern Optics 40, 2281 (1993)

    V . Markel, Coupled-dipole approach to scattering of light from a one-dimensional periodic dipole structure, Journal of Modern Optics 40, 2281 (1993)

    1993

  45. [53]

    M. L. Brongersma, N. J. Halas, and P. Nordlander, Plasmon-induced hot carrier science and technology, Nature Nanotechnology10, 25 (2015)

    2015

  46. [54]

    I. N. Derbenev, A. V . Filippov, A. J. Stace, and E. Besley, Electrostatic interactions between charged dielectric particles in an electrolyte solution, The Journal of Chemical Physics145, 084103 (2016)

    2016

  47. [55]

    Prodan and P

    E. Prodan and P. Nordlander, Plasmon hybridization in spherical nanoparticles, The Journal of Chemical Physics120, 5444 (2004)

    2004

  48. [56]

    J. B. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett.85, 3966 (2000)

    2000

  49. [57]

    V . M. Shalaev, Optical negative-index metamaterials, Nature Photonics1, 41 (2007)

    2007

  50. [58]

    V . A. Markel, Introduction to the Maxwell Garnett approximation: Tutorial, Journal of the Optical Society of America A33, 1244 (2016)

    2016

  51. [59]

    Hudlet, M

    S. Hudlet, M. Saint Jean, C. Guthmann, and J. Berger, Evaluation of the capacitive force between an atomic force microscopy tip and a metallic surface, The European Physical Journal B2, 5 (1998)

    1998

  52. [60]

    Khachatourian, H.-K

    A. Khachatourian, H.-K. Chan, A. J. Stace, and E. Bichoutskaia, Electrostatic force between a charged sphere and a planar surface: A general solution for dielectric materials, The Journal of Chemical Physics140, 074107 (2014)

    2014

  53. [61]

    Banerjee, M

    S. Banerjee, M. Levy, M. Davis, and B. Wilkerson, Exact and approximate capacitance and force expressions for the electrostatic inter- action between two equal-sized charged conducting spheres, IEEE Transactions on Industry Applications53, 2455 (2017)

    2017

  54. [62]

    Mart ´ın-Ruiz, O

    A. Mart ´ın-Ruiz, O. Rodr´ıguez-Tzompantzi, J. R. Maze, and L. F. Urrutia, Magnetoelectric effect of a conducting sphere near a planar topological insulator, Phys. Rev. A100, 042124 (2019)

    2019

  55. [63]

    Mart ´ın-Ruiz, M

    A. Mart ´ın-Ruiz, M. Cambiaso, and L. F. Urrutia, Electromagnetic description of three-dimensional time-reversal invariant ponderable topological insulators, Phys. Rev. D94, 085019 (2016)

    2016

  56. [64]

    Moon and D

    P. Moon and D. E. Spencer,Field Theory Handbook(Springer, Berlin, 1961)

    1961

  57. [65]

    P. M. Morse and H. Feshbach,Methods of Theoretical Physics, V ol. 2 (McGraw-Hill, New York, 1953)

    1953

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