Continuum Mechanics and Thermodynamics

, Volume 30, Issue 2, pp 233–266 | Cite as

An investigation into electromagnetic force models: differences in global and local effects demonstrated by selected problems

  • Felix A. ReichEmail author
  • Wilhelm Rickert
  • Wolfgang H. Müller
Original Article


This study investigates the implications of various electromagnetic force models in macroscopic situations. There is an ongoing academic discussion which model is “correct,” i.e., generally applicable. Often, gedankenexperiments with light waves or photons are used in order to motivate certain models. In this work, three problems with bodies at the macroscopic scale are used for computing theoretical model-dependent predictions. Two aspects are considered, total forces between bodies and local deformations. By comparing with experimental data, insight is gained regarding the applicability of the models. First, the total force between two cylindrical magnets is computed. Then a spherical magnetostriction problem is considered to show different deformation predictions. As a third example focusing on local deformations, a droplet of silicone oil in castor oil is considered, placed in a homogeneous electric field. By using experimental data, some conclusions are drawn and further work is motivated.


Electromagnetic force models Total force Local deformation Experiments 

List of symbols

General   quantities


Position vector in current placement (m)


Position vector in reference placement (m)


Barycentric velocity (m / s)

\({\varvec{v}}_{\text {I}}\)

Barycentric surface velocity (m / s)


Velocity of a singular surface (m / s)

\( w_\bot \)

Normal velocity of a singular surface (m / s)


Normal vector (1)


\(n{\text {th}}\) Legendre polynomial (1)

\(\mathrm {K}\)

Complete elliptic integral of the first kind (1)

\(\mathrm {E}\)

Complete elliptic integral of the second kind (1)

\(\Pi \)

Complete elliptic integral of the third kind (1)

\(\mathrm {B}\)

Incomplete beta function (1)

\(\,{}_2\mathrm {F}_1\)

A hypergeometric function (1)


Characteristic radius of a problem (m)


Characteristic length of a problem (m)


End-to-end distance between magnets (m)

\(\kappa \)

Normed end-to-end distance between magnets, \(\kappa = {d}/{R}\) (1)


Radial spherical coordinate (m)


Dimensionless radial spherical coordinate, \(\tilde{r}= {r}/{R}\) (1)

\(\vartheta \)

Polar spherical angle, \(\vartheta \in [0, {\uppi }]\) (1)


Cosine of polar spherical angle (1)

\(\xi \)

Radial cylindrical coordinate (m)

\(\tilde{\xi }\)

Dimensionless radial cylindrical coordinate, \(\tilde{\xi } = {\xi }/{R}\) (1)


Axial cylindrical coordinate [m]


Dimensionless axial cylindrical coordinate, \(\tilde{z} = {z}/{R}\) (1)

\(\varphi \)

Azimuthal angle, \(\varphi \in [0, 2{\uppi })\) (1)


Volume in current placement (\(\hbox {m}^3\))


Volume in reference placement (\(\hbox {m}^3\))


Displacement field (m)

\({\varvec{u}}_{\text {I}}\)

Surface displacement field (m)


Scale of surface displacement (m)

\(\tilde{u}_\mathrm {P}\)

Dimensionless pole displacement (1)


Radial displacement component w.r.t. \({\varvec{e}}_r\) (m)

\(u_\vartheta \)

Polar displacement component w.r.t. \({\varvec{e}}_\vartheta \) (m)


Deformation gradient \({\varvec{F}} = {\varvec{1}} + {\varvec{u}} \otimes \nabla _X\) (1)


Determinant of deformation gradient (1)

\((\cdot )^\mathrm {I}\)

Indicates interior domain of a problem

\((\cdot )^\mathrm {O}\)

Indicates exterior domain of a problem

\(\tilde{(\cdot )}\)

A normalized dimensionless function (1)

\((\cdot )_{\text {I}}\)

Interface quantity

\((\cdot )^\mathrm {S}\)

Refers to silicone oil

\((\cdot )^\mathrm {C}\)

Refers to castor oil


Cylindrical axial unit vector (1)

\({\varvec{e}}_\xi \)

Cylindrical radial unit vector (1)


Unit tensor of rank two (1)

\({\varvec{1}}_{\text {I}}\)

Interface projector, \({\varvec{1}}_{\text {I}}= {\varvec{1}} - {\varvec{n}} \otimes {\varvec{n}}\) (1)

\(\nabla \)

Nabla operator, (1/m)

\(\nabla _{\text {I}}\)

surface nabla, \(\nabla _{\text {I}}= {\varvec{1}}_{\text {I}}\cdot \nabla \) (1/m)

\({ Continuum\,\, mechanics}\)

\({\varvec{\sigma }}\)

Cauchy stress tensor (N / \(\hbox {m}^2\))


Pressure (N / \(\hbox {m}^2\))

\(\varvec{\sigma }_{\text {I}}\)

Cauchy surface stress tensor (N / m)

\(\sigma _{\text {I}}\)

Surface tension (N / m)

\({\varvec{\varepsilon }}\)

Linear strain tensor (1)

\({\varvec{\varepsilon }}_{\text {I}}\)

Linear surface strain tensor (1)


Mass of a body (kg)

\(\rho \)

Mass density (kg / \(\hbox {m}^3\))

\(\rho _{\text {I}}\)

Surface mass density (kg / \(\hbox {m}^2\))

\(\lambda \)

LamÉ’s first parameter (N / \(\hbox {m}^2\))

\(\mu \)

LamÉ’s second parameter (N / \(\hbox {m}^2\))

\(\lambda _{\text {I}}\)

First elastic surface parameter (N / m)

\(\mu _{\text {I}}\)

Second elastic surface parameter (N / m)

\(\nu \)

Poisson’s ratio (1)

\(\psi \)

Gravitational potential (\(\hbox {m}^2\)/\(\hbox {s}^2\))


Gravitational constant \(G = 6.67408 \star 10^{-11}\) \(\hbox {m}^3\)/(kg \(\hbox {s}^2\))


Gravitational specific force density (m/\(\hbox {s}^2\))

\({\varvec{F}}^\mathrm {tot.}\)

Total force acting on a body (N)


Volumetric force density (N / \(\hbox {m}^3\))

\({\varvec{f}}_{\text {I}}\)

Surface force density (N / \(\hbox {m}^2\))


Scale of surface force density (N / \(\hbox {m}^2\))


Heat flux (N / (m s))


Specific heating (\(\hbox {m}^2\)/\(\hbox {s}^3\))


Specific internal energy (\(\hbox {m}^2\)/\(\hbox {s}^2\))

\(\chi _{\mathrm {v}}\)

Compressibility factor (1)

\(e_{\mathrm {v}}\)

Relative volume change (1)

\(\gamma \)

Pressure-related factor (1)

\({ Electrodynamics}\)


Magnetic flux density (T)


Potential of total electric current (A / m)

\(\varvec{\mathfrak {H}}\)

Potential of free electric current (A / m)


Minkowski magnetization (A / m)


Electric field (V/m)


Potential of total electric charge (C / \(\hbox {m}^2\))

\(\varvec{\mathfrak {D}}\)

Potential of free electric charge (C / \(\hbox {m}^2\))


Polarization (C / \(\hbox {m}^2\))


Electric disturbance potential (V)

\(\mathcal {V}\)

Scaled electric disturbance potential (1)


External electric field (V/m)

\({\varvec{E}}^\mathrm {dist.}\)

Electric disturbance field (V/m)


External electric field strength (V/m)


Magnetization strength of a magnet (A / m)

\(\beta \)

Direction factor of magnetization (1)

\(\mu _0\)

Vacuum permeability (N / A\(^2\))

\(\mu _\mathrm {r}\)

Relative permeability (1)

\(\epsilon _0\)

Vacuum permittivity \({\mathrm {A}^2 \mathrm {s}^2}/({\mathrm {N} \mathrm {m}^2}\))

\(\epsilon _\mathrm {r}\)

Relative permittivity (1)


Total electric charge density (C / \(\hbox {m}^3\))

\(q^{\text {f}}\)

Free electric charge density (C / \(\hbox {m}^3\))

\(q^{\text {r}}\)

Bound electric charge density (C / \(\hbox {m}^3\))

\(q^{\text {f}}_{\text {I}}\)

Singular free electric charge density (C / \(\hbox {m}^2\))

\(q^{\text {r}}_{\text {I}}\)

Singular bound electric charge density (C / \(\hbox {m}^2\))


Total electric current density (A / \(\hbox {m}^2\))

\({\varvec{J}}^{\text {f}}\)

Free electric current density (A / \(\hbox {m}^2\))

\({\varvec{J}}^{\text {r}}\)

Bound electric current density (A / \(\hbox {m}^2\))

\({\varvec{J}}_{\text {I}}\)

Singular total electric current density (A / m)

\({\varvec{J}}^{\text {f}}_{\text {I}}\)

Singular free electric current density (A / m)

\({\varvec{J}}^{\text {r}}_{\text {I}}\)

Singular bound electric current density (A / m)

\({\varvec{j}}^{\text {f}}\)

Free diffusive electric current density (A / \(\hbox {m}^2\))

\({\varvec{j}}^{\text {f}}_{\text {I}}\)

Singular free diffusive electric current density (A / m)

\({ Coupling\,\, of\,\, mechanics\,\, and\,\, electromagnetism}\)

\({\varvec{\sigma }}^\text {(EM)}\)

Electromagnetic stress tensor (N / \(\hbox {m}^2\))

\({\varvec{g}}^\text {(EM)}\)

Electromagnetic momentum density (N / \(\hbox {m}^2\))

\({\varvec{f}}^\text {(EM)}\)

Electromagnetic volumetric force density (N / \(\hbox {m}^3\))

\({\varvec{f}}_{\text {I}}^\text {(EM)}\)

Electromagnetic surface force density (N / \(\hbox {m}^2\))

\((\cdot )^\mathrm {L}\)

Quantity of generalized Lorentz force model

\((\cdot )^{\mathrm {A}_i}\)

Quantity of an Abraham force model

\((\cdot )^{\mathrm {M}_i}\)

Quantity of a Minkowski force model

\((\cdot )^{\mathrm {EL}}\)

Quantity of EinsteinLaub force model


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  1. 1.
    Abraham, M.: Zur Elektrodynamik bewegter Körper. Rendiconti del Circolo Matematico di Palermo (1884–1940) 28(1), 1–28 (1909)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol. 9. Dover (1972)Google Scholar
  3. 3.
    Barnett, S.M.: Resolution of the Abraham–Minkowski dilemma. Phys. Rev. Lett. 104(7), 070401 (2010)ADSCrossRefGoogle Scholar
  4. 4.
    Barnett, S.M., Loudon, R.: On the electromagnetic force on a dielectric medium. J. Phys. B At. Mol. Opt. Phys. 39(15), 671–684 (2006)ADSCrossRefGoogle Scholar
  5. 5.
    Bethune-Waddell, M., Chau, K.J.: Simulations of radiation pressure experiments narrow down the energy and momentum of light in matter. Rep. Prog. Phys. 78(12), 122401 (2015)ADSCrossRefGoogle Scholar
  6. 6.
    Chu, L.J., Haus, H.A., Penfield, P.: The force density in polarizable and magnetizable fluids. Proc. IEEE 54(7), 920–935 (1966)CrossRefGoogle Scholar
  7. 7.
    Datsyuk, V.V., Pavlyniuk, O.R.: Maxwell stress on a small dielectric sphere in a dielectric. Phys. Rev. A 91(2), 023826 (2015)ADSCrossRefGoogle Scholar
  8. 8.
    Dziubek, A.: Equations for two-phase flows: a primer. ArXiv e-prints (2011)Google Scholar
  9. 9.
    Einstein, A., Laub, J.: Über die im elektromagnetischen Felde auf ruhende Körper ausgeübten ponderomotorischen Kräfte. Ann. Phys. 331(8), 541–550 (1908)CrossRefzbMATHGoogle Scholar
  10. 10.
    Fitzpatrick, R.: Classical Electromagnetism. The University of Texas at Austin, Austin (2006)Google Scholar
  11. 11.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products, 7th edn. Academic Press, Cambridge (2007)zbMATHGoogle Scholar
  12. 12.
    Griffiths, D.J.: Resource letter em-1: electromagnetic momentum. Am. J. Phys. 80(1), 7–18 (2012)ADSCrossRefGoogle Scholar
  13. 13.
    Guhlke, C.: Theorie der elektrochemischen Grenzfläche. Ph.D. thesis, Technische Universität Berlin (2015)Google Scholar
  14. 14.
    Hiramatsu, Y., Oka, Y.: Determination of the tensile strength of rock by a compression test of an irregular test piece. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 3(2), 89–90 (1966)CrossRefGoogle Scholar
  15. 15.
    Hutter, K.: On thermodynamics and thermostatics of viscous thermoelastic solids in the electromagnetic fields. A Lagrangian formulation. Arch. Ration. Mech. Anal. 58, 339–368 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hutter, K., Ven, A.A.F., Ursescu, A.: Electromagnetic Field Matter Interactions in Thermoelastic Solids and Viscous Fluids. Springer, Berlin (2006)Google Scholar
  17. 17.
    Jackson, J.D.: Classical Electrodynamics, 2nd edn. Wiley, Hoboken (1975)zbMATHGoogle Scholar
  18. 18.
    Kovetz, A.: Electromagnetic Theory. Oxford University Press, Oxford (2000)zbMATHGoogle Scholar
  19. 19.
    Liebold, C., Müller, W.H.: Are microcontinuum field theories of elasticity amenable to experiments? A review of some recent results. In: Differential Geometry and Continuum Mechanics, pp. 255–278. Springer Nature (2015)Google Scholar
  20. 20.
    magnets4you GmbH: Bar magnet STM-20x34-N (2016).
  21. 21.
    Mahdy, M.C.M.: It should be Einstein-Laub equations inside matter. arXiv preprint arXiv:1211.0155 (2012)
  22. 22.
    Mansuripur, M.: Resolution of the Abraham–Minkowski controversy. Opt. Commun. 283(10), 1997–2005 (2010)ADSCrossRefGoogle Scholar
  23. 23.
    Mansuripur, M.: Electromagnetic force and torque in Lorentz and Einstein-Laub formulations. In: SPIE NanoScience + Engineering. International Society for Optics and Photonics (2014)Google Scholar
  24. 24.
    Minkowski, H.: Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern. Math. Ann. 68(4), 472–525 (1910)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    MTS Systems Corporation, Eden Prairie, Minnesota 55344-2290 USA: Tytron\(^{{\rm TM}}\) 250 Microforce Load Unit (2002).
  26. 26.
    Müller, I.: Thermodynamics, Interaction of Mechanics and Mathematics Series. Pitman, Trowbridge (1985)Google Scholar
  27. 27.
    Müller, W.H.: An Expedition to Continuum Theory. Solid mechanics and its applications series. Springer, Berlin (2014)CrossRefzbMATHGoogle Scholar
  28. 28.
    Obukhov, Y.N.: Electromagnetic energy and momentum in moving media. Ann. Phys. 17(9–10), 830–851 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Raikher, Y.L., Stolbov, O.V.: Deformation of an ellipsoidal ferrogel sample in a uniform magnetic field. J. Appl. Mech. Tech. Phys. 46(3), 434–443 (2005)ADSCrossRefGoogle Scholar
  30. 30.
    Reich, F.A., Rickert, W., Stahn, O., Müller, W.H.: Magnetostriction of a sphere: stress development during magnetization and residual stresses due to the remanent field. Contin. Mech. Thermodyn. 29(2), 535–557 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Reich, F.A., Stahn, O., Müller, W.H.: The magnetic field of a permanent hollow cylindrical magnet. Contin. Mech. Thermodyn. 28(5), 1435–1444 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Shah, D.O.: Improved Oil Recovery by Surfactant and Polymer Flooding. Academic Press, Cambridge (1977)Google Scholar
  33. 33.
    Shevchenko, A., Kaivola, M.: Electromagnetic force density and energy-momentum tensor in an arbitrary continuous medium. J. Phys. B At. Mol. Opt. Phys. 44(17), 175401 (2011)ADSCrossRefGoogle Scholar
  34. 34.
    Slattery, J.C., Sagis, L., Oh, E.S.: Interfacial Transport Phenomena. Springer, Berlin (2007)zbMATHGoogle Scholar
  35. 35.
    Steigmann, D.J.: On the formulation of balance laws for electromagnetic continua. Math. Mech. Solids 14(4), 390–402 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Steinmann, P.: On boundary potential energies in deformational and configurational mechanics. J. Mech. Phys. Solids 56(3), 772–800 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Stratton, J.A.: Electromagnetic Theory. McGraw-Hill, New York (1941)zbMATHGoogle Scholar
  38. 38.
    Torza, S., Cox, R.G., Mason, S.G.: Electrohydrodynamic deformation and burst of liquid drops. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 269(1198), 295–319 (1971)ADSCrossRefGoogle Scholar
  39. 39.
    Truesdell, C.A., Toupin, R.: The classical field theories. In: Handbuch der Physik, Bd. III/1, pp. 226–793; appendix, pp. 794–858. Springer, Berlin (1960). With an appendix on tensor fields by J.L. EricksenGoogle Scholar
  40. 40.
    Wang, C.: Comment on “resolution of the Abraham–Minkowski dilemma”. arXiv preprint arXiv:1202.2575 (2012)
  41. 41.
    Wolfram Research, Inc.: Mathematica, 11 edn. (2016)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Felix A. Reich
    • 1
    Email author
  • Wilhelm Rickert
    • 1
  • Wolfgang H. Müller
    • 1
  1. 1.Institut für Mechanik, Kontinuumsmechanik und MaterialtheorieTechnische Universität BerlinBerlinGermany

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