Continuum Mechanics and Thermodynamics

, Volume 30, Issue 2, pp 233–266

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

Original Article

## Abstract

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.

## Keywords

Electromagnetic force models Total force Local deformation Experiments

## General   quantities

$${\varvec{x}}$$

Position vector in current placement (m)

$${\varvec{X}}$$

Position vector in reference placement (m)

$${\varvec{v}}$$

Barycentric velocity (m / s)

$${\varvec{v}}_{\text {I}}$$

Barycentric surface velocity (m / s)

$${\varvec{w}}$$

Velocity of a singular surface (m / s)

$$w_\bot$$

Normal velocity of a singular surface (m / s)

$${\varvec{n}}$$

Normal vector (1)

$$P_n$$

$$n{\text {th}}$$ 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)

R

Characteristic radius of a problem (m)

H

Characteristic length of a problem (m)

d

End-to-end distance between magnets (m)

$$\kappa$$

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

r

Radial spherical coordinate (m)

$$\tilde{r}$$

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

$$\vartheta$$

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

x

Cosine of polar spherical angle (1)

$$\xi$$

Radial cylindrical coordinate (m)

$$\tilde{\xi }$$

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

z

Axial cylindrical coordinate [m]

$$\tilde{z}$$

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

$$\varphi$$

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

V

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

$$V_0$$

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

$${\varvec{u}}$$

Displacement field (m)

$${\varvec{u}}_{\text {I}}$$

Surface displacement field (m)

$$\hat{u}$$

Scale of surface displacement (m)

$$\tilde{u}_\mathrm {P}$$

Dimensionless pole displacement (1)

$$u_r$$

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

$$u_\vartheta$$

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

$${\varvec{F}}$$

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

J

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

$${\varvec{e}}_z$$

Cylindrical axial unit vector (1)

$${\varvec{e}}_\xi$$

Cylindrical radial unit vector (1)

$${\varvec{1}}$$

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 / $$\hbox {m}^2$$)

p

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)

m

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 ($$\hbox {m}^2$$/$$\hbox {s}^2$$)

G

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

$${\varvec{g}}$$

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

$${\varvec{F}}^\mathrm {tot.}$$

Total force acting on a body (N)

$${\varvec{f}}$$

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

$${\varvec{f}}_{\text {I}}$$

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

$$\hat{f}$$

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

$${\varvec{q}}$$

Heat flux (N / (m s))

$$\hat{r}$$

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

u

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)

## $${ Electrodynamics}$$

$${\varvec{B}}$$

Magnetic flux density (T)

$${\varvec{H}}$$

Potential of total electric current (A / m)

$$\varvec{\mathfrak {H}}$$

Potential of free electric current (A / m)

$${\varvec{M}}$$

Minkowski magnetization (A / m)

$${\varvec{E}}$$

Electric field (V/m)

$${\varvec{D}}$$

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

$$\varvec{\mathfrak {D}}$$

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

$${\varvec{P}}$$

Polarization (C / $$\hbox {m}^2$$)

V

Electric disturbance potential (V)

$$\mathcal {V}$$

Scaled electric disturbance potential (1)

$${\varvec{E}}_0$$

External electric field (V/m)

$${\varvec{E}}^\mathrm {dist.}$$

Electric disturbance field (V/m)

$$E_0$$

External electric field strength (V/m)

$$M_0$$

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)

q

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 / $$\hbox {m}^2$$)

$${\varvec{J}}$$

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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© 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