# Electromagnetic Force and Momentum in Classical Macroscopic Dipolar Media

Arthur D. Yaghjian
Electromagnetics Research, 115 Wright Road, Concord, MA 01742 USA
<EMAIL_ADDRESS>
Abstract – Using realistic classical models of microscopic electric-charge
electric dipoles and Amperian (circulating-electric-current) magnetic dipoles,
it is proven that the Einstein-Laub (as opposed to the Minkowski) macroscopic
electromagnetic force on a macroscopic-continuum volume of these classical
dipoles approximating the actual dipole moments in that volume equals the sum
of the microscopic electromagnetic forces on the discrete classical dipoles in
that volume. The internal (hidden) momentum of the discrete Amperian magnetic
dipoles is rigorously derived and properly included in the determination of
the macroscopic force from the averaging of the microscopic forces.
Consequently, the Abraham/Einstein-Laub, rather than the Minkowski,
macroscopic electromagnetic-field momentum density gives the total microscopic
electromagnetic-field momentum in that volume. The kinetic momentum is found
for the volume of the macroscopic continuum from Newton’s relativistic
equation of motion. It is shown that the difference between the kinetic
momentum and the canonical momentum (which is merely another name for the
Minkowski-force momentum) in a volume of the macroscopic continuum is equal to
the sum of the “hidden electromagnetic momenta” within the circulating-
electric-current magnetic dipoles and within hypothetical circulating-
magnetic-current electric dipoles replacing the electric-charge electric
dipoles in the classical macroscopic continuum. To obtain the correct
unambiguous value of the force on a volume inside the continuum from the
force-momentum expression, it is mandatory that the surface of that volume be
hypothetically separated from the rest of the continuum by a thin free-space
shell. Two definitive experiments performed in the past with time varying
fields and forces are shown to conclusively confirm the Einstein-Laub/Abraham
formulation over the Minkowski formulation.

## I. Introduction

Although the determination of the detailed fields and polarizations of atoms
and molecules requires quantum physics, most bulk materials below optical (or
even higher) frequencies are accurately described by the classical Maxwell
macroscopic equations for dipolar continua [1, sec. 77]. Moreover, the
microscopic (molecular) dipoles producing the macroscopic dipolarization can
be adequately modeled pragmatically by classical electric-charge electric
dipoles and Amperian (circulating-electric-current) magnetic dipoles,
irrespective of their actual quantum origin. Indeed, most of the widely used
physics and engineering textbooks in electromagnetics, such as [1, 2, 3, 4,
5], confine themselves predominantly to classical electromagnetic theory with
classical models of electric and magnetic dipoles.

Nonetheless, since Maxwell published his electromagnetic equations, it has
remained uncertain as to how to correctly determine the time varying classical
macroscopic force on a volume of dipolar material subject to time varying
electromagnetic fields. In particular, uncertainty continues as to whether to
use the Abraham definition [6, 7] of macroscopic electromagnetic-field
momentum (which is also contained in the Einstein-Laub [8] macroscopic force-
momentum equation) or the Minkowski definition [9] of macroscopic
electromagnetic-field momentum (or some other definition of macroscopic
electromagnetic-field momentum), each of which leads to a different
instantaneous time-domain macroscopic electromagnetic force (even when their
predicted average macroscopic electromagnetic force is the same) [10, 11, 12,
13].

A major obstacle preventing the determination of the correct macroscopic
electromagnetic force and electromagnetic-field momentum in polarized material
approximated by a classical macroscopic continuum has been the uncertainty of
the relationship between the forces on individual dipoles and the forces on a
distribution of these dipoles composing the macroscopic continuum. A second
major obstacle has been the absence of a definitive determination of the
electromagnetic time-domain force on classical microscopic Amperian magnetic
dipoles, which approximate the magnetic dipoles found in nature. In this
paper, these obstacles are overcome and it is determined that the correct
macroscopic electromagnetic force and electromagnetic-field momentum for
classical macroscopic dipolar continua are given by the Einstein-Laub
macroscopic force and the Abraham/Einstein-Laub (rather than Minkowski – or
any other) macroscopic electromagnetic-field momentum. An underlying important
requirement of all the derivations is that the surface of any volume of the
dipolar material under consideration lie in a hypothetical thin free-space
shell separating the volume from the rest of the material, so that the volume
contains a discrete number of dipoles.

## II. Microscopic Force and Momentum

We assume that we are dealing with a macroscopic dipolar continuum (solid or
fluid) whose molecules or inclusions have electric and magnetic dipole moments
that can be modeled electromagnetically by classical microscopic electric
charge and electric current with fields that obey the following Maxwell
differential equations in SI (mksA) units

$\nabla\times{\bf e}({\bf r},t)+\frac{\partial{\bf b}({\bf r},t)}{\partial
t}=0$ (1a) $\frac{1}{\mu_{0}}\nabla\times{\bf b}({\bf
r},t)-\epsilon_{0}\frac{\partial{\bf e}({\bf r},t)}{\partial t}={\bf j}({\bf
r},t)$ (1b) $\nabla\cdot{\bf b}({\bf r},t)=0$ (1c)
$\epsilon_{0}\nabla\cdot{\bf e}({\bf r},t)=\varrho({\bf r},t)$ (1d)

where ${\bf e}({\bf r},t)$ and ${\bf b}({\bf r},t)$ are the primary
microscopic electric and magnetic fields at the position ${\bf r}$ and time
$t$, scalar $\varrho({\bf r},t)$ and vector ${\bf j}({\bf r},t)$ are the
microscopic electric-charge and electric-current densities, and $\epsilon_{0}$
and $\mu_{0}$ are the free-space permittivity and permeability. Note that
since there are no polarization densities in (1), it follows that the
microscopic electric displacement vector ${\bf d}$ (secondary electric field)
is given by ${\bf d}=\epsilon_{0}{\bf e}$ and the microscopic secondary
magnetic field ${\bf h}$ is given by ${\bf h}={\bf b}/\mu_{0}$. The charge and
current densities, $\varrho({\bf r},t)$ and ${\bf j}({\bf r},t)$, can be
considered continuous functions of ${\bf r}$, even though they can form
charges and currents of discrete dipoles occupying indefinitely small regions
of space.

With the help of these microscopic Maxwell equations, the microscopic Lorentz-
force density

${\bf f}({\bf r},t)=\varrho({\bf r},t){\bf e}({\bf r},t)+{\bf j}({\bf
r},t)\times{\bf b}({\bf r},t)$ (2)

can be shown to satisfy the equation [2, sec. 2.5]

${\bf f}({\bf r},t)+\epsilon_{0}\frac{\partial}{\partial t}[{\bf e}({\bf
r},t)\times{\bf b}({\bf
r},t)]=\nabla\cdot\mbox{\boldmath${\overline{\mathcal{T}}}$\unboldmath}({\bf
r},t)$ (3)

where the microscopic electromagnetic stress dyadic
$\mbox{\boldmath${\overline{\mathcal{T}}}$\unboldmath}({\bf r},t)$ is defined
with the help of the unit dyadic $\overline{\bf I}$ as

$\mbox{\boldmath${\overline{\mathcal{T}}}$\unboldmath}=\epsilon_{0}\left({\bf
e}{\bf e}-\frac{1}{2}\overline{\bf I}|{\bf
e}|^{2}\right)+\frac{1}{\mu_{0}}\left({\bf b}{\bf b}-\frac{1}{2}\overline{\bf
I}|{\bf b}|^{2}\right).$ (4)

If the Lorentz force is written as the time rate of change of a “Lorentz
momentum” density ${\bf g}_{L}({\bf r},t)$, that is, ${\bf f}({\bf
r},t)=\partial{\bf g}_{L}({\bf r},t)/\partial t$, and ${\bf g}_{f}({\bf
r},t)=\epsilon_{0}[{\bf e}({\bf r},t)\times{\bf b}({\bf r},t)]$ designates the
microscopic ”electromagnetic-field momentum” density, then (3) can be
rewritten as

$\nabla\cdot\mbox{\boldmath${\overline{\mathcal{T}}}$\unboldmath}({\bf
r},t)=\frac{\partial}{\partial t}[{\bf g}_{L}({\bf r},t)+{\bf g}_{f}({\bf
r},t)]$ (5)

which shows that
$\nabla\cdot\mbox{\boldmath${\overline{\mathcal{T}}}$\unboldmath}({\bf r},t)$
is the time rate of change of the total microscopic electromagnetic momentum
density.

Integrating (5) over a volume $V$ with surface $S$ that encloses all the
charge-current, and assuming (as a thought experiment) that all the charge is
held fixed so there is no current (just electrostatic charge and field) until
the charge is released at $t=0$, we have ${\bf g}_{f}({\bf r},t\leq 0)=0$ and
$\int_{V}{\bf f}({\bf r},t\leq 0)dV=0$ (since the charge and field are
electrostatic for $t\leq 0$). Then integrating over time from $0$ to $t$
yields

$\int\limits_{0}^{t}\int\limits_{V}\nabla\cdot\mbox{\boldmath${\overline{\mathcal{T}}}$\unboldmath}({\bf
r},t^{\prime})dVdt^{\prime}=\int\limits_{0}^{t}\int\limits_{S}{\bf\hat{n}}\cdot\mbox{\boldmath${\overline{\mathcal{T}}}$\unboldmath}({\bf
r},t^{\prime})dSdt^{\prime}=\int\limits_{V}[{\bf g}_{L}({\bf r},t)+{\bf
g}_{f}({\bf r},t)]dV$ (6)

where ${\bf\hat{n}}$ is the unit normal to $S$ pointing out of $V$ and
$\int_{V}{\bf g}_{L}({\bf r},t)dV=\int_{0}^{t}\int_{V}{\bf f}({\bf
r},t^{\prime})dVdt^{\prime}$, which equals $0$ for $t\leq 0$. This equation
shows that
${\bf\hat{n}}\cdot\mbox{\boldmath${\overline{\mathcal{T}}}$\unboldmath}({\bf
r},t)$ represents the total electromagnetic momentum flow in the
$-{\bf\hat{n}}$ direction since ${\bf\hat{n}}$ points out of $V$. We see that
if the surface $S$ is far enough away that the radiated fields have not had
time to reach $S$ in the time $t$, then the surface integral in (6) is zero,
that is, the total microscopic electromagnetic momentum in $V$ is zero and

$\int\limits_{V}[{\bf g}_{L}({\bf r},t)+{\bf g}_{f}({\bf r},t)]dV=0$ (7)

for all time $t$ as long as no radiation has crossed $S$. Since $t$ can take
on any value as long as the surface $S$ of $V$ is chosen large enough, and
both $\int_{V}{\bf g}_{L}({\bf r},t\leq 0)dV=0$ and ${\bf g}_{f}({\bf r},t\leq
0)=0$, equation (7) expresses the conservation of total microscopic
electromagnetic momentum in $V$ and confirms that indeed the microscopic
electromagnetic-field momentum density ${\bf g}_{f}({\bf r},t)$ can be treated
as a legitimate physical momentum (because ${\bf g}_{L}({\bf r},t)$ is a
physical electromagnetic-force-produced momentum).

So far, nothing has been said about the kinetic momentum of the charge
carriers. Certainly, the microscopic electromagnetic force density ${\bf
f}({\bf r},t)$ will, in general, change the kinetic momentum and energy of the
charge carriers in $V$ but this does not affect the validity of the purely
electromagnetic-momentum relationships in (3)–(7). Kinetic momentum, as well
as “canonical momentum,” are introduced in Section IV, where the two momenta
are shown to be related by “hidden electromagnetic momenta.” It is assumed
throughout that the macroscopic electromagnetic fields and momenta of the
thermal motion of the molecules are either negligible or lie outside of the
bandwidth of the applied and induced macroscopic electromagnetic fields and
momenta.

## III. Macroscopic Force and Momentum

The ambiguity in the macroscopic force on a volume of a dipolar material can
be demonstrated directly from Maxwell’s homogeneous (no macroscopic free
charge and current, only polarization) equations for a macroscopic dipolar
continuum [2, ch. 1]

$\nabla\times{\bf E}+\frac{\partial{\bf B}}{\partial t}=0$ (8a)
$\nabla\times{\bf H}-\frac{\partial{\bf D}}{\partial t}=0$ (8b)
$\nabla\cdot{\bf B}=0$ (8c) $\nabla\cdot{\bf D}=0$ (8d)

with the constitutive relations

${\bf D}=\epsilon_{0}{\bf E}+{\bf P},\;\;\;\;{\bf B}=\mu_{0}({\bf H}+{\bf
M}).$ (9)

The vectors ${\bf P}$ and ${\bf M}$ are the macroscopic electric polarization
and the magnetic polarization (magnetization) densities and the $({\bf r},t)$
dependence of all the fields and polarizations have been suppressed. The term
“macroscopic” refers to fields and sources obtained by spatially averaging the
microscopic fields and sources at each instant of time over electrically small
volume elements $\Delta V$ that (in the medium) contain many discrete
(isolated to an indefinitely small region of space) dipoles. The term “dipolar
continuum”, which can be solid or fluid, simply means that the medium obeys
the Maxwell dipolar equations in (8,9). Thus, the combined term “macroscopic
dipolar continuum” refers to a medium composed of discrete dipoles that, upon
spatial averaging, obeys, to a good approximation, the Maxwell dipolar
equations in (8,9).111Contrary to what is sometimes stated in the historical
literature, Maxwell (and not the “Maxwellians”) determined all the equations
in (8) [14, 15, 16] for the mathematically defined fields of an ideal dipolar
continuum where the polarization densities are continuous functions of
position throughout the medium rather than composed of discrete dipoles as in
a macroscopic dipolar continuum [17]. It is unequivocally shown in [17], [18,
sec. 2.1.10] that Maxwell’s equations in (8) for the mathematically defined
fields of an ideal dipolar continuum also apply (approximately) to macroscopic
dipolar media if and only if the surfaces $\Delta S$ of the defining
macroscopic volumes $\Delta V$ lie in free space and do not intersect the
discrete dipoles. It follows that the force and momentum expressions obtained
from (8) for a volume $V$ with a surface $S$ that lies within the polarization
densities ${\bf P}$ or ${\bf M}$ are unambiguously defined if and only if the
surface $S$ is placed within the free space of a hypothetical thin shell that
separates $V$ from the rest of the continuum (created by removing the
polarization densities within the shell without changing the adjacent
polarization densities), so that the the total bound charge and current
densities in every $V$ are zero, that is, there are a discrete number of
electric and magnetic dipoles in $V$ [17], [18, secs. 2.1.1 and 2.1.10]. For
an ideal Maxwellian continuum, the thin free-space shells ensure that any
delta functions in the polarization charge and current densities at the
surface of the volume $V$ are taken into account. This requirement for
unambiguous field and force-momentum expressions, namely that the surfaces of
the volumes do not cut through the dipoles, is also stated by Einstein and
Laub [8], Landau and Lifshitz [1, secs. 6 and 29], and De Groot and Suttorp
[19, pp. 195–196]. These hypothetical thin free-space shells containing $S$
are assumed throughout the present paper and are crucial to a consistent
formulation and determination of dipolar electromagnetic force and momentum.
The same macroscopic Maxwell equations in (8,9) can be derived, for example,
using electric-current (Amperian) magnetic dipoles or magnetic-charge magnetic
dipoles as long as $[{\bf E},{\bf B}]$ or $[{\bf E},\mu_{0}{\bf H}]$ are
chosen as the initial primary fields in free space, respectively. In fact, for
the sake of mathematical simplicity, Maxwell uses ideal continuous
differential volume elements of magnetic-charge ${\bf M}$ to define the
primary magnetic field ${\bf H}$ and then defines the secondary magnetic field
as ${\bf B}=\mu_{0}({\bf H}+{\bf M})$ — written here in modern SI units [14,
arts. 385,386], [15].

Adding the equations that result by crossing ${\bf E}$ or ${\bf D}$ into (8a)
and ${\bf B}$ or ${\bf H}$ into (8b) to obtain four possible electromagnetic-
field momenta, then making use of the constitutive relations in (9), one can
obtain an unlimited number of different macroscopic force-momentum density
equations depending on the chosen stress dyadic. Restricting ourselves to five
force-momentum density equations, then integrating them over a volume $V$, one
obtains

$\displaystyle\int\limits_{V}\left[-(\nabla\cdot{\bf P}){\bf
E}+\left(\nabla\times{\bf M}+\frac{\partial{\bf P}}{\partial
t}\right)\times{\bf B}+\epsilon_{0}\frac{\partial}{\partial t}({\bf
E}\times{\bf B})\right]dV$
$\displaystyle=\int\limits_{V}\nabla\cdot\overline{\bf T}_{\\!\rm
Amp}dV=\int\limits_{S}{\bf\hat{n}}\cdot\overline{\bf T}_{\\!\rm Amp}dS$ (10a)
$\displaystyle\int\limits_{V}\left[{\bf P}\cdot\nabla{\bf
E}+\mu_{0}\frac{\partial{\bf P}}{\partial t}\times{\bf H}+\mu_{0}{\bf
M}\cdot\nabla{\bf H}-\frac{1}{c^{2}}\frac{\partial{\bf M}}{\partial
t}\times{\bf E}+\frac{1}{c^{2}}\frac{\partial}{\partial t}({\bf E}\times{\bf
H})\right]dV$ $\displaystyle=\int\limits_{V}\nabla\cdot\overline{\bf
T}_{\\!\rm EL}dV=\int\limits_{S}{\bf\hat{n}}\cdot\overline{\bf T}_{\\!\rm
EL}dS$ (10b) $\displaystyle\int\limits_{V}\left[{\bf P}\cdot\nabla{\bf
E}+\mu_{0}\frac{\partial{\bf P}}{\partial t}\times{\bf H}+\mu_{0}{\bf
M}\cdot\nabla{\bf H}-\frac{1}{c^{2}}\frac{\partial{\bf M}}{\partial
t}\times{\bf E}-\frac{1}{2}\nabla({\bf P}\cdot{\bf E}+\mu_{0}{\bf M}\cdot{\bf
H})+\frac{1}{c^{2}}\frac{\partial}{\partial t}({\bf E}\times{\bf H})\right]dV$
$\displaystyle=\int\limits_{V}\nabla\cdot\overline{\bf T}_{\\!\rm
A}dV=\int\limits_{S}{\bf\hat{n}}\cdot\overline{\bf T}_{\\!\rm A}dS\hskip
14.22636pt$ (10c) $\displaystyle\int\limits_{V}\left[(\nabla\times{\bf
M})\times{\bf B}+\Big{(}\frac{1}{\epsilon_{0}}\nabla\times{\bf
P}\Big{)}\times{\bf D}+\frac{1}{2}\nabla\Big{(}\frac{1}{\epsilon_{0}}{\bf
P}\cdot{\bf D}+{\bf M}\cdot{\bf
B}\Big{)}-\nabla\cdot\Big{(}\frac{1}{\epsilon_{0}}{\bf D}{\bf P}+{\bf B}{\bf
M}\Big{)}+\frac{\partial}{\partial t}({\bf D}\times{\bf B})\right]dV$
$\displaystyle=\int\limits_{V}\nabla\cdot\overline{\bf T}_{\\!\rm
M}dV=\int\limits_{S}{\bf\hat{n}}\cdot\overline{\bf T}_{\\!\rm M}dS\hskip
28.45274pt$ (10d) $\displaystyle\int\limits_{V}\left[-\mu_{0}(\nabla\cdot{\bf
M}){\bf H}+\left(\frac{1}{\epsilon_{0}}\nabla\times{\bf
P}-\mu_{0}\frac{\partial{\bf M}}{\partial t}\right)\times{\bf
D}+\mu_{0}\frac{\partial}{\partial t}({\bf D}\times{\bf H})\right]dV$
$\displaystyle=\int\limits_{V}\nabla\cdot\overline{\bf T}_{\\!\rm
H}dV=\int\limits_{S}{\bf\hat{n}}\cdot\overline{\bf T}_{\\!\rm H}dS$ (10e)

with the respective macroscopic electromagnetic stress dyadics defined as

$\overline{\bf T}_{\\!\rm Amp}=\epsilon_{0}\left({\bf E}{\bf
E}-\frac{1}{2}\overline{\bf I}|{\bf E}|^{2}\right)+\frac{1}{\mu_{0}}\left({\bf
B}{\bf B}-\frac{1}{2}\overline{\bf I}|{\bf B}|^{2}\right)$ (11a)
$\overline{\bf T}_{\\!\rm EL}=\left({\bf D}{\bf
E}-\frac{\epsilon_{0}}{2}\overline{\bf I}|{\bf E}|^{2}\right)+\left({\bf
B}{\bf H}-\frac{\mu_{0}}{2}\overline{\bf I}|{\bf H}|^{2}\right)$ (11b)
$\overline{\bf T}_{\\!\rm A}=\overline{\bf T}_{\\!\rm M}=\left({\bf D}{\bf
E}-\frac{1}{2}\overline{\bf I}({\bf D}\cdot{\bf E})\right)+\left({\bf B}{\bf
H}-\frac{1}{2}\overline{\bf I}({\bf B}\cdot{\bf H})\right)$ (11c)
$\overline{\bf T}_{\\!\rm H}=\frac{1}{\epsilon_{0}}\left({\bf D}{\bf
D}-\frac{1}{2}\overline{\bf I}|{\bf D}|^{2}\right)+\mu_{0}\left({\bf H}{\bf
H}-\frac{1}{2}\overline{\bf I}|{\bf H}|^{2}\right)$ (11d)

where the subscripts $\rm Amp$, $\rm EL$, $\rm A$, $\rm M$, and $\rm H$ stand
for the five macroscopic force-momentum equations in (10) with the Amperian
(as defined in [20, sec. A.1.6.4]), Einstein-Laub [8, sec. 3], Abraham [6,
eqs. (8) and (Va)], [7, eq. (18b) with the electromagnetic-field momentum
density vector defined between eqs. (21) and (22)], Minkowski [9, eqs.
(75),(94–97)], and Minkowski-with-${\bf H}$ (as defined herein by the present
author) macroscopic electromagnetic-field momentum densities, namely
$\epsilon_{0}{\bf E}\times{\bf B}$, ${\bf E}\times{\bf H}/c^{2}$ (for both
Einstein-Laub and Abraham), ${\bf D}\times{\bf B}$, and $\mu_{0}{\bf
D}\times{\bf H}$, respectively, and with the corresponding stress dyadics in
(10,11). Note that the Abraham and Minkowski stress dyadics are identical.
(Four versions of the Poynting theorem can also be expressed with the four
electromagnetic-field momenta used as the energy flux vector [21].)

The forces on the left-hand sides of (10) are given in terms of the
polarization densities ${\bf P}$ and ${\bf M}$ and the primary fields as
determined by the electromagnetic-field momentum vectors. The electromagnetic-
field momentum densities and electromagnetic stress dyadics labeled as
Einstein-Laub, Abraham, and Minkowski are defined by these authors in their
original papers. The designation of “Amperian” given to (10) as well as its
electromagnetic-field momentum density and electromagnetic stress dyadic is
taken from the textbook by Fano, Chu, and Adler [20, sec. A.1.6.4]. The
Amperian formulation is referred to as the Lorentz formulation by Mansuripur
[12]. I am not aware of any publication by either Ampere or Lorentz that
contains the equation (10) or the corresponding electromagnetic-field momentum
density and electromagnetic stress dyadic in (10,11a). The equation in (10)
with ${\bf D}\times{\bf H}$ is included here for the sake of completeness in
discussing the four alternative forms of the electromagnetic-field momentum
densities.

The volume integrals on the left-hand sides of (10) without the
electromagnetic-field momentum terms are the different macroscopic
electromagnetic forces (${\bf F}_{\\!\rm Amp}$, ${\bf F}_{\\!\rm EL}$, ${\bf
F}_{\\!\rm A}$, ${\bf F}_{\\!\rm M}$, or ${\bf F}_{\\!\rm H}$) corresponding
to each formulation. According to Einstein and Laub [8], the equation
containing ${\bf F}_{\\!\rm EL}$ and the Abraham macroscopic electromagnetic-
field momentum ${\bf E}\times{\bf H}/c^{2}$ [6, 7] as well as the stress
dyadic $\overline{\bf T}_{\\!\rm EL}$ is correct, although Einstein later
decided in favor of the Minkowski formulation over the Abraham formulation
[22]. According to Minkowski [9, eqs. (75),(94–97)], the equation with
$\overline{\bf T}_{\\!\rm M}$ and ${\bf D}\times{\bf B}$ as the macroscopic
electromagnetic-field momentum is correct. In fact, any one of these five
equations in (10) may or may not be correct depending upon whether or not the
macroscopic force ${\bf F}_{\\!\rm Amp}$, ${\bf F}_{\\!\rm EL}$, ${\bf
F}_{\\!\rm A}$, ${\bf F}_{\\!\rm M}$, or ${\bf F}_{\\!\rm H}$ (or some other
force) equals the sum of all the electromagnetic microscopic forces in the
volume $V$ with its surface $S$ in free space (or in a thin free-space shell
surrounding $V$).

With the surface $S$ in (10) lying in a thin free-space shell separating $V$
from the rest of the continuum (see Footnote 1), the value of each of the four
macroscopic electromagnetic stress dyadics on $S$ is the same, and thus the
value of each of their four surface integrals is the same and equals the value
of the total microscopic electromagnetic momentum flow across $S$ into $V$,
namely
$\int_{S}{\bf\hat{n}}\cdot\mbox{\boldmath${\overline{\mathcal{T}}}$\unboldmath}dS$,
because the macroscopic fields equal the microscopic fields in the free-space
shell of a sufficiently densely packed dipolar macroscopic continuum. This
implies that the values of each of the four volume integrals on the left-hand
sides of (10) are also equal (provided any delta functions in $\nabla{\bf E}$,
$\nabla{\bf H}$, $\nabla\cdot{\bf P}$, $\nabla\cdot{\bf M}$, $\nabla\times{\bf
P}$, and $\nabla\times{\bf M}$ at the interface between the material in $V$
and the free space of the thin shell surrounding $V$ are properly included in
the volume integrations).222For step functions $u(n)$ in ${\bf P}$ and ${\bf
M}$ and delta functions $\delta(n)$ in $\nabla{\bf E}$ and $\nabla{\bf H}$,
the integration of the resulting products $u(n)\delta(n)$ at the interface is
evaluated using values of ${\bf P}$, ${\bf M}$, ${\bf E}$, and ${\bf H}$ that
change rapidly but continuously across the interface in accordance with
Maxwell’s equations. This leads to $\int u\delta dn=\int u(du/dn)dn=1/2$.
Also, each of the four volume integrals of the divergences of the stress
dyadics in (10) are equal in value, provided any delta functions in the
divergences of the fields of the stress dyadics at the interface between the
material in $V$ and the free space of the thin shell surrounding $V$ are
properly included in the volume integrations.333It should be noted that the
tangential ${\bf E}$ and ${\bf H}$ fields need not be continuous across the
free-space/continuum interface, for example, if the continuum has “extreme”
constitutive parameters [23] or if the continuum is strongly spatially
dispersive [1, sec.103], [24, 25]. In strongly spatially dispersive continua,
the Poynting vector does not necessarily represent the total energy flow.
Nonetheless, all the force-momentum expressions derived in the present paper
hold for temporally and spatially dispersive dipolar continua. There are no
prohibitive restrictions on the constitutive relations except for their
satisfying (9). Moreover, because the surface $S$ of $V$ lies in a thin free-
space shell, the $\nabla({\bf P}\cdot{\bf E}+\mu_{0}{\bf M}\cdot{\bf H})$ term
in (10) and the $[\nabla({\bf P}\cdot{\bf D}/\epsilon_{0}+{\bf M}\cdot{\bf
B}),\nabla\cdot({\bf D}{\bf P}/\epsilon_{0}+{\bf B}{\bf M})]$ terms in (10)
integrate to zero by means of the gradient and divergence integral theorems
(since ${\bf P}$ and ${\bf M}$ are zero in free space). Also, the divergence
integral theorem converts the $-(\nabla\cdot{\bf P}){\bf E}$ and
$-\mu_{0}(\nabla\cdot{\bf M}){\bf H}$ terms in (10) and (10) to ${\bf
P}\cdot\nabla{\bf E}$ and $\mu_{0}{\bf M}\cdot\nabla{\bf H}$, respectively. In
other words, the five macroscopic force-momentum equations in (10) can be
reduced to four macroscopic force-momentum equations such that

$\displaystyle{\bf F}_{\\!\rm Amp}+\epsilon_{0}\frac{d}{dt}\int\limits_{V}{\bf
E}\times{\bf B}dV={\bf F}_{\\!\rm
EL}+\frac{1}{c^{2}}\frac{d}{dt}\int\limits_{V}{\bf E}\times{\bf H}dV={\bf
F}_{\\!\rm M}+\frac{d}{dt}\int\limits_{V}{\bf D}\times{\bf B}dV={\bf
F}_{\\!\rm H}+\mu_{0}\frac{d}{dt}\int\limits_{V}{\bf D}\times{\bf H}dV$
$\displaystyle={\bf F}+\epsilon_{0}\frac{d}{dt}\int\limits_{V}{\bf
e}\times{\bf b}\,dV=\int\limits_{S}{\bf\hat{n}}\cdot\overline{\bf
T}dS=\int\limits_{V}\nabla\cdot\overline{\bf T}dV\hskip 28.45274pt$ (12)

where, as defined above, ${\bf F}_{\\!\rm Amp}$, ${\bf F}_{\\!\rm EL}={\bf
F}_{\\!\rm A}$, ${\bf F}_{\\!\rm M}$, and ${\bf F}_{\\!\rm H}$ denote the
macroscopic-force volume integrals on the left-hand sides of (10), (10), (10),
(10), and (10), respectively, and $\overline{\bf T}$ can be any one of the
electromagnetic stress dyadics $\overline{\bf T}_{\\!\rm Amp}$, $\overline{\bf
T}_{\\!\rm EL}$, $\overline{\bf T}_{\\!\rm A}=\overline{\bf T}_{\\!\rm M}$,
$\overline{\bf T}_{\\!\rm H}$, or ${\overline{\mathcal{T}}}$. The last force-
momentum equation in (III. Macroscopic Force and Momentum) is the microscopic
force-momentum equation obtained from (3) with the total microscopic force in
$V$ given by

${\bf F}(t)=\int_{V}{\bf f}({\bf r},t)dV$ (13)

where the surface $S$ of $V$ in the microscopic distribution of molecules
meanders slightly to avoid cutting through the dipoles so that $S$ encloses a
discrete number of dipoles. By bringing the time derivatives outside of the
integrals in (III. Macroscopic Force and Momentum), it is assumed that $V$ and
its surface $S$ do not change with time. If they do change with time, the
partial time derivatives must remain inside the integral signs.

The volume forces in (10,III. Macroscopic Force and Momentum) can now be
written explicitly as

${\bf F}_{\\!\rm Amp}(t)=\int\limits_{V}\left[{\bf P}\cdot\nabla{\bf
E}+\left(\nabla\times{\bf M}+\frac{\partial{\bf P}}{\partial
t}\right)\times{\bf B}\right]dV$ (14a) ${\bf F}_{\\!\rm EL}(t)={\bf F}_{\\!\rm
A}(t)=\int\limits_{V}\left[{\bf P}\cdot\nabla{\bf E}+\mu_{0}\frac{\partial{\bf
P}}{\partial t}\times{\bf H}+\mu_{0}{\bf M}\cdot\nabla{\bf
H}-\frac{1}{c^{2}}\frac{\partial{\bf M}}{\partial t}\times{\bf E}\right]dV$
(14b) ${\bf F}_{\\!\rm M}(t)=\int\limits_{V}\left[(\nabla\times{\bf
M})\times{\bf B}+\Big{(}\frac{1}{\epsilon_{0}}\nabla\times{\bf
P}\Big{)}\times{\bf D}\right]dV$ (14c) ${\bf F}_{\\!\rm
H}(t)=\int\limits_{V}\left[\mu_{0}{\bf M}\cdot\nabla{\bf
H}+\left(\frac{1}{\epsilon_{0}}\nabla\times{\bf P}-\mu_{0}\frac{\partial{\bf
M}}{\partial t}\right)\times{\bf D}\right]dV$ (14d)

where again it is emphasized that any delta functions in the integrands across
the free-space/continuum interface of $V$ must be included in the evaluation
of the integrals in (14). The Einstein-Laub and Abraham electromagnetic forces
are equal, which has to be the case, because they have the same
electromagnetic-field momentum and with $S$ lying in free space all the
electromagnetic-stress dyadic integrals have the same value (as explained
above).

It is now apparent what each of these forces in (14) represents physically.
The Amperian force ${\bf F}_{\\!\rm Amp}(t)$ is equal to the sum of the forces
exerted by the primary fields ${\bf E}$ and ${\bf B}$ on the electric-charge
polarization density ${\bf P}$ (or, alternatively, on the equivalent electric-
charge density $-\nabla\cdot{\bf P}$) and on the equivalent electric-current
density ($\nabla\times{\bf M}+\partial{\bf P}/\partial t$). The Einstein-Laub
and Abraham forces ${\bf F}_{\\!\rm EL}(t)={\bf F}_{\\!\rm A}(t)$ are equal to
the sum of the forces exerted by the primary fields ${\bf E}$ and ${\bf H}$ on
the electric-charge polarization density ${\bf P}$ (or, alternatively, on the
equivalent electric-charge density $-\nabla\cdot{\bf P}$), and the magnetic-
charge polarization density ${\bf M}$ (or, alternatively, on the equivalent
magnetic-charge density $-\mu_{0}\nabla\cdot{\bf M}$), and on the electric-
and magnetic-polarization current densities $\partial{\bf P}/\partial t$ and
$-\mu_{0}\partial{\bf M}/\partial t$. The Minkowski force ${\bf F}_{\\!\rm
M}(t)$ is equal to the sum of the forces exerted by the primary fields ${\bf
D}$ and ${\bf B}$ on the equivalent electric- and magnetic-current densities
$\nabla\times{\bf M}$ and $\nabla\times{\bf P}/\epsilon_{0}$. The Minkowski-
with-${\bf H}$ force ${\bf F}_{\\!\rm H}(t)$ is equal to the sum of the forces
exerted by the primary fields ${\bf D}$ and ${\bf H}$ on the magnetic-charge
polarization density ${\bf M}$ (or, alternatively, on the equivalent magnetic-
charge density $-\mu_{0}\nabla\cdot{\bf M}$) and on the equivalent magnetic-
current density ($\nabla\times{\bf P}/\epsilon_{0}-\mu_{0}\partial{\bf
M}/\partial t$).

For periodic fields, the time derivatives of all four macroscopic
electromagnetic-field momenta in (10,III. Macroscopic Force and Momentum)
average to zero and all the time-average macroscopic/microscopic forces are
equal to the time-average value of the total macroscopic/microscopic
electromagnetic momentum flow across $S$ into $V$, that is

$\Big{\langle}\int\limits_{V}\nabla\cdot\overline{\bf
T}dV\Big{\rangle}=\int\limits_{V}\nabla\cdot\langle\overline{\bf T}\rangle
dV=\Big{\langle}\int\limits_{S}{\bf\hat{n}}\cdot\overline{\bf
T}dS\Big{\rangle}=\int\limits_{S}{\bf\hat{n}}\cdot\langle\overline{\bf
T}\rangle dS=\langle{\bf F}_{\\!\rm Amp}\rangle=\langle{\bf F}_{\\!\rm
EL}\rangle=\langle{\bf F}_{\\!\rm A}\rangle=\langle{\bf F}_{\\!\rm
M}\rangle=\langle{\bf F}_{\\!\rm H}\rangle=\langle{\bf F}\rangle$ (15)

where $\langle\,\rangle$ denotes the time average. Still the surface $S$ of
$V$ must lie in free space or in a hypothetical thin free-space shell
separating $V$ from the rest of the continuum in order for the time-average
macroscopic forces in (15) to equal the sum of all the time-average
microscopic forces $\langle{\bf F}\rangle$ on the discrete microscopic dipoles
in $V$.

### A. Forces on individual electric and magnetic dipoles

For general time varying (as opposed to time-averaged periodic) fields, the
determination of the total classical electromagnetic force on the microscopic
(molecular) electric and magnetic dipole moments ${\bf p}$ and ${\bf m}$ in an
electrically small macroscopic volume element $\Delta V$ (with surface $\Delta
S$ in free space enclosing $\Delta V$) used to define the macroscopic fields,
polarizations, and forces in a dipolar medium requires classical models to
represent the molecular electric and magnetic dipole moments. The electrically
small $\Delta V$ contain many discrete molecular dipoles and thus it is
implicitly assumed that the sources and fields are bandlimited to a maximum
frequency $f_{\rm max}$ with a minimum free-space or macroscopic-continuum
wavelength $\lambda_{\rm min}$ such that the maximum dimension of $\Delta V$
is much less than $\lambda_{\rm min}$ but with $\Delta V$ still containing
many discrete molecular dipole moments. Thus, the maximum dimension of $\Delta
V$ is electrically small ($k_{\rm max}\Delta a\ll 1$, where $k_{\rm
max}=2\pi/\lambda_{\rm min}$ and $\Delta a$ is the circumscribing radius of
$\Delta V$) and the maximum dimension of the individual molecules is much
smaller than the maximum dimension of $\Delta V$. The electromagnetic fields
from thermal motion of the molecules are assumed to produce negligible
macroscopic electromagnetic force and momentum within the operational
bandwidth. The molecules can be rotating, translating, and distorting as long
as at each instant of time all their multipole moments in each $\Delta V$ are
negligible except for electric and magnetic dipole moments, and the
accelerations of the molecules are not large enough to produce significant
radiation-reaction forces (more precisely, radiation reaction forces of the
molecules in $\Delta V$ are of higher order than $\Delta V$, that is, they
equal $o(\Delta V)$). Then at each instant of time, the molecular dipole
moments in each $\Delta V$ can be represented by the dipole moments of
classical models of electric and magnetic dipoles. Spatially averaging these
dipole moments and their fields at each instant of time using electrically
small macroscopic volume elements $\Delta V$ containing large numbers (in
principle, an infinite number) of these dipoles yields macroscopic
polarization densities and fields that are well-behaved functions of position
and time.

####  1\. Electric dipoles

Since all electric dipoles of molecules in nature are produced by the
separation of electrical charge, all realistic classical electric-dipole
models give the same force exerted by external source-free fields $[{\bf
E}_{e}({\bf r},t),{\bf B}_{e}({\bf r},t)]$ because the total internal force on
the electrically small classical electric dipole is zero. That is, the
quasielectrostatic force exerted on the positive charge by the negative charge
is equal and opposite the quasielectrostatic force exerted on the negative
charge by the positive charge, so that the total force exerted on a single
electrically small electric dipole ${\bf p}(t)$ equals the force exerted
directly by the external fields. A straightforward derivation of this force
from (2) gives [18, eq. (2.156)], [26]

${\bf F}_{ed}(t)={\bf p}(t)\cdot\nabla{\bf E}_{e}({\bf
r},t)+\mu_{0}\frac{d{\bf p}(t)}{dt}\times{\bf H}_{e}({\bf r},t)$ (16)

with the electrically small electric dipole located at a position ${\bf r}$
within $\Delta V$ at the time $t$. The external polarization densities ${\bf
P}_{e}$ and ${\bf M}_{e}$ are assumed zero at the position of the dipole so
that the source-free external electric and magnetic fields at the position of
the dipole are related by ${\bf B}_{e}=\mu_{0}{\bf H}_{e}$ and ${\bf
D}_{e}=\epsilon_{0}{\bf E}_{e}$. Note that the second term on the right-hand
side of (16) is zero for static electric dipole moments.

####  2\. Magnetic dipoles

Since all magnetic dipoles of molecules in nature are produced by electric
currents (circulating electric charges — as magnetic charge does not
exist),444Numerous experiments indicate that the intrinsic magnetic dipole
moments of elementary particles such as the electron, proton, muon, and
neutron are produced by circulating electric currents [4, p. 191]. probably
the simplest, most appealing, rigorously manageable stable classical model for
the electric-current-produced (Amperian) microscopic magnetic dipole moments
of molecules are electrically small perfect electric conductors (PEC’s), where
the term PEC is used here in the sense of a superconductor whose internal
electric and magnetic fields are zero even in the case of static fields [26].
Magnetic dipole moments can be induced by externally applied fields on a
singly connected PEC (for example, on a PEC sphere) and a stable static
magnetic dipole moment can exist on a doubly connected PEC without an
externally applied field, for example, on a PEC wire loop. Notably, Weber and
Maxwell [14, arts. 836–845] explained both diamagnetism and ordinary magnetism
(paramagnetism or ferro(i)magnetism) by means of PEC wire loops with no
initial static electric current in the case of diamagnetism, and predominantly
initial static electric current in the case of ordinary magnetism [15, 17,
26]. Of course, electric dipole moments are also induced on PEC’s by
externally applied fields.

It is rigorously proven from (2) in [26] that, remarkably, when arbitrarily
time varying external fields are applied to electrically small PEC’s (for
example, PEC wire loops), no matter how electrically small the PEC’s, there
is, in addition to a direct external electromagnetic-field force, an internal
“hidden momentum” electromagnetic force, namely $(-1/c^{2})\partial[{\bf
m}(t)\times{\bf E}_{e}({\bf r},t)]/\partial t$, induced indirectly by the
external fields, where $-{\bf m}(t)\times{\bf E}_{e}({\bf r},t)$ is the
microscopic hidden momentum of the Amperian magnetic dipole moment. The proof
in [26] is crucial because it is the only rigorous derivation of $-{\bf
m}\times{\bf E}$ as the hidden momentum for Amperian dipoles subject to
arbitrarily time varying external fields.555Unfortunately, the term “hidden
momentum” has a somewhat mysterious connotation, whereas this momentum simply
arises from the force exerted on the electric charge-current of the PEC by the
internal fields produced by the same electric charge-current [26]. Some
authors, such as Boyer [27, 28], prefer the term “internal momentum” to
“hidden momentum.” Although not mysterious, it may be nonetheless surprising
that this self-force internal momentum does not vanish for electrically small
PEC’s. The reason for this nonvanishing self force is that the electromagnetic
quasistatic fields of the PEC do not uncouple into quasistatic electric fields
and quasistatic magnetic fields. The electromagnetic fields remain coupled to
produce a “hidden momentum” force even as the size of the PEC approaches zero
[26]. Also, the derivation in [26] of (17) applies to electrically small
conductors with finite conductivity as well as PEC’s except that conductors
with finite conductivity cannot support an intrinsic static magnetic dipole
moment. Thus, the total force exerted by the external fields on a single
electrically small PEC Amperian magnetic dipole moment ${\bf m}(t)$ located at
the position ${\bf r}$ is given by [18, eq. (2.163)], [26]

${\bf F}_{md}(t)=\mu_{0}{\bf m}(t)\cdot\nabla{\bf H}_{e}({\bf
r},t)-\frac{1}{c^{2}}\frac{\partial{\bf m}(t)}{\partial t}\times{\bf
E}_{e}({\bf r},t)].$ (17)

This is the same force that would be exerted directly by the external fields
on an electrically small magnetic-charge magnetic dipole moment ${\bf m}(t)$
[18, eq. (2.166)] (if magnetic charge existed), for which the internal forces
cancel like those of the electric-charge electric dipole. Indeed, it is the
same force that Einstein and Laub [8] found assuming magnetic-charge models
for the magnetic dipoles. The “hidden-momentum” electromagnetic force on an
Amperian magnetic dipole has resulted in its total force exerted by the
external electric and magnetic fields equal to the force that would be
experienced by a magnetic-charge magnetic dipole [26]. Note that the second
term on the right-hand side of (17) is zero for static magnetic dipole
moments.

The molecules of the dipolar material with different dipole moments ${\bf
p}(t)$ and ${\bf m}(t)$ located at different positions ${\bf r}$ at time $t$
can be moving within the aforementioned acceleration limits that prevent
significant radiation-reaction forces (as long as ${\bf p}(t)$ and ${\bf
m}(t)$ are the instantaneous dipole moments in the laboratory frame of
reference). Even if all the molecular dipoles are modeled by lossless
scatterers, when brought together to form a macroscopic continuum, energy
losses can be exhibited in the continuum by, for example, assuming the
discrete lossless dipoles are connected to one another and to the other
molecules of the continuum by lossy, linear or nonlinear, compressible and
torsional springs. However, as mentioned above, the macroscopic
electromagnetic forces and momenta of thermal motion are assumed negligible
within the operational bandwidth. Although the molecules can be moving within
the aforementioned acceleration limits that prevent significant radiation-
reaction forces, at any one time $t$, each of the defining electrically small
volume elements $\Delta V$ is chosen to contain a discrete number of dipoles
(so that each $\Delta S$ lies in free space). Of course, if the dipolar
continuum is moving with constant velocity in free space, the dipole moments,
forces, and momenta may be determined more conveniently in an inertial
reference frame moving with the dipolar continuum, and the corresponding
forces and momenta in the laboratory frame can be found from the Lorentz
relativistic transformations.

### B. Macroscopic dipolar forces determined from microscopic dipole forces

To obtain an expression for the macroscopic dipolar forces from the
microscopic electric-dipole and magnetic-dipole forces in (16) and (17),
consider a distribution of many of these dipoles at each instant of time $t$
in a defining electrically small macroscopic volume element $\Delta V$ with
its surface $\Delta S$ in free space surrounding $\Delta V$; see Footnote 1.
The forces between all the isolated electrically small electric-charge
electric dipoles and the PEC combined electric and magnetic dipoles in $\Delta
V$ are quasistatic forces that cancel.666The electric and magnetic dipole
moments ${\bf p}$ and ${\bf m}$ induced by external fields on PEC’s are not
isolated, but the force on the electrically small PEC has already been proven
[26], as explained above, to be equal to the sum of the electric- and
magnetic-dipole forces in (16) and (17). An alternative heuristic way of
showing that the isolated molecules in $\Delta V$ do not exert a net force on
each other is to consider each molecule lying in a small spherical hole within
a spherical $\Delta V$ of uniform polarization which is separated by a thin
free-space spherical shell from the rest of the macroscopic continuum. For an
electrically small $\Delta V$, the quasistatic fields in the hole produced by
the polarization in $\Delta V$ will be zero (the fields of the surface
polarization charges and currents on the surfaces of $\Delta V$ and the hole
cancel) and thus no force will be exerted by the polarization in $\Delta V$ on
the dipole moments of the molecules within the hole. In addition, the
radiation-reaction forces of the molecules in $\Delta V$ are assumed to be of
higher order than $\Delta V$, that is, $o(\Delta V)$. Thus only the external
fields from sources outside $\Delta V$ will produce an appreciable net
electromagnetic force on the dipoles in $\Delta V$. The average electric
polarization of all the ${\bf p}$’s in $\Delta V$ is ${\bf P}({\bf r},t)$ so
that the external force on this electric polarization (all the discrete dipole
moments ${\bf p}$) in $\Delta V$ is given by (16) with ${\bf P}\Delta V$
substituted for ${\bf p}$, namely

$\Delta{\bf F}_{ed}({\bf r},t)=\left[{\bf P}({\bf r},t)\cdot\nabla{\bf
E}_{e}({\bf r},t)+\mu_{0}\frac{\partial{\bf P}({\bf r},t)}{\partial
t}\times{\bf H}_{e}({\bf r},t)\right]\Delta V$ (18)

where ${\bf r}$ is a point in $\Delta V$.

The external electric and magnetic fields in (18) are the fields in $\Delta V$
with ${\bf P}$ and ${\bf M}$ in $\Delta V$ removed, that is, the cavity
fields, ${\bf E}_{e}={\bf E}_{c}={\bf E}-{\bf E}_{s}$ and ${\bf H}_{e}={\bf
H}_{c}={\bf H}-{\bf H}_{s}$ where ${\bf E}$ and ${\bf H}$ are the spatially
averaged macroscopic electric and magnetic fields and ${\bf E}_{s}$ and ${\bf
H}_{s}$ are the electric and magnetic fields produced by the equivalent
surface charge and current densities ${\bf\hat{n}}\cdot{\bf P}$ and ${\bf
M}\times{\bf\hat{n}}$ at the outer surface of the free-space shell that
surrounds the electrically small $\Delta V$ [18, p. 46]. (Incidentally, for
spherical $\Delta V$, the ${\bf E}_{s}$ and ${\bf H}_{s}$ are approximately
uniform throughout $\Delta V$ so that $\nabla{\bf E}_{s}$ and $\nabla{\bf
H}_{s}$ are approximately zero throughout $\Delta V$.) Because these surface
charge and current densities have equal and opposite counterpart surface
charge and current densities for the volume elements adjacent to $\Delta V$,
they create equal and opposite forces on the electrically small $\Delta V$ and
its adjacent volume elements that cancel when the forces on all the $\Delta V$
in a volume $V$ are summed to obtain the integrated force in the dipolar
material in $V$777The volume elements $\Delta V$ that border the
polarization/free-space interface of $V$ have one side without an adjacent
volume element. This merely changes the values of ${\bf E}_{s}$ and ${\bf
H}_{s}$ but not the cancellation argument leading from (18) to (19). However,
it is emphasized that the equivalent surface charge/current densities at the
surface of $V$ contribute to the force in (22a) on the volume $V$ of dipolar
material since these forces are taken into account by delta functions in
$\nabla{\bf E}$ and $\nabla{\bf H}$ in the integrand of (22a).; thus the ${\bf
E}_{s}$ and ${\bf H}_{s}$ fields can be ignored and (18) becomes effectively
for volume elements inside the dipolar macroscopic continuum

$\Delta{\bf F}_{ed}({\bf r},t)=\left[{\bf P}({\bf r},t)\cdot\nabla{\bf E}({\bf
r},t)+\mu_{0}\frac{\partial{\bf P}({\bf r},t)}{\partial t}\times{\bf H}({\bf
r},t)\right]\Delta V$ (19)

a relatively uncomplicated well-defined expression for the macroscopic force
density, which when integrated to get the force on the electric polarization
density ${\bf P}$ in a volume $V$, gives the sum of the microscopic forces on
all the discrete dipole moments ${\bf p}$ in $V$.

Similarly, (17) yields the macroscopic force on the macroscopic magnetic
polarization (magnetization) ${\bf M}$ in $\Delta V$

$\Delta{\bf F}_{md}({\bf r},t)=\left[\mu_{0}{\bf M}({\bf r},t)\cdot\nabla{\bf
H}_{e}({\bf r},t)-\frac{1}{c^{2}}\frac{\partial{\bf M}({\bf r},t)}{\partial
t}\times{\bf E}_{e}({\bf r},t)\right]\Delta V$ (20)

that equals the sum of the microscopic forces on the discrete magnetic dipole
moments ${\bf m}$ in $\Delta V$.

By the same argument that led from (18) to (19), the force density in (20) can
be re-expressed effectively as

$\Delta{\bf F}_{md}({\bf r},t)=\left[\mu_{0}{\bf M}({\bf r},t)\cdot\nabla{\bf
H}({\bf r},t)-\frac{1}{c^{2}}\frac{\partial{\bf M}({\bf r},t)}{\partial
t}\times{\bf E}({\bf r},t)\right]\Delta V$ (21)

for the purpose of integrating over a volume $V$ to get the macroscopic force
on ${\bf M}$ equal to the sum of the microscopic forces on the discrete
magnetic dipole moments ${\bf m}$ in $V$. It is emphasized that (20)–(21)
holds for both diamagnetic and paramagnetic/ferro(i)magnetic magnetization
${\bf M}$.

Within a volume $V>\Delta V$ with the surface $S$ of $V$ lying in a thin free-
space shell, the volume elements in (19) and (21) used to define the
electromagnetic forces in $V$ can change shape slightly and be shifted so that
${\bf r}$ can take any value in $V$. (Because each $\Delta V$ contains a
discrete number of dipoles, there will be a small spatial jitter in this
macroscopic force density with the jitter becoming smaller with the larger the
number of dipoles per unit volume. This jitter can be smoothed by various
techniques such as moving averages.) Thus the total macroscopic
electromagnetic force ${\bf F}_{\\!e\ell}(t)$ on the macroscopic polarization
densities ${\bf P}$ and ${\bf M}$ in a volume $V$ is determined approximately
by integrating the sum of the macroscopic electric-dipole and magnetic-dipole
force densities in (19) and (21) to get

${\bf F}_{\\!e\ell}(t)=\int\limits_{V}\left[{\bf P}({\bf r},t)\cdot\nabla{\bf
E}({\bf r},t)+\mu_{0}\frac{\partial{\bf P}({\bf r},t)}{\partial t}\times{\bf
H}({\bf r},t)+\mu_{0}{\bf M}({\bf r},t)\cdot\nabla{\bf H}({\bf
r},t)-\frac{1}{c^{2}}\frac{\partial{\bf M}({\bf r},t)}{\partial t}\times{\bf
E}({\bf r},t)\right]dV$ (22a) which, with the surface $S$ of $V$ in free space
and any delta functions in the spatial derivatives across the free-
space/continuum interface of $V$ properly included in the integrations (see
Footnote 2), equals the sum of the microscopic forces on the discrete dipole
moments ${\bf p}$ and ${\bf m}$ in $V$, that is ${\bf F}_{\\!e\ell}(t)={\bf
F}(t).$ (22b) (The integration in (22a) becomes a more accurate representation
of the sums in (19) and (21) as the number of dipoles per cubic minimum
wavelength becomes larger and $\Delta V$ becomes electrically smaller.) We see
from (14b) and (22a,22b) that ${\bf F}_{\\!e\ell}(t)={\bf F}_{\\!\rm
EL}(t)={\bf F}(t).$ (22c)

Since we have proven that the macroscopic electromagnetic force ${\bf
F}_{\\!e\ell}={\bf F}_{\\!\rm EL}$ equals the sum of the microscopic
electromagnetic forces in $V$, namely ${\bf F}$ in (13), we also have from
(III. Macroscopic Force and Momentum) that

$\frac{1}{c^{2}}\frac{d}{dt}\int\limits_{V}{\bf E}\times{\bf
H}\,dV=\epsilon_{0}\frac{d}{dt}\int\limits_{V}{\bf e}\times{\bf b}\,dV$ (23)

that is, the macroscopic and microscopic electromagnetic-field momenta in $V$
are equal. If the total electromagnetic momentum entering the volume is zero,
that is, $\int_{S}{\bf\hat{n}}\cdot\overline{\bf T}dS=0$ (for example, if the
surface $S$ lies outside the externally applied pulse), then (III. Macroscopic
Force and Momentum) and (22c) show that the macroscopic force on the volume
$V$ of material is given by

${\bf F}_{\\!e\ell}(t)={\bf F}_{\\!\rm EL}(t)={\bf
F}(t)=-\frac{1}{c^{2}}\frac{d}{dt}\int\limits_{V}{\bf E}\times{\bf
H}\,dV\neq-\frac{d}{dt}\int\limits_{V}{\bf D}\times{\bf B}\,dV.$ (24)

That is, we have shown that the macroscopic electromagnetic force exerted by
an electromagnetic pulse on a volume $V$ of dipolar material is given by the
negative time derivative of the Abraham macroscopic electromagnetic-field
momentum and not by the negative time derivative of the Minkowski (or any
other) macroscopic electromagnetic-field momentum. Integrating the equality in
(24) over a time interval shows that the macroscopic electromagnetic impulse
applied to the charge-current in this volume $V$ plus the change in the
Abraham macroscopic electromagnetic-field momentum in $V$ during that time
interval is zero if no net electromagnetic momentum enters or leaves the
volume $V$.

The macroscopic electromagnetic force in (22) is the Einstein-Laub [8]
macroscopic electromagnetic force in (10,14b) with the associated
Abraham/Einstein-Laub electromagnetic-field momentum ${\bf E}\times{\bf
H}/c^{2}$ in (10,10)). However, Einstein and Laub simply assume microscopic
magnetic-charge magnetic dipoles analogously to microscopic electric-charge
electric dipoles even though magnetic dipoles in matter are produced by
circulating electric currents. They then generalize without proof their force
expressions on microscopic electric and magnetic dipoles to the corresponding
expressions with macroscopic electric and magnetic polarization densities and
fields. Also, Einstein and Laub did not indicate or speculate on the possible
existence of internal momentum (today referred to as hidden momentum) induced
by the external fields applied to Amperian (circulating-electric-current)
magnetic dipoles. In fact, Einstein [22, p. 591] later wrote that the
electromagnetic tensor that he and Laub had derived was “wrong” and that the
correct tensor was that of Minkowski.

Mansuripur [12] also simply assumes that “magnetism is no longer associated
with an electric current density, but rather with bound magnetic-charge and
bound magnetic-current densities” to argue for the Einstein-Laub macroscopic
force (and the associated Abraham macroscopic electromagnetic-field momentum)
but, again, without proving that the macroscopic force is equal to the sum of
the microscopic electromagnetic forces on realistic models of electric and
magnetic dipoles in $V$. On the one hand, assuming magnetism is produced by
magnetic-charge separation denies the experimental results of modern physics
that magnetic charge and magnetic-charge dipoles do not exist in nature; and
on the other hand, avoids treating the problem of magnetic polarization
properly because of the past difficulties with dealing rigorously with
realistic models of Amperian magnetic dipoles.

Here in the present work, we have rigorously proven that the macroscopic
electromagnetic force in (14b,22) on the volume $V$ of macroscopic electric
and magnetic polarization ${\bf P}$ and ${\bf M}$ is equal to the sum of the
electromagnetic forces on the discrete microscopic electric-charge electric
dipoles and electric-current (Amperian) magnetic dipoles in $V$ using
realistic classical models for the discrete microscopic electric and magnetic
dipoles. The macroscopic force in (14b,22) was first obtained in [18, eq.
2.173] using the particular example of a PEC sphere to argue for the hidden
momentum from arbitrary time varying fields rather than the general proof
given in [26] for the hidden-momentum force on arbitrarily shaped microscopic
PEC models of magnetic dipoles. In summary, it is proven that for these
realistic classical models of electric and magnetic dipoles found in nature,
the physically meaningful macroscopic electromagnetic force and momentum
(equal to the sum of the microscopic-dipole electromagnetic forces and
momenta) are given in (III. Macroscopic Force and Momentum) with the Einstein-
Laub macroscopic electromagnetic force ${\bf F}_{\\!\rm EL}(t)$ in (14b,22)
and the Abraham/Einstein-Laub macroscopic electromagnetic-field momentum
density ${\bf E}\times{\bf H}/c^{2}$. Also, as explained above, the
macroscopic polarization densities ${\bf P}$ and ${\bf M}$ in (22a) can be
lossless or lossy, independent of whether the microscopic models of the
discrete dipoles are lossless or lossy, and their constitutive relations are
unrestricted except for obeying (9).

The rigorously derived macroscopic force density in (22a) shows that internal
(hidden) momentum force is produced by the applied fields in macroscopic
magnetization ${\bf M}$ as well as in the microscopic momentum of the Amperian
dipoles comprising the magnetization, and, in particular, in the magnetization
of artificial molecules (inclusions) of metamaterials containing magnetic (or
magnetodielectric) material. That is, the force density in the magnetization
${\bf M}$ of inclusions is given by $-\partial{\bf M}/\partial t\times{\bf
E}/c^{2}$ rather than ${\bf M}\times(\partial{\bf E}/\partial t)/c^{2}$ and,
thus, includes a macroscopic hidden momentum $-\partial({\bf M}\times{\bf
E})/(c^{2}\partial t)$. This macroscopic hidden momentum for the magnetization
${\bf M}$ is a consequence of the microscopic hidden momentum which is
rigorously found for the classical conductor Amperian models of the discrete
magnetic dipoles comprising ${\bf M}$ and is not lost in the averaging process
required to obtain the macroscopic magnetization ${\bf M}$ from the
microscopic Amperian magnetic dipole moments ${\bf m}$.

## IV. Kinetic and Canonical Momenta

It is emphasized that the force ${\bf F}_{\\!e\ell}(t)={\bf F}_{\\!\rm EL}(t)$
in (III. Macroscopic Force and Momentum,14b,22) equals the total
electromagnetic force on realistic models of the microscopic dipoles in $V$.
If the dipoles are rigidly attached to a rigid lattice structure or any other
rigid material with a fixed volume inside $V$ whose surface $S$ in free space
surrounds all of the material, then this electromagnetic force is transferred
to the rigid material. Also, if the rigid material inside $V$ is held fixed,
the opposite force must be exerted by whatever outside agent (other than the
given applied electromagnetic fields) is holding the rigid material inside $V$
fixed. On the other hand, even if the outer surface of the material inside $V$
is held fixed by an outside agent, but the charge carriers accelerate
appreciably (yet not enough to produce radiation-reaction forces greater than
$o(\Delta V)$) with respect to the fixed surface of material inside $V$ or
they collide with other molecules of the material that can accelerate with
respect to its fixed surface, producing a total change in kinetic momentum
${\bf G}_{\rm k}(t)$ of the material inside $V$, then the force ${\bf
F}_{\\!ag}(t)$ exerted by the outside agent holding the surface of the
material inside $V$ fixed will satisfy Newton’s relativistic equation of
motion

${\bf F}_{\\!ag}(t)+{\bf F}_{\\!e\ell}(t)={\bf F}_{\\!ag}(t)+{\bf
F}_{\\!EL}(t)=\frac{d{\bf G}_{\rm k}(t)}{dt}.$ (25)

Thus ${\bf F}_{\\!ag}(t)$ can be expressed with the help of (III. Macroscopic
Force and Momentum) as

${\bf F}_{\\!ag}(t)=\frac{d{\bf G}_{\rm
k}(t)}{dt}+\frac{1}{c^{2}}\frac{d}{dt}\int\limits_{V}{\bf E}\times{\bf
H}\,dV-\int\limits_{S}{\bf\hat{n}}\cdot\overline{\bf T}dS$ (26)

where $d{\bf G}_{\rm k}(t)/dt$ is the time rate of change of the kinetic
momentum of all the material inside the volume $V$. If outside the surface $S$
of $V$ there is only free space, the volume $V$ in (26) can be replaced by the
volume $V_{\infty}$ of all space such that $\overline{\bf T}$ is zero on
$S_{\infty}$ for a finite pulse. Then (26) becomes simply

${\bf F}_{\\!ag}(t)=\frac{d{\bf G}_{\rm
k}(t)}{dt}+\frac{1}{c^{2}}\frac{d}{dt}\int\limits_{V_{\infty}}{\bf
E}\times{\bf H}\,dV.$ (27)

Silveirinha [13, eq. (10b)] obtains a form of (26) with the microscopic
electromagnetic-field momentum replacing the macroscopic electromagnetic-field
momentum in (26) but without proving which macroscopic force (namely, ${\bf
F}_{\\!\rm EL}$) equals the sum of the microscopic forces. For the Minkowski
formulation, ${\bf E}\times{\bf H}/c^{2}$ in the volume integral of (26) would
be replaced by ${\bf D}\times{\bf B}$.

In general, ${\bf F}_{\\!ag}(t)$ in (25–27) is the total outside-agent force
exerted on the material inside any volume $V$ outside of which exists only
free space even if the material inside $V$ is allowed to accelerate and
deform, as long as the macroscopic fields and polarizations are those in the
moving material inside $V$ (as seen in the laboratory frame) and ${\bf G}_{\rm
k}(t)$ is the total change in kinetic momentum of the material inside $V$. The
momentum ${\bf G}_{\rm k}(t)$ can be considered as a macroscopic kinetic
momentum equal to the change in the total kinetic momentum of all the material
(charged and uncharged particles) inside $V$. This change in total kinetic
momentum is brought about by the applied electromagnetic and outside-agent
forces. The equation (26) says that the outside-agent force exerted on the
material inside the volume $V$ equals the time rate of change of the kinetic
plus electromagnetic-field momentum inside $V$ plus the time rate of change of
the total electromagnetic momentum entering $V$; recall that ${\bf\hat{n}}$ is
the unit normal pointing out of $V$. Similarly, (27) says that the outside-
agent force exerted on the material in the volume $V$ equals the time rate of
change of the kinetic momentum of the material in $V$ plus the time rate of
change of the electromagnetic-field momentum throughout all space
$V_{\infty}$. Unfortunately, $d{\bf G}_{\rm k}(t)/dt$ is usually unknown and
may be difficult to determine, although in some cases where the material has a
high rigidity and an outside-agent force keeps the surface of the volume $V$
of material fixed, it may be reasonable to assume $d{\bf G}_{\rm k}(t)/dt$ is
negligible. If no outside force is supplied by an agent to the material inside
the volume $V$, then ${\bf F}_{\\!ag}(t)=0$ and we have from (25–27)

${\bf F}_{\\!\rm EL}(t)=\frac{d{\bf G}_{\rm
k}(t)}{dt}=\int\limits_{S}{\bf\hat{n}}\cdot\overline{\bf
T}dS-\frac{1}{c^{2}}\frac{d}{dt}\int\limits_{V}{\bf E}\times{\bf
H}dV=-\frac{1}{c^{2}}\frac{d}{dt}\int\limits_{V_{\infty}}{\bf E}\times{\bf
H}\,dV.$ (28)

Although it has been shown that the Einstein-Laub macroscopic force and
Abraham/Einstein-Laub macroscopic electromagnetic-field momentum are the ones
that equal the sum of the microscopic forces and the sum of the microscopic
electromagnetic-field momenta, it may be revealing to relate the Minkowski and
Einstein-Laub macroscopic forces and electromagnetic-field momenta. Toward
this end, use (III. Macroscopic Force and Momentum) to write

${\bf F}_{\\!\rm EL}(t)-{\bf F}_{\\!\rm M}(t)=\frac{d}{dt}\int\limits_{V}{\bf
D}\times{\bf B}\,dV-\frac{1}{c^{2}}\frac{d}{dt}\int\limits_{V}{\bf
E}\times{\bf H}dV.$ (29)

If the outside-agent force is zero, ${\bf F}_{\\!\rm EL}=d{\bf G}_{\rm k}/dt$
and (29) can be rewritten as

$\frac{d{\bf G}_{\rm k}(t)}{dt}-{\bf F}_{\\!\rm
M}(t)=\frac{d}{dt}\int\limits_{V}{\bf D}\times{\bf
B}\,dV-\frac{1}{c^{2}}\frac{d}{dt}\int\limits_{V}{\bf E}\times{\bf H}dV.$ (30)

Furthermore, expressing the Minkowski force in terms of the time rate of
change of a “Minkowski-force momentum,” that is

${\bf F}_{\\!\rm M}(t)=\frac{d{\bf G}_{\rm M}(t)}{dt}$ (31)

one obtains

$\frac{d{\bf G}_{\rm k}(t)}{dt}-\frac{d{\bf G}_{\rm
M}(t)}{dt}=\frac{d}{dt}\int\limits_{V}{\bf D}\times{\bf
B}\,dV-\frac{1}{c^{2}}\frac{d}{dt}\int\limits_{V}{\bf E}\times{\bf H}\,dV$
(32)

or, assuming the initial macroscopic fields are zero and ${\bf G}_{\rm k}(t)$
is the change in kinetic momentum from the initial kinetic momentum, and ${\bf
G}_{\rm M}(0)=0$, then

${\bf G}_{\rm k}(t)-{\bf G}_{\rm can}(t)=\int\limits_{V}\left({\bf
D}\times{\bf B}-{\bf E}\times{\bf H}/c^{2}\right)dV=\int\limits_{V}\left({\bf
P}\times{\bf B}-{\bf M}\times{\bf E}/c^{2}\right)dV$ (33)

where the Minkowski-force momentum ${\bf G}_{\rm M}(t)$ has been renamed ${\bf
G}_{\rm can}(t)$ to correspond to the microscopic “canonical momentum” of
Lembessis et al. [29, 30] and to the macroscopic “medium canonical momentum”
of Barnett and Loudon [31, 32]. Specifically, (33) corresponds to the
macroscopic equation [32, eq. 4.4] and to the microscopic equation [29, eq.
(50)], where it should be noted that [29] treats only single discrete electric
dipoles (and no magnetic dipoles).

In the context of the macroscopic formulation of the present paper, the vector
$({\bf P}\times{\bf B}-{\bf M}\times{\bf E}/c^{2})$ in (33) is the sum of the
macroscopic hidden momentum density ($-{\bf M}\times{\bf E}/c^{2}$) of
Amperian (circulating-electric-current) magnetic dipoles and the macroscopic
hidden momentum density (${\bf P}\times{\bf B}$) of hypothetical circulating-
magnetic-current electric dipoles replacing the electric-charge electric
dipoles.888The ${\bf P}\times{\bf B}$ can be shown to be the macroscopic
hidden momentum density of hypothetical circulating-magnetic-current electric
dipoles in the same way that it is shown from [26] that $-{\bf M}\times{\bf
E}/c^{2}$ is the macroscopic hidden momentum density of circulating-electric-
current magnetic dipoles. In the absence of any outside-agent force, the
kinetic momentum is simply equal to the electromagnetic-force momentum
imparted to the dipoles by the applied fields [${\bf F}_{\\!EL}(t)={\bf
F}_{\\!e\ell}(t)=d{\bf G}_{\rm k}(t)/dt$]. Therefore, the canonical
(Minkowski-force) momentum is simply the direct momentum (that is, the
electromagnetic momentum without the internal “hidden” momentum) imparted by
the applied fields to electric and magnetic dipole moments created by magnetic
and electric circulating current, respectively [${\bf G}_{\rm M}(t)={\bf
G}_{\rm can}(t)={\bf G}_{\rm k}(t)-\int_{V}({\bf P}\times{\bf B}-{\bf
M}\times{\bf E}/c^{2})dV$], where $\int_{V}({\bf P}\times{\bf B}-{\bf
M}\times{\bf E}/c^{2})dV$ is the internal “hidden” momentum of the magnetic-
and electric-circulating-current dipoles in $V$. Thus, the difference between
the Abraham and Minkowski electromagnetic forces in (29) equals the sum of the
hidden-momentum forces on these electric-current magnetic dipoles and
hypothetical magnetic-current electric dipoles. This makes sense because the
${\bf E}$ and ${\bf H}$ primary fields of the Abraham formulation require
electric- and magnetic-charge electric and magnetic dipoles, respectively,
(that exhibit no hidden momentum) to derive the Maxwell equations and
constitutive relations in (8,9), whereas the ${\bf D}$ and ${\bf B}$ primary
fields of the Minkowski formulation require electric- and magnetic-current
magnetic and electric dipoles, respectively, (that exhibit the hidden momentum
in (33)) to derive the Maxwell equations and constitutive relations in (8,9).
However, neither (33) nor the references [29, 30, 31, 32] prove, as is done in
the above analysis of the present paper, that it is the time derivative of the
Abraham/Einstein-Laub macroscopic electromagnetic-field momentum, and not the
time derivative of the Minkowski macroscopic electromagnetic-field momentum,
along with any one of the stress dyadics in (4,11), that determine the correct
macroscopic electromagnetic force ${\bf F}_{\rm EL}$ on a volume of bulk
dipolar material (namely the force that equals the sum of the microscopic
electromagnetic forces on realistic discrete electric-charge electric dipoles
and electric-current (Amperian) magnetic dipoles in that volume of material).
It should also be emphasized that in the context of classical physics, the
canonical momentum is defined by (29–33) and [32, eq. 4.4], whereas
fundamentally the “canonical momentum” is a concept that arises in quantum
electrodynamical scattering of light by an atom [29, 30]. Lembessis et al.
[29, 30] refer to the microscopic quantum electrodynamical canonical momentum
as a “Röntgen-type interaction term.” However, none of these papers, or any
other previous papers, as far as I am aware, have associated the difference
between the kinetic and canonical momenta with the internal “hidden”
electromagnetic momentum of electric-current magnetic dipoles and magnetic-
current electric dipoles.

## V. Experimental Verification of the Einstein-Laub/Abraham Formulation

Through the years there have been many experiments done to measure the time-
average macroscopic electromagnetic radiation force, especially the force on
mirrors and on the surfaces or membranes between two fluids. None of the time-
average force experiments can distinguish between the Einstein-Laub/Abraham
formulation and the Minkowski formulation because there is no difference
between the two for time-average forces, as shown in Section III.

There are two noteworthy experiments that have measured the time varying force
on dipolar-material bodies, namely on a magnetodielectric toroid [33] and on a
high-permittivity dielectric toroid [34]. The James magnetodielectric toroid
experiment verifies the Abraham macroscopic electromagnetic-field momentum as
well as the Einstein-Laub electromagnetic force obtained herein, and the
Walker et al. high-permittivity dielectric toroid experiment verifies the
Einstein-Laub macroscopic electromagnetic force obtained herein.

### A. The magnetodielectric toroid

A simplified schematic of the magnetodielectric ferrite toroid taken from
James’s thesis [33] (with Shockley as his thesis advisor) is shown in Fig. 1.
Actually, two ferrite toroids were used in this experiment, one with a
relative permeability of $\mu_{\rm r}\approx 16$ and a relative permittivity
of $\epsilon_{\rm r}\approx 7$, the other with a relative permeability of
$\mu_{\rm r}\approx 43$ and a relative permittivity of $\epsilon_{\rm
r}\approx 7.6$. A current $I_{0}=I\sin({\omega}\pm{\omega}_{0})t$ flows in the
axial ($z$) direction through the center of the toroid. A voltage
$V_{0}=V\sin{\omega}t$ is applied between the metalized inner and outer
circular cylindrical surfaces of the toroid. The value of the beat frequency
${\omega}_{0}/(2\pi)$ was kept at about $3$ kHz, which approximately equaled
the mechanical resonant frequency of the combined toroid and piezoelectric
measurement transducer that was in contact with the toroid. The value of
${\omega}/(2\pi)$ was varied from $10$ to $32$ kHz. This voltage and current
produce time varying magnetic fields in the azimuthal ($\phi$) direction and
time varying electric fields in the radial ($r$) direction. Because the
voltage and current are time varying, each produce both a radial electric
field and an azimuthal magnetic field in the toroid. These quasielectrostatic
and quasimagnetostatic fields are relatively straightforward to determine. The
voltage sets up a radial quasielectrostatic field which in turn produces an
azimuthal quasimagnetostatic field that varies asymmetrically with $z$ across
the toroid. Similarly, the current sets up an azimuthal quasimagnetostatic
field which in turn produces a radial quasielectrostatic field that varies
asymmetrically with $z$ across the toroid. In all, one has a total time
varying radial electric field $E_{r}(r,z,t)$ and a total time varying
azimuthal magnetic field $H_{\phi}(r,z,t)$. These electric and magnetic fields
produce an axial ($z$-directed) force on the toroid that James was able to
measure with a piezoelectric transducer. The predominant measureable time-
varying force occurs at the resonant beat frequency ${\omega}_{0}/(2\pi)$.

Fig. 1: Simplified schematic of the magnetodielectric ferrite toroid used in
the James experiment [33].

The basic equation that James uses to decide experimentally between the
Abraham and Minkowski electromagnetic-field momenta corresponds to our
equation (26), that is

${\bf F}_{\\!ag}(t)=\frac{d{\bf G}_{\rm k}(t)}{dt}+\frac{d}{dt}{\bf
G}_{f}(t)-\int\limits_{S}{\bf\hat{n}}\cdot\overline{\bf T}dS$ (34)

where ${\bf F}_{\\!ag}(t)=F_{\\!ag}(t){\bf\hat{z}}$ is James’s transducer-
measured force on the toroid. The surface integral of the stress dyadic
$\overline{\bf T}$ in (34) is evaluated analytically over the surface $S$ in
free space just outside the material of the toroid, using the theoretically
determined expressions for $E_{r}(r,z,t)$ and $H_{\phi}(r,z,t)$, to also give
a force contribution in the axial ($z$) direction. The kinetic-momentum force
$d{\bf G}_{\rm k}(t)/dt$ turns out to be effectively negligible at the
${\omega}_{0}$ resonant beat frequency that dominates the measured force
$F_{\\!ag}(t)$, so that (34) effectively reduces to

${\bf F}_{\\!ag}(t)=\frac{d}{dt}{\bf
G}_{f}(t)-\int\limits_{S}{\bf\hat{n}}\cdot\overline{\bf T}dS.$ (35)

The ${\bf G}_{f}(t)=G_{f}(t){\bf\hat{z}}$ is the electromagnetic-field
momentum, which equals $\int_{V}{\bf E}\times{\bf H}/c^{2}dV$ for the Abraham
momentum and $\int_{V}{\bf D}\times{\bf B}dV$ for the Minkowski momentum with
$V$ the volume of the toroid inside $S$. Since ${\bf
D}=\epsilon_{0}\epsilon_{\rm r}{\bf E}$ and ${\bf B}=\mu_{0}\mu_{\rm r}{\bf
H}$, the Minkowski electromagnetic-field momentum equals $\mu_{\rm
r}\epsilon_{\rm r}$ times the Abraham electromagnetic-field momentum, so that
(35) can be rewritten as

${\bf F}_{\\!ag}(t)=\left\\{\begin{array}[]{c}1\\\ \mu_{\rm r}\epsilon_{\rm
r}\\\ \end{array}\right\\}\frac{1}{c^{2}}\frac{d}{dt}\int\limits_{V}{\bf
E}\times{\bf H}\,dV-\int\limits_{S}{\bf\hat{n}}\cdot\overline{\bf T}dS$ (36)

where the $1$ in the curly bracket applies to the Abraham electromagnetic-
field momentum and the $\mu_{\rm r}\epsilon_{\rm r}$ in the curly bracket
applies to the Minkowski electromagnetic-field momentum. The values of
$\mu_{\rm r}\epsilon_{\rm r}$ were about $112$ and $327$ for the two different
ferrite toroids that were used.

James’s measurements of ${\bf F}_{\\!ag}(t)$ in (36) along with his
theoretically evaluated stress-dyadic force reveal that the Abraham
electromagnetic-field momentum satisfies the equation in (36) much more
closely than the Minkowski electromagnetic-field momentum. This is a strong
confirmation of the correctness of the Abraham electromagnetic-field momentum
because both the electric and magnetic fields producing the forces and momenta
are time varying and the ratio of the two momenta are equal to $112$ for one
of the toroids and $327$ for the other. Since the right-hand side of the
experimentally verified (top) equation in (36) is equal to the negative of the
Einstein-Laub force, as seen in (III. Macroscopic Force and Momentum,14b),
James’s experimental results also confirm the Einstein-Laub electromagnetic
force as opposed to the Minkowski electromagnetic force. Implicitly, these
experimental results for magnetodielectric toroids in favor of the Abraham
macroscopic electromagnetic-field momentum also confirm the existence of the
internal “hidden-momentum” electromagnetic force within Amperian magnetic
dipoles.

### B. The high-permittivity dielectric toroid

The paper by Walker et al. [34] replaces the magnetodielectric toroid in Fig.
1 with a high-permittivity dielectric toroid ($\epsilon_{\rm r}\approx 3620$,
$\mu_{\rm r}=1$), and replaces the time varying axial current $I_{0}(t)$ with
an axial static magnetic field ${\bf H}_{0}=H_{0}{\bf\hat{z}}$ penetrating the
entire toroid. The voltage $V_{0}(t)$ remains a sinusoidal time varying
voltage applied across inner and outer conducting surfaces of the toroid.
Similarly to the magnetodielectric toroid, this voltage produces a radial
quasielectrostatic field $E_{\rm r}(r,t)$ which in turn produces an azimuthal
quasimagnetostatic field $H_{\phi}(r,z,t)$ that varies asymmetrically with $z$
across the toroid. The azimuthal torque produced by the fields $E_{\rm
r}(r,t)$, $H_{\phi}(r,z,t)$, and $H_{0}$ on the toroid was measured using a
torsion pendulum arrangement. Since the asymmetric azimuthal
quasimagnetostatic field $H_{\phi}(r,z,t)$ exerts no azimuthal torque on the
toroid, this field can be ignored and, indeed, the authors do not mention or
determine this field.

The theory in the Walker et al. paper begins with an expression for force
density taken from Marx and Gyorgyi [35]

$\frac{(\mu_{\rm r}\epsilon_{\rm r}-1)}{c^{2}}\frac{d}{dt}({\bf E}\times{\bf
H})=\frac{(\epsilon_{\rm r}-1)}{c^{2}}\frac{d}{dt}[E_{\rm
r}(r,t)H_{0}]{\bf\hat{\mbox{\boldmath{$\phi$}}}}$ (37)

which is just the azimuthal (${\bf\hat{\mbox{\boldmath{$\phi$}}}}$) component
of our Einstein-Laub force density in (14b) because all the force densities in
the integrand of (14b) are zero except for

$\mu_{0}\frac{\partial{\bf P}}{\partial t}\times{\bf H}=\frac{(\mu_{\rm
r}\epsilon_{\rm r}-1)}{c^{2}}\frac{d}{dt}({\bf E}\times{\bf
H})=\frac{(\epsilon_{\rm r}-1)}{c^{2}}\frac{d}{dt}[E_{\rm
r}(r,t)H_{0}]{\bf\hat{\mbox{\boldmath{$\phi$}}}}$ (38)

and ${\bf P}\cdot\nabla{\bf E}$, which is in the radial direction.

Essentially, Walker et al. evaluated $E_{r}(r,t)$ theoretically and integrated
the right-hand side of (38) around the toroid to get the total value of the
torque exerted on the toroid by the fields. They then found that this
theoretically predicted value of the torque agreed to within 10% with the
measured value of the torque. This is a highly significant result since the
corresponding value of the azimuthal torque determined by the Minkowski force
in (14c) is equal to zero. Thus the experimental results of Walker et al.
conclusively rule out the Minkowski macroscopic electromagnetic force while
confirming the Einstein-Laub macroscopic electromagnetic force.

## VI. Conclusion

We consider a macroscopic (characterized by sources and fields spatially
averaged at each instant of time over electrically small volume elements)
dipolar continuum (medium obeying the Maxwell dipolar equations) with
molecular dipole moments realistically modeled by classical microscopic
discrete electric-charge electric dipoles and circulating-electric-current
magnetic dipoles (assuming the dipoles are densely packed enough in the
defining electrically small volume elements that the macroscopic fields and
polarizations can be sufficiently smoothed). It is rigorously proven that the
sum of the electromagnetic forces on a volume of these realistic classical
microscopic dipoles is equal to the macroscopic force given by the Einstein-
Laub formulation and, thus, the macroscopic Abraham/Einstein-Laub
electromagnetic-field momentum, rather than the macroscopic Minkowski
electromagnetic-field momentum (or the two other possible electromagnetic-
field momenta), gives the correct electromagnetic-field momentum equal to the
sum of the electromagnetic-field momentum of all the microscopic dipoles in
the volume. For periodic fields, the time derivatives of all four macroscopic
electromagnetic-field momenta within the volume average to zero and all four
of the associated time-average macroscopic forces are equal to the time-
average total microscopic/macroscopic electromagnetic momentum flow into the
volume.

A key to deriving the correct macroscopic electromagnetic force on a volume of
dipolar material from the sum of the individual electromagnetic forces on the
discrete microscopic dipoles within the volume of material is to realize that
the macroscopic polarizations, fields, and forces within a volume $V$ can be
defined consistently and unambiguously if and only if the surface $S$ of the
volume $V$ (like the surfaces $\Delta S$ of the averaging volume elements
$\Delta V$) does not intersect the dipoles so that there are a discrete number
of dipoles inside the surface $S$ (like inside the surfaces $\Delta S$). This
implies that the surface $S$ of the volume $V$ in the macroscopic dipolar
continuum must lie in a hypothetical thin free-space shell separating $V$ from
the rest of the continuum. With this unambiguous definition of fields and
polarization densities using macroscopic volume elements surrounded by
hypothetical thin free-space shells, it follows that the forces produced on
the volume elements by the surface-charge and surface-current densities on
either side of the thin shells cancel so that the macroscopically defined
fields can be used in the force expressions rather than the cavity fields that
exist in the absence of each polarized volume element.

Another key to the derivation of the correct macroscopic electromagnetic force
is the rigorous proof given in [26] that for arbitrarily time varying
externally applied electromagnetic fields, the force on a conductor model of a
microscopic electric-current magnetic dipole, such as a wire loop, contains an
internal-momentum (the so-called hidden-momentum) electromagnetic force
(induced indirectly by the externally applied fields) that, when added to the
direct force exerted by the externally applied fields, equals the same
electromagnetic force that an equal-moment, microscopic magnetic-charge
magnetic dipole would experience in the same externally applied fields. (Past
derivations of this “hidden momentum” have been confined to quasistatic rather
than arbitrarily time varying electromagnetic fields and dipole moments; see
[26] for details.) Moreover, after the force on the microscopic dipoles is
averaged in a macroscopic volume element $\Delta V$ to get the macroscopic
force density, this macroscopic force density on the magnetization contains
the analogous macroscopic hidden-momentum force density.

With the correct macroscopic electromagnetic force determined for a volume of
dipolar material in an applied external electromagnetic field, the time rate
of change of the total kinetic momentum of the material in the volume can be
determined through Newton’s relativistic equation of motion in terms of the
correct macroscopic electromagnetic force and any other forces applied to the
material in the volume by an outside agent. The difference between the kinetic
momentum and the canonical momentum (which is the same as the Minkowski-force
momentum) in a volume of dipolar material is shown to equal the sum of the
“hidden electromagnetic momentum” of the circulating-electric-current magnetic
dipoles and the “hidden electromagnetic momentum” of hypothetical circulating-
magnetic-current electric dipoles replacing the electric-charge electric
dipoles within the volume of material. These electric-current magnetic dipoles
and magnetic-current electric dipoles that exhibit hidden momentum are
required in the Minkowski $({\bf D},{\bf B})$ formulation of Maxwell’s
equations and the constitutive relations, as opposed to the Abraham $({\bf
E},{\bf H})$ formulation that requires electric- and magnetic-charge dipoles,
which exhibit no hidden momentum.

Most experiments in the past measuring radiation forces in media revealed only
the time-average electromagnetic forces for which there is no difference
between the Einstein-Laub/Abraham and Minkowski formulations. However, two
well-conceived and well-conducted past experiments that measured the time
varying macroscopic electromagnetic forces and momenta are shown to decidedly
confirm the force and momentum expressions of the Einstein-Laub/Abraham
formulation and to rule out those of the Minkowski formulation.

## Acknowledgement

This work was supported under the U.S. Air Force Office of Scientific Research
Contract # FA9550-22-1-0293 through Dr. Arje Nachman.

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