A macroscopic force density **F**(**r** ) is the force per unit
volume acting on a medium in the neighborhood of **r**. Fundamentally,
the electromagnetic force density is the result of forces acting on
those microscopic particles embedded in the material that are charged,
or that have electric or magnetic dipole moments. The forces acting
on these individual particles are passed along
through interparticle forces to the macroscopic material as a whole.
In the limit where that volume becomes small, the force density can
then be regarded as the sum of the microscopic forces over a volume
element * V*.

Of course, the linear dimensions of * V* are large compared to the
microscopic scale.

Strictly, the forces in this sum should be evaluated using the
microscopic fields. However, we can gain insight concerning the form
taken by the force density by using the macroscopic fields in this
evaluation. This is the basis for the following discussions of the
force densities associated with unpaired charges and with conduction
currents (the Lorentz force density) and with the polarization and
magnetization of media (the Kelvin force density).

To be certain that the usage of macroscopic fields in describing
the force densities is consistent with that implicit in the
constitutive laws already introduced to describe conduction,
polarization, and magnetization, the electromagnetic force densities
should be derived using energy arguments. These derivations are
extensions of those of Secs. 11.6 and 11.7 for forces. We end this
section with a discussion of the results of such derivations and of
circumstances under which they will predict the same total forces or
even material deformations as those derived here.

### The Lorentz Force Density

Without restricting the generality of
the resulting force density, suppose that the electrical force on a
material is due to two species of charged particles. One has *N*_{+}
particles per unit volume, each with a charge *q*_{+}, while the other
has density *N*_{-} and a charge equal to *-q*_{-}. With **v** denoting
the velocity of the macroscopic material and **v**_{}
representing the respective velocities of the carriers relative to
that material, the Lorentz force law gives the force on the individual
particles.

Note that *q*_{-} is a positive number.

In typical solids and fluids, the charged particles are either
bonded to the material or migrate relative to the material, suffering
many collisions with the neutral material during times of interest.
In either case, the inertia of the particles is inconsequential, so that
on the average, the forces on the individual particles are passed along
to the macroscopic material. In either situation, the force density on the
material is the sum of (2) and (3), respectively, multiplied by the
charged particle densities.

Substitution of (2) and (3) into this expression gives the * Lorentz
force density*

where _{u} is the unpaired charge density (7.1.6) and **J**
is the current density.

Because the material is in motion, with velocity **v**, the current
density **J** has not only the contribution familiar from Sec. 7.1
(7.1.4) due to the migration of the carriers relative to the
material, but one due to the net charge carried by the moving material
as well.

In EQS systems, the first term in (5) usually outweighs the
second, while in MQS systems (where the unpaired charge density is
negligible), the second term tends to dominate.

The derivation and Fig. 11.9.1 suggest why the electric term is
proportional to the * net* charge density. In a given region, the
force density resulting from the positively charged particles tends to
be canceled by that due to the negatively charged particles, and the
net force density is therefore proportional to the difference in
absolute magnitudes of the charge densities. We exploited this fact
in Chap. 7 to let electrically induced material motions evidence the
distribution of the unpaired charge density. For example, in
Demonstration 7.5.1, the unpaired charge density was restricted to an
interface, and as a result, the motion of the fluid was suppressed by
constraining the interface. A more recent example is the force on the
upper electrode in the capacitor transducer of Example 11.6.1. Here
again the force density is confined to a thin region on the surface
of the conducting electrode.

** Figure 11.9.1 **The electric Lorentz
force density _{u}**E** is proportional to the * net* charge
density because the charges individually pass their force to the
material in which they are embedded.

** Figure 11.9.2 **The magnetic Lorentz force density
**J x **_{o} **H**.
The magnetic term in (6), pictured in Fig. 11.9.2 as acting on
a current-carrying wire, is also familiar. This force density
was responsible for throwing the metal disk into the air in the
experiment described in Sec. 10.2. The force responsible for the
levitation of the pancake coil in Demonstration 11.7.1 was also the
net effect of the Lorentz force density, acting either over the volume
of the coil conductors or over that of the conducting sheet below.
In MQS systems, where the contribution of the "convection" current
_{u}**v** is negligible, the current density is typically due to
conduction. Note that this means that the velocity of the charge
carriers is determined by the electric field they experience in the
conductor, and not simply by the motion of the conductor. The current
density **J** in a moving conductor is generally not in the direction
of motion.^{9}

^{9}Indeed, it is fortunate that the carriers do not
have the same velocity as the material, for if they did, it would not
be possible to use the magnetic Lorentz force density for
electromechanical energy conversion. If we recognize that the rate at
which a force **f** does work on a particle that moves at the velocity
**v** is **v** **f**, then it follows from the Lorentz force law,
(1.1.1), that the rate of doing work on individual particles through the
agent of the magnetic field is **v** (**v** x _{o} **H**).
The cross-product is perpendicular to **v**, so this rate of doing
work must be zero.

### The Kelvin Polarization Force Density

If microscopic particles
carrying a net charge were the only contributors to a macroscopic
force density, it would not be possible to explain the forces on
polarized materials that are free of unpaired charge. Example 11.6.2
and Demonstration 11.6.2 highlighted the polarization force. The
experiment was carried out in such a way that the dielectric material
did not support unpaired charge, so the force is not explained by the
Lorentz force density.

In EQS cases where _{u} = 0, the macroscopic force
density is the result of forces on the microscopic particles with
dipole moments. The resulting force density is fundamentally
different from that due to unpaired charges; the forces **p**
**E** on the individual microscopic particles are
passed along by interparticle forces to the medium as a whole. A
comparison of Fig. 11.9.3 with Fig. 11.9.1 emphasizes this point.
For a single species of particle, the force density is the force on a
single dipole multiplied by the number of dipoles per unit volume
*N*_{p}. By definition, the polarization density **P** = N_{p} **p**,
so it follows that the force density due to polarization is

This is often called the * Kelvin polarization force density*.

** Figure 11.9.3 **The electric Kelvin force density
results because the force on the individual dipoles is passed on to
the neutral medium.

#### Example 11.9.1. Force on a Dielectric Material

In Fig. 11.9.4, the cross-section of a pair of electrodes that are
dipped into a liquid dielectric is shown. The picture might be of a
cross-section from the experiment of Demonstration 11.6.2. With the
application of a potential difference to the electrodes, the
dielectric rises between the electrodes. According to (7), what is
the distribution of force density causing this rise?

** Figure 11.9.4 **In terms of the Kelvin force
density, the dielectric liquid is pushed into the field region between
capacitor plates because of the forces on individual dipoles in the
fringing field.
For the liquid dielectric, the polarization constitutive law is
taken as linear [(6.4.2) and (6.4.4)]

so that with the understanding that is a function of position
(uniform in the liquid, _{o} in the gas, and taking a step at the
interface), the force density of (7) becomes

By using a vector identity

**A** **A**

=
( x **A**) x **A** + (**A**
**A**)
and invoking the EQS approximation where * x ***E** = 0,
this expression is written as

A second identity

^{11} * ( ) =
+ *

converts this expression into one that will now prove useful in
picturing the distribution of force density.

Provided that the interface is well removed from the fringing
fields at the top and bottom edges of the electrodes, the electric
field is uniform not only in the dielectric and gas above and below
the interface between the electrodes, but through the interface as
well. Thus, throughout the region between the electrodes, there is no
gradient of **E**, and hence, according to (7), no Kelvin force
density. The Kelvin force density is therefore confined to the
fringing field region where the fluid surrounds the lower edges of the
electrodes. In this region, is uniform, so the force density
reduces to the first term in (11). Expressed by this term, the
direction and magnitude of the force density is determined by the
gradient of the scalar **E** **E**. Thus, where **E** is varying
in the fringing field, it is directed generally upward and into the
region of greater field intensity, as suggested by Fig. 11.9.4. The
force on the dipole shown by the inset lends further credence to the
dipolar origins of the force density.

Although there is no physical basis for doing so, it might seem
reasonable to take the force density caused by polarization as being
_{p} **E**. After all, it is the polarization charge density
_{p} that was used in Chap. 6 to represent the effect of the
media on the macroscopic electric field intensity **E**. The
experiment of Demonstration 11.6.2, pictured in Fig. 11.6.7,
makes it clear that this force density is not
correct. With the interface well removed from the fringing fields,
there is no polarization charge density anywhere in the liquid, either
at the interface or in the fringing field. If _{p}**E** were the
correct force density, it would be zero throughout the fluid volume
except at the interfaces with the conducting electrodes. There, the
forces are perpendicular to the surface of the electrodes. Such a
force distribution could not cause the fluid to rise.

### The Kelvin Magnetization Force Density

Forces caused by magnetization are probably the most commonly
experienced electromagnetic forces. They account for the attraction
between a magnet and a piece of iron. In Example 11.7.1, this
force density acts on the disk of magnetizable material.

Given that the magnetizable material is made up of microscopic
dipoles, each experiencing a force of the nature of (11.8.22), and
that the magnetization density **M** is the number of these per unit
volume multiplied by **m**, it follows from the arguments of the
preceding section that the force density due to magnetization is

This is sometimes called the * Kelvin magnetization force density*.

#### Example 11.9.2. Force Density in a Magnetized Fluid

With the dielectric liquid replaced by a ferrofluid having a
uniform permeability , and the electrodes replaced by the pole
faces of an electromagnet, the physical configuration shown in Fig.
11.9.4 becomes the one of Fig. 11.9.5, illustrating the magnetization
force density. In such fluids^{[1]}, the magnetization
results from an essentially permanent suspension of magnetized
particles. Each particle comprises a magnetic dipole and passes its
force on to the liquid medium in which it is suspended. Provided that
the magnetization obeys a linear law, the discussion of the
distribution of force density given in Example 11.9.1 applies equally
well here.

** Figure 11.9.5 **In an experiment that is the magnetic
analog of that shown in Fig. 11.9.4, a magnetizable liquid is pushed
upward into the field region between the pole faces by the forces on
magnetic dipoles in the fringing region at the bottom.

### Alternative Force Densities

We now return to comments made at the beginning of
this section. The fields used to express the Lorentz and
Kelvin force densities are macroscopic. To assure consistency between
the averages implied by these force densities and those already
inherent in the constitutive laws, an energy principle can be used.
The approach is a continuum version of that exemplified for
lumped parameter systems in Secs. 11.7 and 11.8. In the lumped
parameter systems, electrical terminal relations were used to
determine a total energy, and energy conservation was used to
determine the force. In the continuum
system^{[2]},
the electrical constitutive law is used to find an energy density, and
energy conservation used, in turn, to find a force density. This energy
method, like the one exemplified in Secs. 11.7 and 11.8 for lumped
parameter systems, describes systems that are loss free. In making
practical use of the result, it is assumed that it will be applicable
even if there are losses. A more general method, which invokes a
* principle of virtual power*^{[3]}, allows for dissipation but
requires more empirical
information than the polarization or magnetization constitutive law as
a starting point.

Force densities derived from more rigorous arguments than given
here can have very different distributions from the superposition of
the Lorentz and Kelvin force densities. We would expect that the
arguments break down when the microscopic particles become so
densely packed that the field experienced by one is significantly
altered by its nearest neighbor. But surely the difference between
the magnetic force density of Lorentz and Kelvin (LK)

we have derived here and the * Korteweg-Helmholtz force density* (KH) for
incompressible media

cited in the literature^{[2]}
is not due to interactions between microscopic particles. This latter
force density is often obtained for an * incompressible* material
from energy arguments. [Note that with *-* and **H**
**H**, respectively, playing the roles of *dL/d* and *i*^{2},
the magnetization term in (14) takes a form found for the
force on a magnetizable material in Sec. 11.7.]

In Example 11.9.2 (where **J** = 0), we found the force density of
(13) to be confined to the fringing field. By contrast, (14)
gives no force density in the fringing region (where is
uniform),
but rather puts it all at the interface. According to this latter
equation, through the agent of a surface force density (a force
density that is a spatial impulse at the interface), the field pulls
upward on the interface.

The question may then be asked whether, and how, the two force
density expressions can be reconciled. The answer is that if _{o}
**M** = ( - _{o})**H**, they predict the same motion for any
volume-conserving material deformations such as those of an
incompressible fluid. We shall demonstrate this for the case of a
liquid, such as shown in Fig. 11.9.5, but allowing for the action of a
current **J** as well. As the first step in the derivation, we shall
show that (13) and (14) differ by the gradient of a scalar, *
(***r**).

To see this, use a vector
identity

This expression differs from (14) by the last term, which indeed
takes the form where

Now consider Newton's force law for an elemental volume of material.
Using the Korteweg-Helmholtz force density, (14), it takes the form

where *p* is the internal fluid pressure and **F**_{m} is the sum of
all other mechanical contributions to the force density.
Alternatively, using (13) written as (17) as the force density,
this same law is represented by

For an incompressible material, none of the other laws needed to
describe the continuum (such as mass conservation) involve the
pressure.

^{14} For example, for a compressible fluid, the
pressure depends on mass density and temperature, so the pressure
does appear in the physical laws. Indeed, in the constitutive law
relating these values, the pressure has a well-defined value.
However, in an incompressible fluid, the constitutive law relating
the pressure to mass density and temperature is not relevant to the
prediction of material motion. Thus, if (19) is used, *p* appears
only in that equation and if (20) is used, *p*^{'}
p -
appears only in that expression. This means that *p* and *p*^{'}
play identical roles in predicting the deformation. In an
incompressible material, it is the
role of the pressure to adjust itself so that only volume conserving
deformations are allowed.

^{15}Like the "perfectly
permeable material" of magnetic circuits, in which **B** remains
finite as **H** goes to zero, the "perfectly incompressible"
material is one in which the pressure remains finite even as the
material becomes infinitely "stiff" to all but those deformations
that conserve volume.

The two formulations would differ in what
one would call the pressure, but would result in the same material
deformation and velocity. An example is the height of rise of the
fluid between the parallel plates in Fig. 11.9.5.

Included in the class of incompressible deformations are rigid
body motions. If used self-consistently, force densities that differ
by the gradient of a "" will predict
the same motions of rigid bodies. Thus, the net force on a body
surrounded by free space will be the same whether found using the
Lorentz-Kelvin or the Korteweg-Helmholtz force density. The following
example illustrates this concept.

#### Example 11.9.3. Magnetic Force on a Magnetizable Current-Carrying
Material

A block of conducting material having permeability is shown
in Fig. 11.9.6 sandwiched between perfectly conducting plates. A
current source, distributed over the left edges of these electrodes,
drives a constant surface current density *K* in the *+x* direction along
the left edge of the lower electrode. This current passes through the
block in the *y* direction as a current density

** Figure 11.9.6 **The block having uniform
permeability and conductivity carries a uniform current density in
the *y* direction which produces a *z-*directed magnetic field
intensity. Although the force densities of (13) and (14) have very
different distributions in the block, they predict the same net
force.

and is returned to the source in the *-x* direction at the left edge of
the upper electrode. The thickness *a* of the block is small compared
to its other two dimensions, so the magnetic field between the
electrodes is *z* directed and dependent only on *x*. From
Ampère's law it follows that

in the conducting block.

The alternative force densities, (13) and (14), have very
different distributions in the block. Yet we must find that the
net force on the block, found by integrating each over its volume, is
the same. To see that this is so, consider first the sum of the
Lorentz and Kelvin force densities, (13).

There is no *x* component of the magnetic field intensity, so for
this particular configuration, the
magnetization term makes no contribution to (13). Evaluation of the
first term using (19) and (20) then gives

Integration of this force density over the volume amounts to a
multiplication by the cross-sectional area *ad*, and integration on
*x*. Thus, the net force predicted by using the force density of
Lorentz and Kelvin is

Now, the Korteweg-Helmholtz force density given by (14) is evaluated.
The permeability is uniform throughout the interior of the
block, so the magnetization term is again zero there. However,
is a step function at the ends of the block, where *x = -b* and *x =
0*. Thus, is an impulse there and we must take care to
include the contributions from the surface regions in our
integration. Evaluation of the *x* component of (14) using (21) and

Integration of (25) over the volume of the block therefore gives

Note that *H*_{z} is constant through the interface at *x = -b*. Thus,
the integration of the last term can be carried out. Simplification of
this expression gives the same total force as found before, (24).

The distributions of the force densities given by (13) and (14)
are generally different, even very different. It is therefore
natural to ask which of the two is the "right" one. In general, until
the "other" force densities acting on the medium in question are
specified, this question cannot be answered. Here, where a discussion
of continuum mechanics is beyond our purview, we have identified a
class of mechanical deformations (namely, those that are volume
conserving or "incompressible"), where these force densities are
equally valid. In fact, any other force density differing from these
by a term having the form would also be valid. The
combined Lorentz and Kelvin force densities have the
advantage of a satisfying physical interpretation. However, the
derivation has the weakness of making an * ad hoc* use of the
macroscopic fields. Force densities resulting from an energy argument
have the advantage of dealing rigorously with the macroscopic fields.