The energy principle was used in the preceding sections to derive the
macroscopic forces on polarizable and magnetizable materials. The
same principle can also be applied to derive the force distributions,
the force densities. For this purpose, one needs more than a purely
electromagnetic description of the system.
In order to develop the simple model for the force density
distribution, we need the expression for the force on an electric
dipole for polarizable media, and on a magnetic dipole for
magnetizable media. The force on an electric dipole will be derived
simply from the Lorentz force law. We have not stated a
corresponding force law for magnetic charges. Even though these are
not found in nature as isolated charges but only as dipoles, it is
nevertheless convenient to state such a law. This will be done by
showing how the electric force law follows from the energy principle.
By analogy a corresponding law on magnetic charges will be derived
from which the force on a magnetic dipole will follow.
Figure 11.8.1 An electric dipole
experiences a net electric force if the positive charge q is
subject to an electric field E (r + d) that differs
from E(r) acting on the negative charge q.
Force on an Electric Dipole
The force on a stationary
electric charge is given by the Lorentz law with v = 0.
A dipole is the limit of two charges of equal magnitude and opposite
sign spaced a distance d apart, in the limit
with p being finite. Charges q of opposite polarity,
separated by the vector distance d, are shown in Fig. 11.8.1.
The total force on the dipole is the sum of the forces on the
individual charges.
Unless the electric field at the location r + d of the positive
charge differs from that at the location r of the negative charge,
the separate contributions cancel.
In order to develop an expression for the force on the dipole in
the limit where the spacing d of the charges is small compared to
distances over which the field varies appreciably, (2) is written in
Cartesian coordinates and the field at the positive charge expanded
about the position of the negative charge. Thus, the x component is
The first and last terms cancel. In more compact notation, this
expression is therefore
where we have identified the dipole moment p qd. The
other force components follow in a similar fashion, with y and z
playing the role of x. The three components are then summarized in
the vector expression
The derivation provides an explanation of how p
E is evaluated in Cartesian coordinates. The ith component of
(5) is obtained by dotting p with the gradient of the ith
component of E.
Illustration. Force on a Dipole
Suppose that a dipole finds itself in the field
which is familiar from Example 4.1.1. It follows from (5) that the
force is
According to this expression, the ydirected dipole on the x axis in
Fig. 11.8.2 experiences a force in the x direction. The ydirected
force is zero because E_{y} is the same at the respective locations
of the charges. The xdirected force exists because E_{x} goes from
being positive just above the x axis to negative just below. Thus,
the xdirected contributions to the force of each of the charges is in
the same direction.
Figure 11.8.2 Dipole having y
direction and positioned on the x axis in field of (6) experiences
force in the x direction.
Force on Electric Charge Derived from Energy Principle
The force on an electric charge is stated in the Lorentz law. This
law is also an ingredient in Poynting's theorem, and in the
identification of energy and power flow. Indeed, E
J_{u} was recognized from the Lorentz law as the power density
imparted to the current density of unpaired charge. The energy
principle can be used to derive the force law on a microscopic charge
"in reverse". This seems to be the hard way to obtain the
Lorentz law of force on a stationary charge. Yet we go
through the derivation for three reasons.
shown to be consistent with the Lorentz force on a stationary
charge.
(a)  The derivation of force from the EQS energy principle is

(b)  The derivation shows that the field can be
produced by permanently polarized material objects, and yet the
energy principle can be employed in a straightforward manner.

(c)  The same principle can be applied to derive the
microscopic MQS force on a magnetic charge.

Let us consider an EQS field produced by charge distributions and
permanent polarizations P_{p} in free space as sketched in Fig.
11.8.3a. By analogy, we will then have found the force on a magnetic
charge in the field of a permanent magnet, Fig. 11.8.3b. The
Poynting theorem identifies the rate of energy imparted to the
polarization per unit volume as
Figure 11.8.3 (a) Electric charge brought into
field created by permanent polarization. (b) Analogous magnetic
charge brought into field of permanent magnet.
Because P_{p} is a permanent polarization, P_{p} / t =
0, and the permanent polarization does not contribute to the change
in energy associated with introducing a point charge. Hence, as
charge is brought into the vicinity of the permanent polarization,
the change of energy density is
where stands for the differential change of _{o} E. The
change of energy is
where V includes all of space. The electroquasistatic E field is
the negative gradient of the potential
Introducing this into (10), one has
where we have "integrated by parts," using an
identity.^{7}
^{7}
( ) A = (A )  (
A)
The first integral can be written as an
integral over the
surface enclosing the volume V. Since V is all of space, the
surface is at infinity. Because E vanishes at infinity at
least as fast as 1/r^{3} (1/r^{2} for E, 1/r for , where
r is the distance from the origin of a coordinate system mounted
within the electroquasistatic structure), the surface integral
vanishes. Now
from Gauss' law, where _{u} is the change of unpaired
charge. Thus, from (12),
so the change of energy is equal to the charge increment
_{u} dv introduced at r times the potential at r,
summed over all the charges.
Suppose that one introduces only a small test charge q, so that
_{u} dv = q at point r. Then
The change of energy is the potential at the point at which
the charge is introduced times the charge. This form of the energy
interprets the potential of an EQS field as the work to be done in
bringing a charge from infinity to the point of interest.
If the charge is introduced at r + r, then the change in
total energy associated with introducing that charge is
Introduction of a charge q at r, subsequent removal of the
charge, and introduction of the charge at r + r is
equivalent to the displacement of the charge from r to r +
r. If there is a net energy decrease, then work must have
been done by the force f exerted by the field on the charge.
The work done by the field on the charge is
and therefore
Thus, the Lorentz law for a stationary charge is implied by the EQS
laws.
Before we attack the problem of force on a magnetic charge, we
explore some features of the electroquasistatic case. In (17), q
is a small test charge. Electric test charges are available as
electrons. But suppose that in analogy with the magnetic case, no
free electric charge was available. Then one could still produce a
test charge by the following artifice. One could polarize a very
longthin rod of crosssection a, with a uniform polarization
density P along the axis of the rod (Sec. 6.1). At one end of
the rod, there would be a polarization charge q = Pa, at the other
end there would be a charge of equal magnitude and opposite sign. If
the rod were of very long length, while the end with positive charge
could be used as the "test charge," the end of opposite charge
would be outside the field and experience no force. Here the charge
representing the polarization of the rod has been treated as
unpaired.
We are now ready to derive the force on a magnetic charge.
Force on a Magnetic Charge and Magnetic Dipole
The attraction of a magnetizable
particle to a magnet is the result of the force exerted by a magnetic
field on a magnetic dipole. Even in this case, because the particle
is macroscopic, the force is actually the sum of forces acting on the
microscopic atomic constituents of the material. As pointed out in
Secs. 9.0 and 9.4, the magnetization characteristics of macroscopic
media such as the iron particle relate back to the magnetic moment of
molecules, atoms, and even individual electrons. Given that a particle
has a magnetic moment m as defined in Sec. 8.2, what is the force
on the particle in a magnetic field intensity
H? The particle can be comprised of a macroscopic material such
as a piece of iron. However, to distinguish between forces on
macroscopic media and microscopic particles, we should consider here
that the force is on an elementary particle, such as an atom or
electron.
We have shown how one derives the force on an electric charge in an
electric field from the energy principle. The electric field could
have been produced by permanently polarized dielectric bodies. In
analogy, one could produce a magnetic field by permanently magnetized
magnetic bodies. In the EQS case, the test charge could have been
produced by a long, uniformly and permanently polarized cylindrical
rod. In the magnetic case, an "isolated" magnetic charge could be
produced by a long, uniformly and permanently magnetized rod of
crosssectional area a. If the magnetization density is M,
then the analogy is
where, for the uniformly magnetized rod, and the magnetic charge
is located at one end of the rod, the charge q_{m} at the
other end of the rod (Example 9.3.1). The force on a magnetic charge
is thus, in analogy with (18),
which is the extension of the Lorentz force law for a stationary
electric charge to the magnetic case. Of course, the force on a
dipole is, in analogy to (5) (see Fig. 11.8.4),
where m is the magnetic dipole moment.
Figure 11.8.4 Magnetic dipole
consisting of positive and negative magnetic charges q_{m}.
We have seen in Example 8.3.2 that a magnetic dipole of moment m
can be made up of a circulating current loop with magnitude m = ia,
where i is the current and a the area of the loop. Thus, the
force on a current loop could also be evaluated from the Lorentz law
for electric currents as
with i the total current in the loop. Use of vector identities
indeed yields (22) in the MQS case. Thus, this could be an alternate
way of deriving the force on a magnetic dipole. We prefer to derive
the law independently via a Lorentz force law for stationary magnetic
charges, because an important dispute on the validity of the magnetic
dipole model rested on the correct
interpretation of the force law^{[13]}. While the details of
the dispute are beyond the scope of this textbook, some of the issues
raised are fundamental and may be of interest to the reader who wants
to explore how macroscopic formulations of the electrodynamics of
moving media based on magnetization represented by magnetic charge
(Chap. 9) or by circulating currents are reconciled.
The analogy between the polarization and magnetization was emphasized
by Prof. L. J. Chu^{[2]}, who taught the introductory electrical
engineering course in electromagnetism at MIT in the fifties. He
derived the force law for moving magnetic charges, of which (21) is
the special case for a stationary charge. His approach was soon
criticized by Tellegen^{[3]}, who pointed out that the accepted
model of magnetization is that of current loops being the cause of
magnetization. While this in itself would not render the
magnetic charge model invalid, Tellegen pointed out that the force
computed from (23) in a dynamic field does not lead to (22), but to
Because the force is different depending upon whether one uses the
magnetic charge model or the circulating current model for the
magnetic dipole, so his reasoning went, and because the circulating
current model is the physically correct one, the magnetic charge
model is incorrect. The issue was finally settled^{[4]} when it
was shown that the force (24) as computed by Tellegen was incorrect.
Equation (23) assumes that i could be described as constant around
the current loop and pulled out from under the integral. However, in
a timevarying electric field, the charges induced in the loop cause a
current whose contribution precisely cancels the second term in (24).
Thus, both models lead to the same force on a magnetic dipole and it
is legitimate to use either model. The magnetic model has the advantage
that a stationary dipole contains no "moving parts," while the
current model does contain moving charges. Hence the
circulating current formalism is by necessity more complicated and
more likely to lead to error.
Comparison of Coulomb's Force on an Electron to the Force
on its Magnetic Dipole
Why is it possible to accurately describe the motions of an electron
in vacuum by the Lorentz force law without including the magnetic
force associated with its dipole moment? The answer is that the
magnetic dipole effect on the electron is relatively small. To
obtain an estimate of the magnitude of the magnetic dipole effect, we
compare the forces produced by a typical (but large) electric field
achievable without electrical breakdown in air on the charge e of the
electron, and by a typical (but large) magnetic field gradient acting
on the magnetic dipole moment of the electron. Taking for E
the value 10^{6} V/m, with e 1.6 x 10^{19} coulomb,
A B of 1 tesla (10,000 gauss) is a typical large flux density
produced by an iron core electromagnet. Let us assume that a flux
density variation of this order can be produced over a distance of
1 cm, which is, in practice, a rather high gradient. Yet taking this
value and a moment of one Bohr magneton (9.0.1), we obtain from
(22) for the force on the electron
Note that the electric force associated with the net charge is much
greater than the magnetic one due to the magnetic dipole moment.
Because of the large ratio f_{e}/f_{m} for fields of realistic
magnitudes, experiments designed to detect magnetic dipole effects on
fundamental particles did not utilize particles having a net charge,
but rather used neutral atoms (most notably, the SternGerlach
experiment^{8}.
^{8}W. Gerlach and O. Stern, "Uber die
Richtungsquantelung im Magnetfeld," Ann. d. Physik, 4th series,
Vol. 74, (1924), pp. 673699.). Indeed, a stray electric field on the
order of 10^{6} V/m would deflect an electron as strongly as a
magnetic field gradient of the very large magnitude assumed in
calculating (26).
The small magnetic dipole moment of the electron can become very
important in solid matter because macroscopic solids are largely
neutral. Hence, the forces exerted upon the positive and negative
charges within matter by an applied electric field more or less
cancel. In such a case, the forces on the electronic magnetic dipoles
in an applied magnetic field can dominate and give rise to the
significant macroscopic force observed when an iron filing is picked
up by a magnet.
Example 11.8.1. Magnetization Force on a Macroscopic Particle
Suppose that we wanted to know the force exerted on an iron
particle by a magnet. Could the microscopic force, (22), be used?
The energy method derivation shows that, provided the particle is
surrounded by free space, the answer is yes. The particle is taken
as being spherical, with radius R, as shown in Fig. 11.8.5. It is
assumed to have such a large magnetizability that its permeability
can be taken as infinite. Further, the radius R is much smaller
than other dimensions of interest, especially those characterizing
variations in the applied field in the neighborhood of the particle.
Figure 11.8.5 By dint of its
field gradient, a magnet can be used to pick up a spherical
magnetizable particle.
Because the particle is small compared to dimensions over which
the field varies significantly, we can compute its moment by
approximating the local field as uniform. Thus, the magnetic
potential is determined by solving Laplace's equation in the region
around the particle subject to the conditions that H be the
uniform field H_{o} at "infinity" and be constant on the
surface of the particle. The calculation is fully analogous to that
for the electric potential surrounding a perfectly conducting sphere
in a uniform electric field. In the electric analog, the dipole
moment was found to be (6.6.5),
p = 4 _{o} R^{3} E. Therefore, it follows from the
analogy provided by (19) that the magnetic dipole moment at the
particle location is
Directly below the magnet, H has only a z component. Thus,
the dipole moment follows from (27) as
Evaluation of (22) therefore gives
where H_{z} and its derivative are evaluated at the location of the
particle.
A typical axial distribution of H_{z} is shown in Fig. 11.8.6
together with two pictures aimed at gaining insight into the origins
of the magnetic dipole force. In the first, the dipole is again
depicted as a pair of magnetic monopoles, induced to form a moment
collinear with the H. Because the field is more intense at the
north pole of the particle than at the south pole, there is then a net
force.
Figure 11.8.6 In an increasing axial field, the
force on a dipole is upward whether the dipole is modeled as a pair
of magnetic charges or as a circulating current.
Alternatively, suppose that the dipole is actually a circulating
current, so that the force is given by (23). Even though the energy
argument makes it clear that the force is again given by (22), the
physical picture is different.
Because H is solenoidal, an intensity that increases with z
implies that the field just off axis has a component that is directed
radially inward. It is this radial component of the flux density
crossed with the current density that results in an upward force
on each segment of the loop.