Poynting’s Theorem with Magnetic Monopoles
Kirk T. McDonald
Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
(March 23, 2013; updated July 14, 2015)
1
Problem
Deduce a version of Poynting’s theorem [1] in macroscopic electrodynamics supposing that
magnetic charges (monopoles) exist in Nature. Discuss its relation to the Lorentz force law
for magnetic charges.1
2
Solution
2.1
Maxwell’s Equations
When Heaviside first presented Maxwell’s equations in vector notation [3] he assumed that
in addition to electric charge and current densities, ρe and Je , there existed magnetic charge
and current densities, ρm and Jm , although there remains no experimental evidence for the
latter.2,3 Maxwell’s equations for microscopic electrodynamics are then (in SI units)
∇ · 0 E = ρe ,
∇·
B
= ρm ,
μ0
−c2∇ × 0 E =
∂ B
+ Jm ,
∂t μ0
∇×
B
∂0 E
+ Je ,
=
μ0
∂t
(1)
√
where c = 1/ 0μ0 is the speed of light in vacuum. In macroscopic electrodynamics we
consider media that contain volume densities of electric- and Ampèrian magnetic-dipole moments, Pe and Me , respectively (often called the densities of polarization and magnetization).
Supposing that magnetic charges exists, the media could also contain volume densities of
(Gilbertian) electric- and magnetic-dipole moments, Pm and Mm , respectively. These densities can be associated with bound charge and current densities, which together with the
“free” charge and current densities ρ̃e , J̃e , ρ̃m and J̃m comprise the total charge and current
densities, and are related by
ρe = ρ̃e − ∇ · Pe ,
ρm = ρ̃m − ∇ · Mm ,
∂Pe
+ ∇ × Me ,
∂t
∂Mm
− c2 ∇ × Pm .
Jm = J̃m +
∂t
Je = J̃e +
1
(2)
(3)
Only in three spatial dimensions do the electric and magnetic fields of electric and magnetic charges
have the same character, such that a single electric and a single magnetic field could describe the effects of
both types of charges [2].
2
Heaviside seems to have regarded magnetic charges as “fictitious,” as indicated on p. 25 of [4].
3
If the interaction of magnetic charges with magnetic moments due to electrical currents is to conserve
energy, the magnetic charges must be at the end of “strings” of magnetic flux, as first postulated by Dirac
[6, 7].
1
It is customary in macroscopic electrodynamics to use versions of Maxwell’s equations in
which only “free” charge and current densities appear. For this we introduce the fields4
De = 0 E + Pe ,
He =
B
− Me ,
μ0
Dm =
E
− c2Pm ,
μ0
Hm =
B
+ Mm ,
μ0
(4)
such that De and Hm , and also He and Dm , have similar forms, and
∇ · De = ρ̃e ,
∇ · Hm = ρ̃m ,
−∇ × Dm =
∂Hm
+ J̃m ,
∂t
∇ × He =
∂De
+ J̃e . (5)
∂t
where in the absence of magnetic charges De and He are the familiar fields D and H.5
2.2
Force Laws
In static situations with no “free” currents J̃e or J̃m the curls of both Dm and He are zero
and these fields can be deduced from scalar potentials Ve and Vm ,
∇ × Dm = 0
⇔
Dm = −∇Ve ,
∇ × He = 0
⇔
He = −∇Vm .
(6)
We can associate potential energies,
Ue = μ0 qe Ve ,
Um = μ0 qm Vm ,
(7)
with electric and magnetic “test” charges qe and qm in scalar potentials due to other charges.
If those other charges are held fixed, the forces on the “test” charges can be written as
Fe = −qe ∇Ve = μ0 qe Dm ,
Fm = −qm∇Vm = μ0 qmHe .
(8)
The magnetic version of eq. (8) was introduced by Poisson [11], and Maxwell [10] reflected
this tradition by calling He = H the magnetic force (per unit magnetic charge) and B
the magnetic induction. Note that eq. (8) holds in media with nonzero, static densities Pe ,
Pm , Me and Mm ; the forces on charges inside static electromagnetic media are not qe E or
qm B/μ0 .6,7 This contrasts with force calculations for the effective magnetic-charge density,
4
The forms (4) were suggested to the author by David Griffiths in a comment on an early draft of this
note. Such “double” D and H fields were anticipated by Heaviside [5], who wrote H for He and h0 for Hm
near his eq. (88). The only explicit statement of eq. (4) in the literature seems to be in sec. 4 of [8], with
the identifications that ē → E, b̄ → B, p → Pe , m → Me , m → Pm , p → Mm , D → De , H → He ,
E → Dm , B → Hm . See Appendix D.2 for a justification of eq. (4) via the concept of electromagnetic
duality
5
The relation B = μ0 (H + M) (or B = H + 4πM in Gaussian units) seems to have been first introduced
by W. Thomson in 1871, eq. (r), p. 401 of [9], and appears in sec. 399 of Maxwell’s Treatise [10].
6
The notion of the force on a static “test” charge inside a macroscopic medium is somewhat contradictory,
in that the macroscopic fields are based on averages over volumes larger than atoms/molecules. People often
suppose the test charge to be inside a cavity whose volume is at least as large as an atom/molecule, but
then the magnitude of the force depends on the shape of the cavity. A more meaningful issue is the force on
a “test” charge that moves through the medium, thereby sampling the microscopic fields in a way that can
be well approximated in terms of the macroscopic fields. See also sec. 8 of [12].
7
In sec. 400 of [10], Maxwell noted that (in Gaussian units) the H field inside a disk-shaped cavity with
axis parallel to B and H inside a magnetic medium has Hcavity = Bcavity = Bmedium = Hmedium + 4πM, so
that in this case one could say that the force on a magnetic charge qm in the cavity is Fm = qm Hcavity =
qm Bmedium . The led Maxwell to the characterization of B as the “actual magnetic force,” which this author
finds misleading.
2
ρm,eff = −∇ · Me , which represent effects of Ampèrian currents, as discussed in Appendix
A.
As noted in [13] and on p. 429 of [14], if a magnetic charge qm could be made to move
around a loop some or all of which lies inside an Ampèrian magnetic material where B
does not equal μ0He (and hence ∇ × B is nonzero around the loop), then energy could
be extracted from the system each cycle if the force were qm B/μ0, and we would have a
perpetual-motion machine. Similarly, if an electric charge qe could be made to move around
a loop some or all of which lies inside a Gilbertian magnetic material where E does not equal
μ0 Dm (and hence ∇ × E is nonzero around the loop), then energy could be extracted from
the system each cycle if the force were qe E, and we would again have a perpetual-motion
machine.
The electromagnetic force on a moving electric charge qe and magnetic charge qm , each
with velocity v, is, in microscopic electrodynamics,8,9,10
Fe = qe (E + v × B) = μ0 qe (Dm + v × Hm ) ,
(9)
v
Fm = qm B − 2 × E = μ0 qm(He − v × De ).
(10)
c
Consistency of the Lorentz force law with special relativity requires that either E and B
or De and He or Dm and Hm appear in Fe and in Fm (see Appendix B). In macroscopic
electrodynamics the Lorentz force law for the force density f on “free” charge and current
densities takes the forms11,12
fe = ρ̃e E + J̃e × B,
or μ0 (ρ̃e Dm + J̃e × Hm ),
(11)
J̃m
× E, or μ0 (ρ̃m He − J̃m × De ).
(12)
c2
It has been verified that B not He deflects high-energy electrically charged particles as they
pass through magnetized iron (with no magnetic charges) [32], which confirms either form
of fe in eq. (11).13 See also [35, 36, 37]. The above argument about perpetual motion then
favors the second forms of eqs. (11)-(12).
fm = ρ̃m B −
Lorentz advocated the form Fe = μ0 qe (De + v × He ) in eq. (V), sec. 12 of [15], although he seems
mainly to have considered its use in vacuum. See also eq. (23) of [16]. That is, Lorentz considered De and
He , rather than E and B, to be the microscopic electromagnetic fields.
9
It is generally considered that Heaviside first gave the Lorentz force law (9) for electric charges in [17],
but the key insight is already visible for the electric case in [3] and for the magnetic case in [18]. The form
of Fm in terms of B and E is implicit in eq. (7) of [19] and explicit in sec. 28B of [20]. See also [21, 22].
10
For the macroscopic equations to appear as in eq. (5), as given, for examples, in sec. 7.3.4 and prob. 7.60
of [23], the Lorentz force law must have the form (10) for magnetic charges. One can also redefine the strength
of magnetic charges, ρm → ρm /μ0 , Jm → Jm /μ0 , which leads to the forms given, for example, in sec. 6.11
of [24]. These alternative definitions echo a debate initiated by Clausius in 1882 [25].
11
A subtlety is that the field B in the first form of eq. (11) is not the total field, but rather the field at
the location of the free current that would exist in its absence. See, for example, [26], especially sec. 4.
12
In 1908-10, Einstein argued that the Lorentz force law should take the form fe = μ0 (ρ̃e De + J̃e × He )
inside materials [27, 28], perhaps based on a misunderstanding discussed in [29], or that discussed in sec. 2.3.1
of [30]. This misunderstanding underlies the recent “paradox” of Mansuripur [31].
13
The magnetization of materials such as iron depends on the character of the magnetic moment of
electrons. An argument due to Fermi [33] that the hyperfine interaction depends on the magnetic field at
the origin, and so can distinguish between Ampèrian and Gilbertian moments of “nuclei.” For the case of
positronium (e+ e− ) the data imply that the moment of the electron is Ampèrian, as discussed in [34].
8
3
Further confirmation of this comes via consideration of energy flow in the electromagnetic
fields.
2.3
Poynting’s Theorem
Poynting’s argument [1] relates the rate of work done by electromagnetic fields on “free”
electric and magnetic currents to both flow of energy and to rate of change of stored energy.
This argument has delicacies of interpretation, discussed, for example, in sec. 2.19 of [38].
The density of the time rate of change of work on (“free”) electric currents is, from eq. (11),
dw̃e
= fe · ve = J̃e · E,
dt
Thus, either
μ0 J̃e · Dm .
or
dw̃e
∂De
= J̃e · E = E · ∇ × He −
dt
∂t
(13)
∂De
∂t
∂De
∂B
= −∇ · (E × He ) − E ·
− He ·
− μ 0 Jm · H e ,
∂t
∂t
= −∇ · (E × He ) + He · ∇ × E − E ·
(14)
where the total current Jm rather than the “free” current J̃m appears in the last line, or
dw̃e
∂De
= μ0 J̃e · Dm = μ0 Dm · ∇ × He −
dt
∂t
= −μ0∇ · (Dm × He ) + μ0He · ∇ × Dm − μ0Dm ·
∂De
.
∂t
∂De
∂Hm
− μ0 He ·
− μ0 J̃m · He .
(15)
∂t
∂t
Similarly, for “free” magnetic currents in macroscopic electrodynamics we have, from eq. (12),
= −μ0∇ · (Dm × He ) − μ0Dm ·
dw̃m
= fm · vm = J̃m · B or μ0J̃m · He .
(16)
dt
A requirement of simplicity of Poynting’s theorem when magnetic charges are included favors
that the time rate of change of the work done on “free” electric and magnetic currents be the
second forms in eqs. (13) and (16), and that the Lorentz force law on macroscopic electric
and magnetic charge and current densities be
fe = μ0(ρ̃e Dm + J̃e × Hm ) → μ0 ρ̃e (Dm + ve × Hm ),
fm = μ0(ρ̃m He − J̃m × De ) → μ0 ρ̃m (He − vm × De ),
(17)
(18)
(as also required not to have magnetic perpetual-motion machines).14,15
14
The form (18) is also affirmed in [8] via considerations of a magnetic current in a “wire” surrounded by
a dielectric medium. The issues here are somewhat different from those for the force on individual moving
charges, but are similar to those considered in [39] for an electrical current in a wire inside a magnetic
medium.
15
It is argued in [40] that a slowly moving magnetic charge perturbs electric polarization of a dielectric
medium in such a way that the velocity-dependent force is −qm v × 0 E, where E = D/ is the electric field
4
Then,
dw̃
∂De
∂Hm
∂u
dw̃e dw̃m
,(19)
=
+
= −μ0 ∇ · (Dm × He ) − μ0 Dm ·
− μ 0 He ·
≡ − ∇·S+
dt
dt
dt
∂t
∂t
∂t
which is the same form as if only electric charges exist, and hence the usual version of
Poynting’s theorem still applies if magnetic charges exist. That is, the Poynting vector,
S = μ 0 Dm × H e
(all media),
(20)
is interpreted as describing the flow of energy in the electromagnetic field, and for linear media
in which De and Dm are both proportional to E, and He and Hm are both proportional to
B, the density u of stored energy associated with the electromagnetic fields is
u = μ0
De · Dm + H e · H m
2
(linear media).
(21)
That Poynting’s theorem retains its usual form when magnetic charges are present is discussed by Heaviside in sec. 19 of [5]. That the form of the Lorentz force law for magnetic
charge and current densities is given by eqs. (17)-(18) is consistent with Heaviside’s argument; for example, his eq. (88), but is not explicitly stated. See also sec. 50, p. 49 of [4].
A peculiar argument that the “ordinary” form of Poynting’s theorem implies the existence
of magnetic charges is given in sec. 7.10 of [41]; thus misunderstanding is clarified in [42].
The extension of Poynting’s theorem to momentum flow, with the implication that μ0 De ×
Hm is the density of stored momentum, as argued by Minkowski [43], remains valid if the
Lorentz force law for magnetic charges is given by eqs. (17)-(18), but not for other forms, as
discussed in sec. V of [40]. See also Appendix C.
In a search for an isolated magnetic charge qm in media that otherwise contain only
electric charges and currents, De → D, Dm → E/μ0 , He → H, Hm → B/μ0, the Lorentz
force law reduces to
Fe = qe (E + ve × B),
A
Fm = μ0qm (H − vm × E).
(22)
Appendix: Effective Magnetic Charge Density
ρm,eff = −∇ · M
So far as is presently known, magnetic charges do not exist, and all magnetic effects can be
associated with electrical currents, as first advocated by Ampère [44].16 For materials with
in the absence of the moving magnetic charge. The argument of [40] seems to this author to be a variant of
sec. 400 of [10] in which it is supposed that the charge resides in a “cavity” whose surface details affect the
fields experienced by the charge. Such arguments assume that the charge occupies a volume at least equal
to one atom/molecule of the medium, which might have seemed reasonable to Maxwell but is not consistent
with our present understanding of the size of elementary charges. The results of [32] show that a moving
electric charges does not create a “cavity” inside a magnetic medium wherein the average B field differs from
the macroscopic average B field in the absence of the charge. We infer that a moving magnetic charge would
experience an average D inside a dielectric medium equal to the macroscopic average D field in the absence
of the charge.
16
For discussion of the experimental evidence that “permanent” magnetization is Ampèrian, see [30].
5
magnetization density Me = M the associated (macroscopic) electrical current density is
Je = ∇ × M,
(23)
and on the surface of such materials there is the surface current density
Ke = n̂ × M,
(24)
where n̂ is the outward unit vector normal to the surface.
Alternatively, we can suppose the magnetization is associated with densities of effective
magnetic charges. Some care is required to use this approach, since a true (Gilbertian)
magnetic charge density ρm would obey ∇·B = μ0ρm as in eq. (1), and the static force density
on these charges would be Fm = μ0ρm He . However, in Nature ∇ · B = 0 = ∇ · μ0 (H + M),
so we can write
(25)
∇ · H = −∇ · M = ρm,eff ,
and identify
ρm,eff = −∇ · M
(26)
as the volume density of effective (Ampèrian) magnetic charges.
Inside linear magnetic media, where B = μH, the Maxwell equation ∇ · B = 0 then
implies that ρm,eff = 0. However, a surface density σ m,eff of effective magnetic charges can
exist on an interface between two media, and we see that Gauss’ law for the field H implies
that
(27)
σ m,eff = (H2 − H1) · n̂,
where unit normal n̂ points across the interface from medium 1 to medium 2. The magnetic
surface charge density can also be written in terms of the magnetization M = B/μ0 − H as
σm,eff = (M1 − M2) · n̂,
(28)
since ∇ · B = 0 insures that the normal component of B is continuous at the interface.
The force on the surface density of effective magnetic charges is
F = σm,eff B,
(29)
since the effective magnetic charges, which are a representation of effects of electrical currents,
couple to the magnetic field B, as in eq. (9).17
The total force on a linear medium is, in this view, the sum of the force on the conduction
current plus the force on the effective magnetic surface charges. Care is required to implement
such a computation of the force, as discussed in [39], where eq. (29) is affirmed by example.
The key result of this Appendix is that while “true” (Gilbertian, and nonexistent in
Nature) magnetic charges qm obey the force law Fm,true = μ0qm H, the effective (Ampèrian)
magnetic charges (which are a representation of effects of electrical currents) obey Fm,eff =
qm,eff B.
Equation (29) is in agreement with prob. 5.20 of [24], recalling the different convention for factors of μ0
used there. However, the Coulomb Committee in their eq. (1.3-4) [45], and Jefimenko in his eq. (14-9.9a,b)
[46], recommends that the field H/μ0 be used rather than B when using the method of effective magnetic
charges, which would imply a force μ0 /μ times that of eq. (29) for linear media.
17
6
For “effective” Ampèrian magnetic charges the magnetic fields obey ∇ · B/μ0 = 0 and
∇ · He = ρm,eff inside magnetic materials, while for “true” Gilbertian magnetic charges the
fields obey ∇ · B/μ0 = ρm,true and ∇ · Hm = 0 inside magnetic materials where there are no
“free,” “true” magnetic charges. Hence, the roles of B/μ0 and H are reversed in magnetic
materials that contain “true” or “effective” magnetic charges. We illustrate this below for
the fields of a uniformly magnetized sphere.
A.1
Fields of a Uniformly Magnetized Sphere
In this subappendix we deduce the static magnetic fields associated with uniform spheres
of radius a with either uniform Gilbertian magnetization density Mm or uniform Ampèrian
(effective) magnetization density Me .
A.1.1
Uniform Ampèrian Magnetization Density Me
The total magnetic moment of the sphere is
4πMe a3
.
(30)
3
We speed up the derivation by noting that the fields inside the sphere are uniform, and the
fields outside the sphere are the same at those of a point magnetic dipole of strength me ,
me =
B(r > a)
3(me · r̂)r̂ − me
Me a3 (2 cos θ r̂ − sin θ θ̂)
= He (r > a) = Hm (r > a) =
=
, (31)
μ0
4πr3
3r3
in a spherical coordinate system with origin at the center of the sphere and z-axis parallel
to Me .
To characterize the fields inside the sphere, we note use the method of effective magnetic
charges (Appendix A). Since Me is constant inside the sphere, there is no net effective
magnetic charge density there, ρe,eff (r < a) = −∇ · Me (r < a) = 0, while there is a nonzero
surface density of effective magnetic charge,
σ e,eff (r = a) = Me · r̂ = Me cos θ.
(32)
The boundary condition on the magnetic field He at the surface of the sphere is that
He,r (r = a+ ) − He,r (r = a− ) = σe,eff (r = a),
(33)
and hence,
He,r (r = a− ) = He,r (r < a) = He (r < a) cos θ = He,r (r = a+ ) − σ e,eff (r = a)
2Me cos θ
Me cos θ
− Me cos θ = −
,
=
3
3
He (r < a) = −
Me
,
3
Hm (r < a) =
(34)
B(r < a)
2Me
. (35)
= He (r < a) + Me (r < a) =
μ0
3
The result (35) for B/μ0 implies that the magnetic field for the idealization of a “point,”
“effective” (Amperian) magnetic dipole me would be
3(me · r̂)r̂ − me 2me 3
B
=
+
δ (r).
μ0
4πr3
3
7
(36)
A.1.2
Uniform Gilbertian Magnetization Density Mm
The total magnetic moment of the sphere for this case is
mm =
4πMm a3
.
3
(37)
As in sec. D.1.1,we speed up the derivation by noting that the fields inside the sphere are
uniform, and the fields outside the sphere are the same at those of a point magnetic dipole
of strength mm ,
3(mm · r̂)r̂ − me
Mm a3 (2 cos θ r̂ − sin θ θ̂)
B(r > a)
= He (r > a) = Hm (r > a) =
=
. (38)
μ0
4πr3
3r3
To characterize the fields inside the sphere, we note use the method of effective magnetic
charges (Appendix A). Since Mm is constant inside the sphere, there is no net true magnetic
charge density there, ρm (r < a) = −∇ · Mm (r < a) = 0, while there is a nonzero surface
density of true magnetic charge,
σ m (r = a) = Mm · r̂ = Mm cos θ.
(39)
The boundary condition on the magnetic field B at the surface of the sphere is that
Br (r = a+ ) − Br (r = a− ) = μ0 σm (r = a),
(40)
and hence,
Br (r = a− )
Br (r < a)
B(r < a) cos θ
Br (r = a+ )
=
=
=
− σ r (r = a)
μ0
μ0
μ0
μ0
2Mm cos θ
Mm cos θ
− Mm cos θ = −
,
=
3
3
B(r < a)
Mm
= He (r < a) = −
,
μ0
3
Hm (r < a) =
(41)
B(r < a)
2Me
+ Me (r < a) =
. (42)
μ0
3
Comparing with eqs. (34)-(35) we see that the roles of B and H are reversed in the case of
uniform true and effective magnetization. In particular, the sign of B inside the magnetized
sphere is opposite for the cases of Ampèrian and Gilbertian magnetization, although B is
the same outside the sphere in the two cases.18
The result (42) for B/μ0 implies that the magnetic field for the idealization of a “point,”
“true” (Gilbertian) magnetic dipole mm would be
3(mm · r̂)r̂ − mm mm 3
B
δ (r).
=
−
μ0
4πr3
3
(43)
For the case of a cylinder with uniform transverse magnetization, see [47], where the interior B field is
equal and opposite for Ampèrian and Gilbertian magnetization.
18
8
B
Appendix: Lorentz Transformations of the Fields
The various electromagnetic fields can be embedded in antisymmetric 4-tensors (six vectors)
that obey Lorentz transformations. The microscopic fields E and B can be written as
components of the tensor F, and of its dual F obtained by the transformation E → c B,
c B → −E,19
⎛
⎞
Ex
⎜ 0
⎜
⎜
⎜ −Ex 0
F=⎜
⎜
⎜
⎜
⎝
Ey
−cBz
−Ey cBz
⎛
Ez
0
−Ez −cBy cBx
⎞
cBx
⎜ 0
⎜
⎜
⎜ −cBx 0
⎟
⎟
cBy ⎟
⎟
⎟,
⎟
−cBx ⎟
⎟
⎠
F = ⎜
⎜
⎜
cBy
Ez
⎜ −cBy −Ez 0
⎝
0
−cBz Ey
cBz ⎟
⎟
−Ey ⎟
⎟
Ex
−Ex 0
⎟,
⎟
⎟
⎟
⎠
(44)
such that the fields in the frame where the frame of eq. (44) has velocity v are
E
cB
v
= γ E − × cB − (γ − 1)(v̂ · E)v̂,
c
v
= γ cB + × E − (γ − 1)(v̂ · cB)v̂.
c
(45)
(46)
The densities of Ampèrian electric and magnetic dipole moments Pe and Me comprise
the tensor Pe , as first noted by Lorentz [48],
⎛
Pe =
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎞
0
Pe,x
−Pe,x 0
Pe,y
Pe,z
Me,z /c
−Me,y /c
−Pe,y −Me,z /c 0
−Pe,z Me,y /c
Me,x /c
−Me,x /c 0
⎟
⎟
⎟
⎟
⎟,
⎟
⎟
⎟
⎠
(47)
such that the fields in the frame where the frame of eq. (47) has velocity v are
Pe
M e
c
v Me
= γ Pe + ×
− (γ − 1)(v̂ · Pe )v̂,
c
c
Me
Me v
= γ
− × Pe − (γ − 1) v̂ ·
v̂,
c
c
c
(48)
(49)
The macroscopic fields De = 0 E+Pe and He = B/μ0 −Me can be written as components
of the tensor Ge ,
⎛
⎞
De,x
⎜ 0
⎜
⎜
⎜ −De,x 0
De,y
⎜
⎜
⎝
0
Ge = 0 F + Pe = ⎜
⎜
−De,y He,z /c
De,z
−He,z /c He,y /c
−De,z −He,y /c He,x /c
−He,x /c
⎟
⎟
⎟
⎟
⎟,
⎟
⎟
⎟
⎠
(50)
0
The terminology that the electromagnetic field tensor F is the dual of the field tensor F was introduced
by Minkowski, eq. (35) of [43].
19
9
such that the fields in the frame where the frame of eq. (50) has velocity v are
De
He
c
v He
= γ De − ×
− (γ − 1)(v̂ · De )v̂,
c
c
He v
He
v̂.
= γ
+ × De − (γ − 1) v̂ ·
c
c
c
(51)
(52)
The densities of Gilbertian electric and magnetic dipole moments Pm and Mm comprise
the tensor Pm ,
⎛
Pm =
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎞
0
Pm,x
−Pm,x 0
Pm,y
Pm,z
Mm,z /c
−Mm,y /c
−Pm,y −Mm,z /c 0
Mm,x /c
−Mm,x /c 0
−Pm,z Mm,y /c
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(53)
such that the fields in the frame where the frame of eq. (53) has velocity v are
Pm
M m
c
v Mm
= γ Pm + ×
− (γ − 1)(v̂ · Pm )v̂,
c
c
Mm v
Mm
= γ
− × Pm − (γ − 1) v̂ ·
v̂,
c
c
c
(54)
(55)
Finally, the macroscopic fields Dm = E/μ0 − c2 Pm and Hm = B/μ0 + Mm can be written
as components of the tensor Gm ,
⎛
Gm =
F
− c2 Pm =
μ0
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎞
0
Dm,x
Dm,y
−Dm,x 0
−cHm,z
−Dm,y cHm,z
0
−Dm,z −cHm,y cHm,x
Dm,z
⎟
⎟
cHm,y ⎟
⎟
⎟,
⎟
−cHm,x ⎟
⎟
⎠
(56)
0
such that the fields in the frame where the frame of eq. (56) has velocity v are
v
× cHm − (γ − 1)(v̂ · Dm )v̂,
c
v
= γ cHm + × Dm − (γ − 1) (v̂ · cHm ) v̂.
c
Dm = γ Dm −
cHm
(57)
(58)
If we accept that the forces on electric and magnetic charges qe and qm in their rest frame
are
Fe = μ0 qe Dm ,
Fm = μ0 qm He ,
(59)
as in eq. (8), then we see by inverting eqs. (52) and (57) that the Lorentz forces in a frame
where the charges have velocity v are as in eqs. (17)-(18).
10
C
Appendix: Momentum Density and Stress Tensor
We can extend an argument of Minkowski [43] as to field momentum by considering the total
force density on electromagnetic media, following eqs. (17)-(18),
f = fe + fm = μ0 (ρ̃e Dm + J̃e × Hm ) + μ0 (ρ̃m He − J̃m × De ) =
dpmech
,
dt
(60)
where pmech is the density of mechanical momentum in the media. Using the Maxwell
equations (5) for the macroscopic fields,
∂De
dpmech
= μ0 Dm (∇ · De ) − Hm × (∇ × He ) + Hm ×
dt
∂t
∂Hm
+μ0 He (∇ · Hm ) − De × (∇ × Dm ) − De ×
∂t
∂
= − (μ0 De × Hm ) + μ0 [Dm (∇ · De ) + He (∇ · Hm )
∂t
−De × (∇ × Dm ) − Hm × (∇ × He )]
∂pEM
+ ∇ · TEM ,
≡ −
∂t
(61)
where
pEM = μ0 De × Hm
(62)
is the density of momentum associated with the electromagnetic field, and for linear media,
TEM,ij = μ0 Dm,i De,j + He,i Hm,j
De · Dm + H e · H m
− δij
2
(63)
is the symmetric Maxwell stress 3-tensor associated with the electromagnetic fields. To arrive
at eq. (63) we note that for linear media,
∂De,j
∂Dm,j
∂Dm,i
− De,j
+ De,j
∂xj
∂xi
∂xj
∂
De · Dm
=
Dm,i De,j − δij
.
∂xj
2
[Dm (∇ · De ) − De × (∇ × Dm )]i = Dm,i
D
D.1
(64)
Appendix: Electromagnetic Duality
Microscopic Electrodynamics
As noted at the beginning of Appendix B, Minkowski [43] introduced the duality transformation
E → c B,
c B → −E,
11
(65)
as a useful construct when discussing Maxwell’s equation in 4-tensor notation. The microscopic form of Maxwell’s equation, eq. (1), is symmetric (invariant) under this transformation
if in addition all electric charges are replaced by magnetic charges, and vice versa.
We now consider the appropriate duality transformation for other quantities than the
fields E and B. An important constraint is that such a transformation should relate quantities
that have the same dimensions. Now, in SI units electric and magnetic charges do not have
the same dimensions, so their duality transformation is not simply qe ↔ qm .
First, we note that in SI units the microscopic density u of electromagnetic field energy
is
0 E 2
B2
u=
.
(66)
+
2
2μ0
√
√
√
That is, 0 E and B/ μ0 have the same dimension, such that E and B/ 0μ0 = c B have
the same dimensions. Hence, Minkowski’s duality transformation (65) is indeed between
quantities with the same dimensions.
We then rewrite the first two Maxwell equations (1) as
√
ρ
B
√
0 E = √ e , ∇ · √ = μ0 ρm ,
(67)
0
μ0
√
√
which indicates that in SI units ρe / 0 has the same dimensions as μ0 ρm , i.e., that qe and
qm /c have the same dimensions. Thus, the duality transformation for charges is20
qm
qm
qe →
(68)
,
→ −qe .
c
c
This is consistent with the microscopic force laws (9)-(10),
v
qm
v
Fm =
cB − × E ,
(69)
Fe = qe E + × c B ,
c
c
c
whose duality transformation is
∇·
Fe ↔ Fm .
(70)
For completeness, we note that the third and fourth Maxwell equations (1) can be written
as
Jm
∂E
∂
(71)
(c B) + μ0 c
,
∇× cB =
+ μ0 c Je ,
∂ct
c
∂ct
which are consistent with the duality transformations (65) and (68).
One should not (in this author’s view) say that the constants 0 and μ0 (which have
different dimensions) are duals of one another. Indeed, since 0μ0 = 1/c2 , they are not
independent quantities, and electrodynamics could be formulated using only one of them
(along with the universal constant c). For example, the static force between two like electric
charges separated by distance r is Fe = qe2 /4π0 r2 , while the force between two magnetic
2
/4πr2 = μ0 c2 (qm /c)2 /4πr2 = (qm /c)2 /4π0 r2 , such that the duality
charges is Fm = μ0 qm
relation (68) leads to the relation (70) without any “duality relation” between 0 and μ0 .
−∇×E =
20
If only electric charges exist in Nature (as is the case so far as we know), the duality transformations
(65) and (68) have the somewhat trivial application that we could redefine electric fields to be magnetic
fields, and vice versa, call all charges magnetic, and say that electric charges do not exist.
12
D.2
Macroscopic Electrodynamics
In macroscopic electrodynamics we consider media with densities Pe of electric dipoles and
Me of (Ampèrian) magnetic moments due to electrical currents. As the dual of electric charge
qe is magnetic charge divided by c, qm /c, the dual of Pe is a density Mm /c of (Gilbertian)
magnetic dipoles due to pairs of opposite magnetic charges, divided by c. Similarly, the
dual of Me is proportional to a density Pm of (Gilbertian) electric dipoles due to currents
of magnetic charges.
Historically, the electric densities Pe and Me were incorporated in the macroscopic fields
such that H/c = 0 cB − Me /c. While we say that
D = 0 E + Pe and H = B/μ0 − Me cB is
the dual of E, it is not the case that μ0 /0 H is the dual of E (nor is B the dual of μ0 /0D,
as claimed in sec. 6.11 of [24]). Rather, we see that duality requires the introduction of a
second D and a second H field according to eq. (4),
De = 0 E + Pe ,
He =
B
− Me ,
μ0
Dm =
E
− c2Pm ,
μ0
Hm =
B
+ Mm ,
μ0
(72)
with the duality transformations
cPe → Mm ,
cDe → Hm ,
Me → −cPm ,
cHe → −Dm ,
cPm → Me ,
Dm → cHe ,
Mm → −cPe ,
Hm → −cDe .
(73)
(74)
A consequence of the minus signs in eq. (73) is that while the Ampèrian magnetic dipole
moment me associated with electric current density Je is
me =
r × Je
dVol → Ie Area,
2
(75)
the Gilbertian electric dipole moment pm associated with magnetic current density Jm is
pm = −
r × Jm
dVol → −Im Area.
2
(76)
The macroscopic force densities and their duality relation are
fe = μ0(ρ̃e Dm + J̃e × Hm ),
fm = μ0 (ρ̃m He − J̃m × De ),
fe ↔ fm .
(77)
The Poynting vector is
S = μ 0 Dm × H e
(all media),
(78)
and for linear media in which De and Dm are both proportional to E, and He and Hm are
both proportional to B, the density u of stored energy associated with the electromagnetic
fields is
u = μ0
De · Dm + H e · H m
2
Both S and u are self-dual.
13
(linear media).
(79)
D.3
Gaussian Units
We have already remarked (footnote 10) that some people write the second Maxwell equation
as ∇·B = ρm in SI units. The macroscopic fields Dm and Hm could also be defined differently
than here. In particular, the definition Dm = 0 E − Pm (which is our Dm divided by c2 )
could be considered, although then the third and fourth Maxwell equations of eq. (5) differ
more in their forms.
Gaussian units were developed to have greater symmetry between electric and magnetic
quantities, which will carry over into the duality relations in these units. Here, we summarize
the main electrodynamic relations in Gaussian units.
The microscopic Maxwell equations are
∂B
∂E
+ 4πJm , c∇ × B =
+ 4πJe , (80)
∇ · E = 4πρe , ∇ · B = 4πρm , −c∇ × E =
∂t
∂t
for which the duality relations are
qe → qm ,
qm → −qe ,
E → B,
B → −E.
(81)
In Gaussian units, electric and magnetic charges have the same dimensions, and electric and
magnetic fields have the same dimensions.
The macroscopic fields are
De = E + 4πPe ,
He = B − 4πMe ,
Dm = E − 4πPm ,
Hm = B + 4πMm , (82)
with the duality transformations
Pe → Mm ,
De → H m ,
Me → −Pm ,
He → −Dm ,
Pm → Me ,
Dm → H e ,
Mm → −Pe ,
Hm → −De .
(83)
(84)
The macroscopic Maxwell equations are
∇ · Hm = 4πρ̃m ,
∇ · De = 4πρ̃e ,
∂Hm
∂De
+ 4π J̃m ,
+ 4π J̃e .
c∇ × He =
−c∇ × Dm =
∂t
∂t
The microscopic force law on charges is
v
v
Fe = qe E + × B ,
Fm = qm B − × E ,
Fe ↔ Fm ,
c
c
while the macroscopic force densities and their duality relations are
fe = ρ̃e Dm +
J̃e
× Hm ,
c
fm = ρ̃m He −
J̃m
× De ,
c
fe ↔ fm .
(85)
(86)
(87)
The Poynting vector is
Dm × H e
(all media),
(88)
4πc
and for linear media in which De and Dm are both proportional to E, and He and Hm are
both proportional to B, the density u of stored energy associated with the electromagnetic
fields is
De · Dm + H e · H m
(linear media).
(89)
u=
8π
Both S and u are self-dual.
S=
14
Acknowledgment
Thanks to David Griffiths and Ömer Zor for e-discussions of this problem.
References
[1] J.H. Poynting, On the Transfer of Energy in the Electromagnetic Field, Phil. Trans.
Roy. Soc. London 175, 343 (1884),
http://physics.princeton.edu/~mcdonald/examples/EM/poynting_ptrsl_175_343_84.pdf
[2] K.T. McDonald, Electrodynamics in 1 and 2 Spatial Dimensions, (Sept. 23, 2014),
http://physics.princeton.edu/~mcdonald/examples/2dem.pdf
[3] O. Heaviside, Electromagnetic Induction and Its Propagation, sec. IV, The Electrician
(Feb. 21, 1885), p. 306; in Electrical Papers, vol. I, p. 447,
http://physics.princeton.edu/~mcdonald/examples/EM/heaviside_ep1_pp429-476.pdf
[4] O. Heaviside, Electromagnetic Theory, vol. I,
http://physics.princeton.edu/~mcdonald/examples/EM/heaviside_electromagnetic_theory_1.pdf
[5] O. Heaviside, On the Forces, Stresses and Fluxes of Energy in the Electromagnetic
Field, Phil. Trans. Roy. Soc. London 183, 423 (1892),
http://physics.princeton.edu/~mcdonald/examples/EM/heaviside_ptrsla_183_423_92.pdf
[6] P.A.M. Dirac, Quantised Singularities in the Electromagnetic Field, Proc. Roy. Soc.
London A 133, 60 (1931),
http://physics.princeton.edu/~mcdonald/examples/QED/dirac_prsla_133_60_31.pdf
[7] K.T. McDonald, Comay’s Paradox: Do Magnetic Charges Conserve Energy? (June 1,
2015), http://physics.princeton.edu/~mcdonald/examples/comay.pdf
[8] X. Artru and D. Fayolle, Dynamics of a Magnetic Monopole in Matter, Maxwell Equations in Dyonic Matter and Detection of Electric Dipole Moments, Prob. Atom. Sci.
Tech. 6, 110 (2001),
http://physics.princeton.edu/~mcdonald/examples/EM/artru_past_6_110_01.pdf
A sign error 6 lines before eq. (6) is corrected in
http://arxiv.org/pdf/physics/0208007.pdf
[9] W. Thomson, Reprints of Papers on Electrostatics and Magnetism (Macmillan, 1884),
http://physics.princeton.edu/~mcdonald/examples/EM/thomson_electrostatics_magnetism.pdf
[10] J.C. Maxwell, A Treatise on Electricity and Magnetism, 3rd ed., vol. II, (Clarendon
Press, 1892), http://physics.princeton.edu/~mcdonald/examples/EM/maxwell_treatise_v2_92.pdf
[11] S.-D. Poisson, Mem. d. l’Acad. V, 247 (1824).
[12] F.N.H. Robinson, Electromagnetic Stress and Momentum in Matter, Phys. Rep. 16,
313 (1975), http://physics.princeton.edu/~mcdonald/examples/EM/robinson_pr_16_313_75.pdf
15
[13] C. Kittel and A. Manoliu, Interaction of a magnetic monopole with a ferromagnetic
domains, Phys. Rev. B 15, 333 (1977),
http://physics.princeton.edu/~mcdonald/examples/EP/kittel_prb_15_333_77.pdf
[14] A.S. Goldhaber and W.P. Trower, Resource Letter MM-1: Magnetic monopoles, Am.
J. Phys. 58, 429 (1990),
http://physics.princeton.edu/~mcdonald/examples/EM/goldhaber_ajp_58_429_90.pdf
[15] H.A. Lorentz, Versuch einer Theorie der Electrischen und Optischen Erscheinungen
Bewegten Körpern (E.J. Brill, 1895),
http://physics.princeton.edu/~mcdonald/examples/EM/lorentz_electrical_theory_95.pdf
http://physics.princeton.edu/~mcdonald/examples/EM/lorentz_electrical_theory_95_english.pdf
[16] H.A. Lorentz, The Theory of Electrons (B.G. Teubner, 1909),
http://physics.princeton.edu/~mcdonald/examples/EM/lorentz_theory_of_electrons_09.pdf
[17] O. Heaviside, On the Electromagnetic Effects due to the Motion of Electrification
through a Dielectric, Phil. Mag. 27, 324 (1889),
http://physics.princeton.edu/~mcdonald/examples/EM/heaviside_pm_27_324_89.pdf
[18] O. Heaviside, Electromagnetic Induction and Its Propagation, sec. XXII, The Electrician
(Jan. 15, 1886), p. 186; in Electrical Papers, vol. I, p. 542,
http://physics.princeton.edu/~mcdonald/examples/EM/heaviside_ep1_pp542-548.pdf
[19] P.A.M. Dirac, The Theory of Magnetic Monopoles, Phys. Rev. 74, 817 (1948),
http://physics.princeton.edu/~mcdonald/examples/QED/dirac_pr_74_817_48.pdf
[20] A. Sommerfeld, Electrodynamics (Academic Press, 1952).
[21] W. Rindler, Relativity and electromagnetism; The force on a magnetic monopole, Am.
J. Phys. 57, 993 (1989), http://physics.princeton.edu/~mcdonald/examples/EM/rindler_ajp_57_993_89.pdf
[22] F. Moulin, Magnetic monopoles and Lorentz force, Nuovo Cim. 116B, 869 (2001),
http://physics.princeton.edu/~mcdonald/examples/EM/moulin_nc_116b_869_01.pdf
[23] D.J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice Hall, Upper Saddle
River, New Jersey, 1999).
[24] J.D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).
[25] R. Clausius, On the different Systems of Measures for Electric and Magnetic Quantities,
Phil. Mag. 13, 381 (1882),
http://physics.princeton.edu/~mcdonald/examples/EM/clausius_pm_13_381_82.pdf
[26] K.T. McDonald, Magnetic Force on a Permeable Wire (March 17, 2002),
http://physics.princeton.edu/~mcdonald/examples/permeable_wire.pdf
16
[27] A. Einstein and J. Laub, Über die elektromagnetishen Grundgleichungen für bewegte
Köper, Ann. Phys. 26, 532 (1908),
http://physics.princeton.edu/~mcdonald/examples/EM/einstein_ap_26_532_08.pdf
Über die im elektromagnetishen Felde auf ruhende Köper ausgeübten ponderomotorishen Kräfte, Ann. Phys. 26, 541 (1908),
http://physics.princeton.edu/~mcdonald/examples/EM/einstein_ap_26_541_08.pdf http://
physics.princeton.edu/~mcdonald/examples/EM/einstein_ap_26_541_08_english.pdf
Einstein and Laub on the Electrodynamics of Moving Media, in The Collected Papers
of Albert Einstein, vol. 2, ed. J. Stachel (Princeton U. Press, 1989), p. 503,
http://physics.princeton.edu/~mcdonald/examples/EM/einstein_laub_08_commentary.pdf
[28] A. Einstein, Sur les force pondéromotrices qui agissent sur des conducteurs ferromagnétique disposés un champ magnétique et parcourus par un courant, Arch. Sci.
Phys. Nat. 30, 323 (1910),
http://physics.princeton.edu/~mcdonald/examples/EM/einstein_aspn_30_323_10.pdf
[29] K.T. McDonald, Biot-Savart vs. Einstein-Laub Force Law (May 21, 2013),
http://physics.princeton.edu/~mcdonald/examples/laub.pdf
[30] K.T. McDonald, Forces on Magnetic Dipoles (Oct. 26, 2014),
http://physics.princeton.edu/~mcdonald/examples/neutron.pdf
[31] K.T. McDonald, Mansuripur’s Paradox (May 2, 2012),
http://physics.princeton.edu/~mcdonald/examples/mansuripur.pdf
[32] F. Rasetti, Deflection of mesons in magnetized iron, Phys. Rev. 66, 1 (1944),
http://physics.princeton.edu/~mcdonald/examples/EM/rasetti_pr_66_1_44.pdf
This paper mentions the earlier history of erratic results on this topic.
[33] E. Fermi, Über die magnetischen Momente der Atomkerne, Z. Phys. 60, 320 (1930),
http://physics.princeton.edu/~mcdonald/examples/QED/fermi_zp_60_320_30.pdf
[34] J.D. Jackson, The Nature of Intrinsic Magnetic Dipole Moments, CERN-77-17 (1977),
http://physics.princeton.edu/~mcdonald/examples/EP/jackson_CERN-77-17.pdf
[35] G.H. Wannier, The Deflection of Charged Particles in a Ferromagnetic Medium, Phys.
Rev. 72, 304 (1947), http://physics.princeton.edu/~mcdonald/examples/EP/wannier_pr_72_304_47.pdf
[36] J.R. Anderson and A.V. Gold, de Haas-van Alphen Effect and Internal Field in Iron,
Phys. Rev. Lett. 10, 227 (1963),
http://physics.princeton.edu/~mcdonald/examples/EM/anderson_prl_10_227_63.pdf
[37] C. Kittel, Internal Magneteic Field in the de Haas-van Alphen Effect in Iron, Phys. Rev.
Lett. 10, 339 (1963),
http://physics.princeton.edu/~mcdonald/examples/EM/kittel_prl_10_339_63.pdf
[38] J.A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941),
http://physics.princeton.edu/~mcdonald/examples/EM/stratton_electromagnetic_theory.pdf
17
[39] K.T. McDonald, Methods of Calculating Forces on Rigid Magnetic Media (March 18,
2002), http://physics.princeton.edu/~mcdonald/examples/magnetic_force.pdf
[40] Y. Hara, Electromagnetic force on a magnetic monopole, Phys. Rev. A 32, 1002 (1985),
http://physics.princeton.edu/~mcdonald/examples/EM/hara_pra_32_1002_85.pdf
[41] R.M. Fano, L.J. Chu and R.B.Adler, Electromagnetic Fields, Energy and Forces (Wiley,
1960).
[42] R.D.H. Tellegen, Magnetic-Dipole Models, Am. J. Phys. 30, 650 (1962),
http://physics.princeton.edu/~mcdonald/examples/EM/tellegen_ajp_30_650_62.pdf
[43] H. Minkowski, Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten
Körper, Nachr. Ges. Wiss. Göttingen 1, 53 (1908),
http://physics.princeton.edu/~mcdonald/examples/EM/minkowski_ngwg_53_08.pdf
http://physics.princeton.edu/~mcdonald/examples/EM/minkowski_ngwg_53_08_english.pdf
[44] A.-M. Ampère, La détermination de la formule qui represénte l’action mutuelle de deux
portions infiniment petites de conducteur Voltaı̈ques, L’académie Royale des Sciences
(Paris, 1822); Théorie mathématique des phénomenes électro-dynamiques, uniquement
déduite de l’expérience, (A. Blanchard, Paris, 1958).
[45] W.F. Brown, Jr., Electric and magnetic forces: A direct calculation. I, Am. J. Phys.
19, 290 (1951), http://physics.princeton.edu/~mcdonald/examples/EM/brown_ajp_19_290_51.pdf
[46] O.D. Jefimenko, Electricity and Magnetism, 2nd ed. (Electret Scientific Co., Star City,
1989).
[47] K.T. McDonald, Long Rod with Uniform Magnetization Transverse to its Axis (Nov.
3, 1999), http://physics.princeton.edu/~mcdonald/examples/magrod.pdf
[48] H.A. Lorentz, Alte und Neue Fragen der Physik, Phys. Z. 11, 1234 (1910),
http://physics.princeton.edu/~mcdonald/examples/EM/lorentz_pz_11_1234_10.pdf
18