We use electromagnetic notations and nomenclature here (with SI units) because
that's the prime application.
However, some of the discussion is really of a more general mathematical nature.
Dipole moments are what you get when you pack a globally neutral finite variation into
an infinitesimal amount of space.
An electric dipole is a charge multiplied into a length, a magnetic dipole is a
current into a surface area.
(2008-05-16)
Electric Dipole Moment (EDM)
On the permanent EDM of asymmetrical molecules.
In 1912, Peter Debye (1884-1966)
pioneered the study of the electric dipole moments (EDM)
of asymmetrical molecules
(i.e., molecules without a center of symmetry).
He was awarded the Nobel prize for chemistry in 1936.
The unit of dipolar moment most commonly used by chemists
is the debye (D) which is defined as a decimal
submultiple of the franklin-centimeter,
the standard cgs unit (esu).
The franklin is a unit of electric charge
also known as statcoulomb (statC) and is worth exactly
0.1 C /299792458.
One debye is equal
to one attofranklin-centimeter
(this particular use of a
metric prefix with a non-SI unit is especially dubious,
as the "atto" prefix was only introduced in 1975).
1 D = 10-18 statC.cm
=
(10-21 J/T) / c
=
3.33564095198... 10-30 C.m
As the elementary charge (e) is
1.602176487(40) 10-19 C,
an electric dipole moment (EDM)
of 1 D
corresponds to two opposite elementary charges separated by a distance of
about 0.2082 Å
(or 0.02082 nm).
Electric dipole moments of a few asymmetrical molecules :
Although many atoms have a permanent magnetic
moment, no permanent electric dipole moment (EDM)
has ever been detected for any atom.
In 2000, the search for a nonzero atomic EDM has led a team at the University of Washington
to one of the most precise measurements ever made
(cf. Romalis et al., Phys. Rev. Lett. 86 (2001) pp.2505-2508).
The EDM of a mercury atom, if it has any,
would correspond to a displacement of its electronic cloud
(80 electrons) less than2 10-30 m.
This is about 18 orders of magnitude less than what's
observed for the simple polar molecules
listed in the above table.
This result was obtained by looking for a possible shift due to strong electric fields of
the precession frequency of
199Hg atoms in a weak magnetic field.
No such frequency shift was observed at a precision of 0.4 nHz.
Even more fundamentally, the same type of investigation was carried out about the electron itself
at the Department of Physics of Imperial College London. The results, published in
Nature in May 2011,
indicate that the charge of the electron has a perfect spherical symmetry to a precision
of 15 orders of magnitude (one part in a million billion)
which was widely advertised as "hair's width compared to the size of the solar system".
The new experimental accuracy
(corresponding to atto-electronvolt energy shifts, on a pulsed beam of ytterbium fluoride)
represents only a relatively small improvement (a factor of 1.5) over the
remarkable result obtained in 2002 by Eugene Commins et al. at UC Berkeley.
However, the British team expects to improve their current accuracy
by a factor of 10 or 100 "over the next few years".
A lack of symmetry is expected to occur at that level of accuracy if our fundamental theories are correct
(concerning, in particular, the breaking of symmetry between matter and antimatter in the early universe).
(2005-05-18)
Force exerted on a dipole by a nonuniform field
A uniform fields exerts a torque but no net force.
The net force an electric field E exerts on an electric dipole
p is:
F = grad (p.E)
- ( div E ) p
In the similar expression for the force exerted on a magnetic dipole m,
the second term vanishes because B
is divergence-free:
F = grad (m.B)
- ( div B ) m
= grad (m.B)
Originally, Coulomb defined what we now call the magnetic induction
B and the magnetic moment m
of a compass needle in terms of each other, using essentially the following
expression of the torque applied by the magnetic field to the needle.
He measured that mechanical torque directly
with the delicate torsion balance
which he invented. (Coulomb would later
use that instrument to establish the
basic law of electrostatics
which now bears his name.)
Torque on a Magnetic Dipolem
m ´ B
Potential Energy of a Dipolem
- m . B
(2005-07-15)
Electric Moment & Magnetic Moment
The electrodynamic fields of dipoles.
The following expressions could be obtained
from the general expressions of electrodynamic
potentials and/or
fields in the
limit of dipolar distributions.
However, I fondly remember establishing both sets of dipolar formulas
(as an undergraduate student, in June of 1975 or 1976) by proving that,
if there are no sources at a nonzero distance from the origin, linear superpositions of these
two are the only "spherical and dipolar" solutions of Maxwell's equations.
Loosely speaking, this is to say that there's no other way to build
solutions of Maxwell's equations where the value of each field component at position
r is a sum of products of k(r)r by a vector
Z(t-r/c) or by one of its derivatives...
The two types of dynamic solutions that emerge from such an analysis are readily
identified from the respective static parts
of the electric and magnetic dipolar fields.
(These well-known static fields are obtained
as the limiting cases of simple distributions
[two point charges, or a current loop]
whose moments are kept constant as their sizes tend to zero.)
My long-forgotten motivation was to use such solutions as rigorous building blocks for
dealing with interference using Huygens' principle.
The potentials listed below both satisfy the Lorenz gauge.
Electric Dipole :
The electric dipolar moment p of a charge distribution is:
p =
òòò
r r dV
-
r òòò
r dV
The second term is zero if there's no net electric charge, in which case the value
of the first term does not depend on the origin chosen for positions.
The dipolar moment of a neutral distribution of point charges is
p = å qi ri.
Electrodynamic Field of an Electric Dipole at the Originp = p(t-r/c) u = r/r
f
=
1
[
u . p
+
u . p'
]
4peo r
r
c
A
=
mo
[
p'
]
4pr
E
=
1
[
3 (u.p)u - p
+
3 (u.p') u - p'
+
u ´
(u ´ p'' )
]
4peo r
r 2
cr
c 2
B
=
- mo
[
u ´ p'
+
u ´ p''
]
4pr
r
c
The counterpart of the above for magnetic dipoles is discussed below.
Magnetic Dipole :
The magnetic dipolar moment m of a current distribution is:
m =
½
òòò
r ´ j dV
-
½
r ´
òòò
j dV
The second term is zero for confined currents, in which case the value
of the first term does not depend on the origin chosen for positions.
The dipolar moment for a current I flowing in a loop of
vectorial areaS
is m = I S.
Quantitative magnetic moments were introduced in 1777 by
Charles de Coulomb (1736-1806)
for compass needles.
Coulomb studied them with the torsion balance which he devised and would later put to good
use to establish the law of electrostatics named after him.
Electrodynamic Field of a Magnetic Dipole at the Originm = m(t-r/c) u = r/r
f
=
0
A
=
- mo
[
u ´ m
+
u ´ m'
]
4pr
r
c
E
=
mo
[
u ´ m'
+
u ´ m''
]
4pr
r
c
B
=
mo
[
3 (u.m)u - m
+
3 (u.m') u - m'
+
u ´
(u ´ m'' )
]
4pr
r 2
cr
c 2
Elsewhere on this site, we discuss
the electromagnetic properties of matter, using the symbols
P and M to denote the changing
densities of electric and magnetic dipoles
per unit of volume.
The above lowercase
symbols p and m
can be construed as denoting those
densities integrated at a point.
In the following sections, we discuss the
electromagnetic fields which are found in the midst of
static dipoles, distributed with densitiesM and P.
(2016-08-24)
The Huygens-Fresnel Principle redux :
Making the Huygens principle,
accommodate polarization.
Let's consider how
an electromagnetic planar wave
(progressing along the direction of the x-axis)
can be generated from a source consisting of a uniform distribution of
synchronized electromagnetic dipoles in the yOz plane.
At time t, each infinitesimal dipole in that plane is equal to the elementary
area dy dz multiplied into an areal moment density
(a vector) which depends on t only.
Using the the equations of the above section,
we compute the fields produces by a synchronized sheet of
dipoles at pulsatance
w.
We first examine the contribution to the fields at location (x,0,0)
of an infinitesimal crown of radius R on the source sheet at
x = 0.
This involves only a constant value of the moments
on the sheet (namely the value at time t-d/c
where d2 = x2+R2 ).
(2008-04-25)
Electromagnetic equivalent of permanent magnets :
Simulating any static magnetization (M) with steady currents.
A particularly simple case is that of a uniform sheet
of magnetic dipoles, namely an open surface
(not necessarily a planar one) where each element of surface
dS carries a normal magnetic moment I dS
proportional to it
(the vector dS points
northward and its
magnitude is equal to the infinitesimal surface area
it represents).
The constant I (the density of magnetization
per unit of surface area) is homogeneous to a current and,
indeed, such a magnetic sheet generates everywhere exactly
the same static magnetic induction as would
a current I circulating around the oriented
loop which borders the surface!
This can be established by triangulating the surface
(the tinier the triangles, the better the approximation).
The coarse triangulation at left is enough to visualize the situation:
Each triangle carries a dipole moment equal to I times its
(vectorial) area,
which is exactly the same as the dipole moment of a triangular circuit
with current I
flowing through its 3 edges. Because all inner
edges in this decomposition belong to two adjoining triangles,
the total current flowing through each of them is zero !
Thus, no inner edge contributes
anything to the magnetic field, which is thus the
same as the field produced by a current I flowing through the
loop bordering the triangulated
surface.
Stacking vertically (with uniform spacing) such horizontal magnetic
sheets,
we see that a uniform distribution of magnetic dipole moments with density
M inside an infinite vertical cylinder is magnetically
equivalent to a long solenoid.
Thus, there's no magnetic field outside the cylinder whereas,
inside the cylinder, we have:
B = moM
More generally, the magnetic field produced by
any static distribution M of magnetic dipoles
is the same as the field produced by a current density:
j = rot M
To establish this, notice that the above can be construed as
the elementary cases (in integrated form)
whose superpositions yield the general case.
Incidentally, this implies that two distributions of static magnetic dipoles which have
the same rotational (curl) generate the same magnetic induction.
Such distributions differ by the gradient of some scalar field,
which is a very special type of "magnetization"
that doesn't produce any magnetic induction !
(2008-04-28)
Distribution of electric dipole moments
Permanent electric dipoles are called electrets.
Just like any static distribution of magnets can be simulated by a distribution of currents,
it can be shown that any static distribution P of electric dipoles
produces the same electric field
as the following distribution of charges:
r =
- div P
In particular, an infinite slab of uniformly distributed electric dipoles creates
the same electric field as two parallel plates with opposing charges, namely:
E = -
P / eo
The minus sign need not be surprising: In an horizontal slab
of vertical dipoles, each dipole contributes only equatorially
with a vector whose direction is opposite
to that of the dipole itself !
What's less intuitive is that all the polar and
equatorial contributions cancel each other perfectly outside the
plane of the slab.
The term electret is used (by analogy with magnet)
to denote something endowed with a remanent electric dipole moment
( ferroelectric substances are analogous to
ferromagnetic ones).
More commonly, a net density of electric dipole moments is induced
by an external electric field in a dielectric material.
The word "electret" was coined in 1885 by
Oliver Heaviside (1850-1925),
to whom we owe so many other electromagnetic terms, including:
conductance (Sept. 1885),
permeability (Sept. 1885),
inductance (Feb. 1886),
impedance (July. 1886),
admittance (Dec. 1887) and
reluctance (May 1888).
Heaviside also used the term "permittance" in June 1887, for what is now known
as susceptance.
In full generality, a dynamic distribution of electric and magnetic dipoles
would create the same electromagnetic fields as the following distribution
of charges and currents.
In the classical
description of the electromagnetic
properties of matter (by H.A. Lorentz)
bound charges and currents are expressed this way.
j =
rot M +
¶P/¶t
r =
- div P
Conversely, any distribution of charges and currents can be shown to have
the same electromagnetic field as some distribution of magnetic and dipole
moments
(which is not uniquely determined
by the above equations).
(2008-05-04)
Uniformly Magnetized or Polarized Spheres
Solid spheres with uniform magnetization or electric polarization.
Bluntly speaking, an observer in the midst of a static distribution of
upward dipoles will receive upward
field contributions from the dipoles of both polar regions
(above and below herself) but downward
field contributions from the equatorial region
(in the vicinity of her own horizontal plane).
The polar and equatorial contributions of
all the dipoles at a fixed distance cancel perfectly, since
the integral of 3 (u.P) u - P
vanishes.
Only the axial component (parallel to P) could require a computation:
E =
Pdr
ò
p
[3 cos2 q
- 1 ]
(2p
sin q )
dq = 0
4p eo r
0
However, even that simple computation is made useless by a simple scaling
argument: Since static dipolar fields vary inversely as the cube of
distances, two spheres of different sizes carrying the same density of
dipoles will create the same field at the center.
Thus, the difference between two such spheres is a thick spherical shell
which contributes absolutely nothing to the field at the center.
Although the singularity at the origin makes it impossible to
obtain the field by integrating down to a zero distance shell-by-shell
(you'd obtain different results for different shapes of the shells)
we can use our previous physical results for uniform
magnetic rods or electric slabs (containing the origin)
and subtract from those quantities the convergent
integrals corresponding to contributions that stay clear from the origin.
With this kind of subtraction, we may obtain the field at the center of a uniform
sphere of electric or magnetic dipoles...
At the center of a uniformly magnetized sphere, the magnetic induction is:
B =
( 2/3 )
moM
At the center of a uniformly polarized sphere, the electric field is:
(2008-05-01)
Skewed analogy between magnetic and electric dipoles:
Matching uniform distributions of dipoles generate opposite fields!
There is a nice paradox in the results of the previous sections for the
fields created by uniform distributions of dipoles:
The mathematical expression of the
electric field (E) created by an electric dipole is
exactly the same as
the expression for the magnetic induction (B) of a
magnetic dipole.
Yet, we've just established that, in the main, uniformly distributed magnetic dipoles
create a magnetic induction parallel to them,
whereas uniformly distributed electric dipoles create an electric field
antiparallel to them.
What's going on here?
Well, the mathematical expressions we obtained
for ideal dipoles (zero-size dipoles specified only by their
dipolar moments) are the limits of the fields created by actual
finite dipoles (either tiny loops of current or
tight pairs of opposite charges).
No paradox occurs with uniform distributions of finite dipoles which are formed
in a physically sensible way:
You can easily stack magnetic dipoles (think of little magnets)
to form a long bar magnet equivalent to a solenoid.
The magnetic induction inside a long solenoid is simply given
by Ampère's law
and there's no induction outside of them which would hinder
a side-by-side assembly to create essentially a uniform
distribution of magnetic dipoles throughout a larger volume of space.
To actually feel with your muscles the problems that would occur if you tried to
assemble magnetic dipoles the other way (stacking slabs, instead of bunching
rods) just try to assemble two short bar magnets sideways
(e.g., two flat disks with their north sides up).
The opposite is true for electric dipoles which can easily form thin
polarized membranes when aggregated side by side
(such things are important in biology).
Those flat membranes are stacked effortlessly into
thick slabs because there's virtually no electric
field outside of them (except near the borders).
Don't even think that elementary electric dipoles would align into
a rod like little magnets do.
Actually, uniform distribution of "zero-size" dipoles (with the fields given
above in the magnetic
and electric cases) yield
indeterminate fields because of divergences at short distances.
To settle the issue, you must go back to the physics:
Although the magnetic field of a magnetic dipole has the same expression
as the electric field of an electric dipole, the respective fields still
retain their particular nature. Thus, we can only use
Ampère's law to integrate a magnetic field
and Gauss's law to integrate an electric field.
Those two laws are totally different and so are the orientations
of the fields they yield in uniform distributions of their respective types of dipoles.
I find it
absolutely wonderful that the distinct characteristics of the electric and
magnetic fields translate so directly into this puzzling reversal of sign...
(2008-04-06)
Relativistic Dipoles
A moving magnetic dipole m
develops an electric moment
v ´ m / c2
