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Electromagnetic Dipoles

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Apollonian circles  &  Bipolar coordinates  (Wikipedia).
Molecular Dipole Moments  in  Hyperphysics  by  Rod Nave.
Physics News in 2001  (APS).  The most spherical thing  (atoms lack EDM).
 
Video:  Electric field of a dipole  by  Yves Pelletier
 
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Electromagnetic Dipoles

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.


 Peter Debye 
 (1884-1966)
Peter Debye
(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 :
  Molecule   C.m Debye   Charge    Displacement  
 Sodium chloride, NaCl  30.02 10-30  9.00   D 28 e6.7 10-12
  Water, H2O     6.17  10-30     1.85   D   10 e3.9  10-12 m
Ammonia, NH3 4.90  10-30 1.47   D 10 e3.1  10-12 m
  Peroxide, H2O2     7.54  10-30     2.26   D   18 e2.6  10-12 m
Hydrazine, N2H4 6.17  10-30 1.85   D 18 e2.1  10-12 m
Hdrogen chloride, HCl 3.60  10-30 1.08   D 18 e1.2  10-12 m
Nitric oxide, NO 0.534 10-30  0.16   D 15 e0.22 10-12
Carbon monoxide, CO 0.374 10-30  0.112 D 14 e0.17 10-12

Hydrogen Peroxide and Polarity  by  Vince Calder
The Dipole Moment of Nitric Oxide  by  C. P. Smyth  &  K. B. McAlpine   (1933).
Microwave Detection of Interstellar NO  by  H. S. Liszt  &  B. E. Turner   (1978).

Atoms are electrically spherical  (EDM = 0)

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  (cfRomalis et al., Phys. Rev. Lett. 86, 2505-2508).  The EDM of a mercury atom, if it has any, would correspond to a displacement of its electronic cloud  (80 electrons)  less than 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.

Electrons are (almost) electrically spherical  (EDM = 0)

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).

Improved measurement of the shape of the electron   Nature  473,  pp. 493-496  (26 May 2011)
by Jony J. Hudson, D. M. Kara, I. J. Smallman, B. E. Sauer, M. R. Tarbutt & Edward A. Hinds
Spherical Electron by Ed Copeland   (in "Sixty Symbols" by Brady Haran)


(2005-05-18)   Force exerted on a dipole by a nonuniform field
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 Dipole  m
m ´ B

Potential Energy of a Dipole  m
- 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 qri.

Electrodynamic Field of an Electric Dipole at the Origin     p = p(t-r/c)   u = r/r
 Curly Bracket f=  1     [       u . p       +       u . p'     ]
vinculum vinculum vinculum
4peo r r c
A= mo    [       p'     ]
vinculum
4p r
 
 Curly Bracket E=  1     [   3 (u.p) u - p   +   3 (u.p' ) u - p'   +   u ´ (u ´ p'' )   ]
vinculum
4peo r  r 2 c r  c 2
B= - mo    [         u ´ p'       +       u ´ p''       ]
vinculum vinculum vinculum
4p r

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 area  S  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 Origin     m = m(t-r/c)   u = r/r
 Curly Bracket f = 0 
 
A= - mo   [      u ´ m       +       u ´ m'     ]
vinculum vinculum vinculum
4p r r c
 
 Curly Bracket E= mo   [        u ´ m'       +       u ´ m''       ]
vinculum vinculum vinculum
4p r
B= mo   [  3 (u.m) u - m   +   3 (u.m' ) u - m'   +   u ´ (u ´ m'' )   ]
vinculum
4p r  r 2 c r  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  densities  M  and  P.


(2016-08-24)   The  Huygens-Fresnel Principle  redux :
A modern view of the  Huygens principle,  accomodating  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 infinifesimal 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+R). 

 Come back later, we're
 still working on this one...

Elements of diffraction theory:  2015-02-07   by  Svetoslav S. Ivanov  (U. of Sofia).
Huygens-Fresnel-Kirchhoff wave-front diffraction formulation.  by  Hal G. Kraus  (March 1989).
 
Wikipedia :   Huygens-Fresnel principle  |  Kirchhoff-Fresnel diffraction formula  |  Kirchhoff integral theorem


(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!

 Triangulation  

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.  QED

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   =   mo M

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

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 conributions 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   =     P  dr   ò  p   [ 3 cos2 q  -  1 ]   ( 2p sin q )  dq     =     0
vinculum
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.  QED

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 mo M

At the center of a uniformly polarized sphere, the electric field is:

E   =   - ( 1/3 P / eo

A uniformly magnetized sphere  by  Richard Fitzpatrick   |   Weisstein


(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

 Come back later, we're
 still working on this one...

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