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{{short description|Electromagnetic phenomenon}}
{{short description|Electromagnetic phenomenon}}
{{about|the electromagnetic phenomenon||Dipole (disambiguation)}}
{{about|a general description of dipoles in physics and related fields|detailed description of electric dipoles|electric dipole moment|detailed description of magnetic dipoles|magnetic dipole||Dipole (disambiguation)}}
[[File:VFPt Dipole field.svg|thumb|right|250px|The magnetic field of a sphere with a north magnetic pole at the top and a  south magnetic pole at the bottom. By comparison, [[Earth's magnetic field|Earth]] has a ''south'' magnetic pole near its north geographic pole and a ''north'' magnetic pole near its South Pole.]]
 
[[File:VFPt dipole point.svg|thumb|Field lines of a point dipole of any type, electric, magnetic, acoustic, etc.]]


In [[physics]], a '''dipole''' ({{etymology|grc|''{{Wikt-lang|grc|δίς}}'' ({{grc-transl|δίς}})|twice||''{{Wikt-lang|grc|πόλος}}'' ({{grc-transl|πόλος}})|axis}})<ref>[https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Ddi%2Fs δίς], Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus</ref><ref>[https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dpo%2Flos πόλος], Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus</ref><ref>{{cite encyclopedia |title=dipole, n. |encyclopedia=[[Oxford English Dictionary]] |edition=2nd |publisher=[[Oxford University Press]] |year=1989}}</ref> is an [[electromagnetic]] phenomenon which occurs in two ways:
In [[physics]], a '''dipole''' ({{etymology|grc|''{{Wikt-lang|grc|δίς}}'' ({{grc-transl|δίς}})|twice||''{{Wikt-lang|grc|πόλος}}'' ({{grc-transl|πόλος}})|axis}})<ref>[https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Ddi%2Fs δίς], Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus</ref><ref>[https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dpo%2Flos πόλος], Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus</ref><ref>{{cite encyclopedia |title=dipole, n. |encyclopedia=[[Oxford English Dictionary]] |edition=2nd |publisher=[[Oxford University Press]] |year=1989}}</ref> is an [[electromagnetic]] phenomenon which occurs in two ways:
* An [[electric dipole moment|electric dipole]] deals with the separation of the positive and negative [[electric charge]]s found in any electromagnetic system. A simple example of this system is a pair of charges of equal magnitude but opposite sign separated by some typically small distance. (A permanent electric dipole is called an [[electret]].)
 
* A [[magnetic dipole]] is the closed circulation of an [[electric current]] system. A simple example is a single loop of wire with constant current through it. A [[bar magnet]] is an example of a magnet with a permanent [[magnetic dipole moment]].<ref>
* An [[electric dipole moment|electric dipole]] formed by the separation of the positive and negative [[electric charge]]s (typically in atomic and molecular systems).  
* A [[magnetic dipole]] represents a sufficiently small [[magnet]] such as those due to [[atom]]s, [[molecule]]s, and [[electron]]s.  
 
The strength of a dipole, whether electric or magnetic, is characterized by its dipole moment, a [[vector quantity]].
Electric dipoles produce an electric field and experience forces and torques in an electric field that are proportional to their electric dipole moment. The same is true of magnetic dipoles with magnetic fields. Further, the equations for the magnetic dipole are nearly identical to their electric counterparts.
 
Electric dipoles are typically represented by a pair of equal but opposite electric charges separated by a small distance. The [[electric dipole moment]] points from the negative charge towards the positive charge and has a magnitude equal to the strength of each charge times the separation between the charges.<ref group="note">To be precise: for the definition of the dipole moment, one should always consider the "dipole limit", where, for example, the distance of the generating charges should ''converge'' to 0 while simultaneously, the charge strength should ''diverge'' to infinity in such a way that the product remains a positive constant.</ref> Magnetic dipoles are typically modeled as a loop of constant current.<ref>
{{cite book
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  | last = Brau | first = Charles A.
  | last = Brau | first = Charles A.
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{{cite book | last = Griffiths | first = David J. | title = Introduction to Electrodynamics | edition = 3rd | publisher = Prentice Hall | year = 1999 | isbn = 0-13-805326-X | url-access = registration | url = https://archive.org/details/introductiontoel00grif_0 }}
{{cite book | last = Griffiths | first = David J. | title = Introduction to Electrodynamics | edition = 3rd | publisher = Prentice Hall | year = 1999 | isbn = 0-13-805326-X | url-access = registration | url = https://archive.org/details/introductiontoel00grif_0 }}
</ref>
</ref> The [[magnetic dipole moment]] points through the loop (according to the [[right hand grip rule]]), with a magnitude equal to the current in the loop times the area of the loop.
== Classification ==


Dipoles, whether electric or magnetic, can be characterized by their dipole moment, a vector quantity. For the simple electric dipole, the [[electric dipole moment]] points from the negative charge towards the positive charge, and has a magnitude equal to the strength of each charge times the separation between the charges. (To be precise: for the definition of the dipole moment, one should always consider the "dipole limit", where, for example, the distance of the generating charges should ''converge'' to 0 while simultaneously, the charge strength should ''diverge'' to infinity in such a way that the product remains a positive constant.)
=== Electric dipole ===
[[File:VFPt dipole electric.svg|thumb|Electric field lines of a '''physical electric dipole''': two opposite charges separated by a finite distance {{math|d}}.]]


For the magnetic (dipole) current loop, the [[magnetic dipole moment]] points through the loop (according to the [[right hand grip rule]]), with a magnitude equal to the current in the loop times the area of the loop.
{{main|Electric dipole}}
Objects having positive and negative charge with no net charge (such as atoms or molecules) can often be modeled as an '''electric dipole'''. For sufficiently large distances (or equivalently sufficiently small objects), the complexities of these object can be ignored so that all of the physics depends on one quantity the [[electric dipole moment]]. In this model, the object is represented as two equal but opposite [[point charge]]s with charge {{math|±q}} and separated by a distance {{math|d}}. The electric dipole moment has a magnitude <math display="block">p = qd</math> and is directed from the negative charge to the positive one.


Similar to magnetic current loops, the [[electron]] particle and some other [[fundamental particle]]s have magnetic dipole moments, as an electron generates a [[magnetic field]] identical to that generated by a very small current loop. However, an electron's magnetic dipole moment is not due to a current loop, but to an [[Intrinsic and extrinsic properties|intrinsic]] property of the electron.<ref>{{cite book|title=Introduction to Quantum Mechanics|last=Griffiths|first=David J.|publisher=Prentice Hall|year=1994|isbn=978-0-13-124405-4}}</ref> The electron may also have an ''electric'' dipole moment though such has yet to be observed (see ''[[Electron electric dipole moment]]'').
A better definition, accounting for the [[vector quantity| vector nature]] of the dipole moment, expresses the electric dipole moment {{math|'''p'''}} in vector form <math display="block">\mathbf{p} = q \mathbf{d}</math> where {{math|'''d'''}} is the [[displacement (geometry)|displacement vector]] pointing from the negative charge to the positive charge. The electric dipole moment vector {{math|'''p'''}} then points in the same direction. With this definition, the dipole direction tends to align itself with an external [[electric field]] (which tends to oppose the flux lines of the external field). Note that this [[sign convention]] is used in physics, while the opposite sign convention for the dipole, from the positive charge to the negative charge, is used in chemistry.<ref name=Atkins>
[[File:DipoleContourPoint.svg|thumb|right|250px|Contour plot of the [[Electrostatics#Electrostatic potential|electrostatic potential]] of a horizontally oriented electrical dipole of infinitesimal size. Strong colors indicate highest and lowest potential (where the opposing charges of the dipole are located).]]
{{cite book
| title=Chemical principles: the quest for insight
| author1=Peter W. Atkins
| author2=Loretta Jones
| url=https://books.google.com/books?id=46cOswEACAAJ
| isbn=978-1-4641-8395-9
| year=2016
| publisher=Macmillan Learning
| edition = 7th
}}</ref>
 
=== Magnetic dipole ===
[[File:VFPt dipole magnetic2.svg|thumb|Magnetic field lines of a '''physical magnetic dipole''' represented by a ring current of finite diameter.]]


A permanent magnet, such as a bar magnet, owes its magnetism to the intrinsic magnetic dipole moment of the electron. The two ends of a bar magnet are referred to as poles (not to be confused with [[magnetic monopole|monopoles]], see ''{{slink|#Classification}}'' below) and may be labeled "north" and "south". In terms of the Earth's magnetic field, they are respectively "north-seeking" and "south-seeking" poles: if the magnet were freely suspended in the Earth's magnetic field, the north-seeking pole would point towards the north and the south-seeking pole would point towards the south. The dipole moment of the bar magnet points from its magnetic [[south pole|south]] to its magnetic [[north pole]]. In a magnetic [[compass]], the north pole of a bar magnet points north. However, that means that Earth's geomagnetic north pole is the ''south'' pole (south-seeking pole) of its dipole moment and vice versa.
{{main|Magnetic dipole}}
A '''magnetic dipole''' is a theoretical description of a sufficiently small magnet such as that of an atom or an electron. All magnets can be described as being a magnetic dipole for sufficiently large distances from the magnet. The strength of a magnetic dipole is determined by a single property: its [[magnetic dipole moment]], {{math|'''m'''}}. The magnetic dipole model accurately predicts many properties of small magnets such as the magnetic field it produces and how it interacts with other magnetic dipoles, and external magnetic fields.


The only known mechanisms for the creation of magnetic dipoles are by current loops or quantum-mechanical [[Spin (physics)|spin]] since the existence of [[magnetic monopole]]s has never been experimentally demonstrated.
Two different models can be used to describe a magnetic dipole. The simplest to understand, but least correct, is to imagine the magnet as 2 equal but opposite poles. The magnetic dipole moment, similar to the electric dipole moment, then is the product of the magnetic charge (also known as pole strength) and the vector distance between the charges. This can give correct results in an easy to understand way, but suffers from being incorrect (magnetic poles do not exist as separate entities) and giving incorrect results in certain cases (for example inside of a magnet).  


== Classification ==
The more correct description of a magnetic dipole is that of a closed loop of [[electric current]] that encloses a flat area {{math|'''a'''}}. The magnetic moment of this dipole then is the product of its area and it current {{math|I}}. This amperian loop model has the advantage of being physically correct, at least for the part of the magnetic field of an atom due to the motion of the electrons around the nucleus of atoms.
[[File:VFPt dipole electric.svg|thumb|250px|Electric field lines of two opposing charges separated by a finite distance.]]
[[File:VFPt dipole magnetic2.svg|250px|right|thumb|Magnetic field lines of a ring current of finite diameter.]]
[[File:VFPt dipole point.svg|thumb|250px|Field lines of a point dipole of any type, electric, magnetic, acoustic, etc.]]


A ''physical dipole'' consists of two equal and opposite point charges: in the literal sense, two poles. Its field at large distances (i.e., distances large in comparison to the separation of the poles) depends almost entirely on the dipole moment as defined above. A ''point (electric) dipole'' is the limit obtained by letting the separation tend to 0 while keeping the dipole moment fixed. The field of a point dipole has a particularly simple form, and the order-1 term in the [[multipole expansion]] is precisely the point dipole field.
=== Physical vs. ideal dipole ===
[[File:VFPt dipole animation electric.gif|thumb|Animation showing the [[electric field]] of an electric dipole. The dipole consists of two point electric charges of opposite polarity located close together. A transformation from a point-shaped dipole to a finite-size electric dipole is shown.]]
A ''physical dipole'' consists of two equal and opposite point charges: in the literal sense, two poles. Its field at large distances (i.e., distances large in comparison to the separation of the poles) depends almost entirely on the dipole moment as defined above. A ''point (electric) dipole'' or ''ideal dipole'' is the limit obtained by letting the separation tend to 0 while keeping the dipole moment fixed. The field of a point dipole has a particularly simple form, and the order-1 term in the [[multipole expansion]] is precisely the point dipole field.


Although there are no known [[magnetic monopole]]s in nature, there are magnetic dipoles in the form of the quantum-mechanical [[Spin (physics)|spin]] associated with particles such as [[electron]]s (although the accurate description of such effects falls outside of classical electromagnetism). A theoretical magnetic ''point dipole'' has a magnetic field of exactly the same form as the electric field of an electric point dipole. A very small current-carrying loop is approximately a magnetic point dipole; the magnetic dipole moment of such a loop is the product of the current flowing in the loop and the (vector) area of the loop.
=== Dominant term in multipole expansion ===


Any configuration of charges or currents has a 'dipole moment', which describes the dipole whose field is the best approximation, at large distances, to that of the given configuration.  This is simply one term in the multipole expansion when the total charge ("monopole moment") is 0—as it ''always'' is for the magnetic case, since there are no magnetic monopoles.  The dipole term is the dominant one at large distances:  Its field falls off in proportion to {{sfrac|1|''r''<sup>3</sup>}}, as compared to {{sfrac|1|''r''<sup>4</sup>}} for the next ([[quadrupole]]) term and higher powers of {{sfrac|1|''r''}} for higher terms, or {{sfrac|1|''r''<sup>2</sup>}} for the monopole term.
{{main|Multipole expansion}}


== Molecular dipoles ==
Any finite size charge distribution near the origin can be expressed equivalently as an infinite sum of infinitesimally small charge distributions at the origin with progressively finer angular features. (Something similar happens for finite size current distributions producing magnetic fields.) One advantage of this [[multipole expansion]] is that for sufficiently large distances from the origin the first non-zero term of this series dominates. For electric and magnetic fields this term is typically the electric and magnetic dipole respectively.
<!-- This section is linked from [[Ammonia]] and redirects from [[Molecular dipole]] -->
{{See also|Chemical polarity|Dipole moments of molecules}}


Many [[molecule]]s have such dipole moments due to non-uniform distributions of positive and negative charges on the various atoms. Such is the case with polar compounds like [[hydrogen fluoride]] (HF), where [[electron density]] is shared unequally between atoms. Therefore, a molecule's dipole is an [[electric dipole]] with an inherent electric field that should not be confused with a [[magnetic dipole]], which generates a magnetic field.
The first term in the multipole expansion is the monopole. It represents the total ''charge'' of the charge distribution and produces spherically symmetric fields (electric field {{math|'''E'''}} for the electric dipole or magnetic field {{math|'''B'''}} for the magnetic dipole) that decrease as {{sfrac|1|''r''<sup>2</sup>}}. [[Magnetic monopole]]s do not exist in nature, therefore they don't contribute to the magnetic field. Electric monopoles (isolated electric charges) exist but do not contribute for the common case of materials with no net electrical charge.


The physical chemist [[Peter Debye|Peter J. W. Debye]] was the first scientist to study molecular dipoles extensively, and, as a consequence, dipole moments are measured in the non-[[SI]] unit named ''[[debye]]'' in his honor.
Any configuration of charges or currents has a 'dipole moment' whose field is the best approximation, at large distances, to that of that configuration. The field of a dipole falls off in proportion to {{sfrac|1|''r''<sup>3</sup>}}, as compared to {{sfrac|1|''r''<sup>4</sup>}} for the next ([[quadrupole]]) term and higher powers of {{sfrac|1|''r''}} for higher terms.


For molecules there are three types of dipoles:
Although there are no known magnetic monopoles in nature, magnetic dipoles exist in the form of the quantum-mechanical [[Spin (physics)|spin]] associated with particles such as [[electron]]s and the 'currents' of electrons around nuclei. A theoretical magnetic ''point dipole'' has a magnetic field of the same form as the electric field of an electric point dipole. A very small current-carrying loop is approximately a magnetic point dipole; the magnetic dipole moment of such a loop is the product of the current flowing in the loop and the (vector) area of the loop.
; Permanent dipoles: These occur when two atoms in a molecule have substantially different [[electronegativity]] : One atom attracts electrons more than another, becoming more negative, while the other atom becomes more positive. A molecule with a permanent dipole moment is called a ''polar'' molecule. See ''{{slink|Intermolecular force#Dipole–dipole interactions}}''.
== Potential of static dipoles ==
; Instantaneous dipoles : These occur due to chance when [[electron]]s happen to be more concentrated in one place than another in a [[molecule]], creating a temporary dipole. These dipoles are smaller in magnitude than permanent dipoles, but still play a large role in chemistry and biochemistry due to their prevalence. See [[London dispersion force|instantaneous dipole]].
[[File:DipoleContourPoint.svg|thumb|Contour plot of the [[Electrostatics#Electrostatic potential|electrostatic potential]] of a horizontally oriented electrical dipole of infinitesimal size. Strong colors indicate highest and lowest potential (where the opposing charges of the dipole are located).]]
; Induced dipoles : These can occur when one molecule with a permanent dipole repels another molecule's electrons, ''inducing'' a dipole moment in that molecule. A molecule is ''polarized'' when it carries an induced dipole. See [[Intermolecular force#Debye (permanent–induced dipoles) force|induced-dipole attraction]].


More generally, an induced dipole of ''any'' polarizable charge distribution ''ρ'' (remember that a molecule has a charge distribution) is caused by an electric field external to ''ρ''. This field may, for instance, originate from an ion or polar molecule in the vicinity of ''ρ'' or may be macroscopic (e.g., a molecule between the plates of a charged [[capacitor]]). The size of the induced dipole moment is equal to the product of the strength of the external field and the dipole [[polarizability]] of ''ρ''.
In [[electromagnetism]], the calculation of the electric and magnetic fields are often made simpler by first calculating the scaler and vector potentials, {{math|Φ}} and {{math|'''A'''}} respectively.


Dipole moment values can be obtained from measurement of the [[dielectric constant]]. Some typical gas phase values given with the unit [[debye]] are:<ref>
The [[electrostatic potential]] at position {{math|'''r'''}} due to an electric dipole at the origin is given by:
{{cite book
| last = Weast | first = Robert C.
| title=CRC Handbook of Chemistry and Physics
| edition = 65th
| publisher=CRC Press
| year=1984
| isbn=0-8493-0465-2
}}</ref>
* [[carbon dioxide]]: 0
* [[carbon monoxide]]: 0.112&nbsp;D
* [[ozone]]: 0.53&nbsp;D
* [[phosgene]]: 1.17&nbsp;D
* [[ammonia]]: 1.42&nbsp;D
* [[water vapor]]: 1.85&nbsp;D
* [[hydrogen cyanide]]: 2.98&nbsp;D
* [[cyanamide]]: 4.27&nbsp;D
* [[potassium bromide]]: 10.41&nbsp;D


[[File:Carbon-dioxide-2D-dimensions.svg|thumb|160 px|The linear molecule CO<sub>2</sub> has a zero dipole as the two bond dipoles cancel.]]
<math display="block"> \Phi_{dip} = \frac{1}{4\pi\varepsilon_0}\,\frac{\mathbf{p}\cdot\hat{\mathbf{r}}}{r^2},</math>
Potassium bromide (KBr) has one of the highest dipole moments because it is an [[ionic compound]] that exists as a molecule in the gas phase.
where {{math|'''p'''}} is the electric dipole moment, and {{math|''ε''<sub>0</sub>}} is the [[permittivity of free space]]. This term appears as the second term in the [[Multipole expansion#Expansion in Cartesian coordinates|multipole expansion]] of an arbitrary electrostatic potential {{math|Φ}}. This term will dominate at large distances if there is no net charge (and if {{math|'''p'''≠0}}).


[[File:H2O 2D labelled.svg|thumb|160 px|The bent molecule H<sub>2</sub>O has a net dipole. The two bond dipoles do not cancel.]]
The [[vector potential]] {{math|'''A'''<sub>dip</sub>}} at position {{math|'''r'''}} of a magnetic dipole moment {{math|'''m'''}} at the origin is
The overall dipole moment of a molecule may be approximated as a [[Euclidean vector#Addition and subtraction|vector sum]] of [[bond dipole moment]]s. As a vector sum it depends on the relative orientation of the bonds, so that from the dipole moment information can be deduced about the [[molecular geometry]].


For example, the zero dipole of CO<sub>2</sub> implies that the two C=O bond dipole moments cancel so that the molecule must be linear. For H<sub>2</sub>O the O−H bond moments do not cancel because the molecule is bent. For ozone (O<sub>3</sub>) which is also a bent molecule, the bond dipole moments are not zero even though the O−O bonds are between similar atoms. This agrees with the Lewis structures for the resonance forms of ozone which show a positive charge on the central oxygen atom.
<math display="block">\mathbf{A}_{dip} = \frac{\mu_0}{4\pi} \frac{\mathbf{m} \times \hat{\mathbf{r}}}{r^2},</math>


[[File:Ozone-resonance-Lewis-2D.svg|center|400px|Resonance Lewis structures of the ozone molecule]]
where {{math|μ<sub>0</sub>}} is the [[permeability of free space]]. This term appears as the second term (first non-zero term) in the multipole expansion of an arbitrary vector potential {{math|'''A'''}} in terms of the current density {{math|'''J'''}} that created it. This term dominates at large distances if {{math|'''m'''≠0}}.
{{multiple image
| align=right
| image1=Cis-1,2-dichloroethene.png
| width1=150
| caption1=''Cis'' isomer, dipole moment 1.90&nbsp;D
| image2=Trans-1,2-dichloroethene.png
| width2=150
| caption2=''Trans'' isomer, dipole moment zero
}}
An example in organic chemistry of the role of geometry in determining dipole moment is the [[cis–trans isomerism|''cis'' and ''trans'' isomers]] of [[1,2-dichloroethene]]. In the ''cis'' isomer the two polar C−Cl bonds are on the same side of the C=C [[double bond]] and the molecular dipole moment is 1.90&nbsp;D. In the ''trans'' isomer, the dipole moment is zero because the two C−Cl bonds are on opposite sides of the C=C and cancel (and the two bond moments for the much less polar C−H bonds also cancel).


Another example of the role of molecular geometry is [[boron trifluoride]], which has three polar bonds with a difference in [[electronegativity]] greater than the traditionally cited threshold of 1.7 for [[ionic bonding]]. However, due to the equilateral triangular distribution of the fluoride ions centered on and in the same plane as the boron cation, the symmetry of the molecule results in its dipole moment being zero.
== Field of a static dipole ==
The [[electric field]], {{math|'''E'''<sub>dip</sub>}}, and [[magnetic field]], {{math|'''B'''<sub>dip</sub>}}, at a location, {{math|'''r'''}}, due to a dipole at the origin with dipole moment, {{math|'''p'''}} for the electric dipole or {{math|'''m'''}} for the magnetic dipole, can be determined from the scalar, {{math|&Phi;<sub>dip</sub>}}, and vector, {{math|'''A'''<sub>dip</sub>}}, potential respectively as:


== Quantum-mechanical dipole operator ==
<math display="block">\begin{align}
Consider a collection of ''N'' particles with charges ''q<sub>i</sub>'' and position vectors '''r'''<sub>''i''</sub>. For instance, this collection may be a molecule consisting of electrons, all with [[electron charge|charge]] −''e'', and nuclei with charge ''eZ<sub>i</sub>'', where ''Z<sub>i</sub>'' is the [[atomic number]] of the ''i''&thinsp;th nucleus.
\mathbf{E}_{dip} & = -\nabla \Phi_{dip} & = \frac {1} {4\pi\varepsilon_0} \ \frac{3(\mathbf{p}\cdot\hat{\mathbf{r}}) \hat{\mathbf{r}} - \mathbf{p}}{r^3}\, & - \, \delta^3(\mathbf{r})\frac{\mathbf{p}}{3\varepsilon_0}, \\
The dipole observable (physical quantity) has the quantum mechanical '''dipole operator''':{{citation needed|date=April 2015}}
\mathbf{B}_{dip} & = \;\; \nabla \times \mathbf{A}_{dip} &  = \frac{\mu_0}{4\pi} \ \frac{3(\mathbf{m} \cdot \hat{\mathbf{r}}) \hat{\mathbf{r}} - \mathbf{m}}{r^3}\, & + \, \delta^3(\mathbf{r})\frac{2\mu_0\mathbf{m}}{3},
: <math>\mathfrak{p} = \sum_{i=1}^N \, q_i \, \mathbf{r}_i \, .</math>
\end{align}</math>


Notice that this definition is valid only for neutral atoms or molecules, i.e. total charge equal to zero. In the ionized case, we have
where {{math|''ε''<sub>0</sub>}} is the permitivity of free space and {{math|''μ''<sub>0</sub>}} is the permeability of free space (both constants). The last term in the equations (containing the [[dirac delta function]]) only contributes at the origin and in most cases can be ignored. Note the similarity in the two equations making them near identical except for the last term.
: <math>\mathfrak{p} = \sum_{i=1}^N \, q_i \, (\mathbf{r}_i - \mathbf{r}_c) ,</math>
where <math> \mathbf{r}_c</math> is the center of mass of the molecule/group of particles.<ref>{{Cite web|url=http://www.av8n.com/physics/electric-dipole.htm#eq-dipole-ref|title = The Electric Dipole Moment Vector -- Direction, Magnitude, Meaning, et cetera}}</ref>


== Atomic dipoles ==
These equations are ''exactly'' the field of a point (ideal) dipole, ''exactly'' the dipole term in the multipole expansion of an arbitrary field, and ''approximately'' the field of any dipole-like configuration at large distances. The field of a real (physical) dipole is continuous everywhere and will be different close to the origin. The delta function represents the strong field pointing in the opposite direction between the point charges for the electric dipole case (and in same direction for magnetic dipole), which is often omitted since one is rarely interested in the field at the dipole's position.
<!-- This section is linked from [[Intermolecular force]] -->
A non-degenerate (''S''-state) atom can have only a zero permanent dipole. This fact follows quantum mechanically from the inversion symmetry of atoms. All 3 components of the dipole operator are antisymmetric under [[Inversion in a point|inversion]] with respect to the nucleus,
: <math> \mathfrak{I} \;\mathfrak{p}\;  \mathfrak{I}^{-1} = -\mathfrak{p}, </math>
where <math>\mathfrak{p}</math> is the dipole operator  and <math>\mathfrak{I}</math> is the inversion operator.


The permanent dipole moment  of an atom in a non-degenerate state (see ''[[Degenerate energy level]]'') is given as the expectation (average) value of the dipole operator,
For further discussions about the internal field of dipoles, see<ref name=":0" /><ref>{{cite book|title=Classical Electrodynamics, 3rd Ed.|last=Jackson|first=John D.|publisher=Wiley|year=1999|isbn=978-0-471-30932-1|pages=148–150}}</ref> or ''{{slink|Magnetic moment#Internal magnetic field of a dipole}}''.
: <math>\left\langle \mathfrak{p} \right\rangle = \left\langle\, S\, | \mathfrak{p} |\, S \,\right\rangle,</math>
where <math> |\, S\, \rangle </math> is an ''S''-state, non-degenerate, wavefunction, which is symmetric or antisymmetric under inversion: <math> \mathfrak{I}\, |\, S\, \rangle = \pm|\, S\, \rangle</math>. Since the product of the wavefunction (in the ket) and its [[complex conjugate]] (in the bra) is always symmetric under inversion and its inverse,
: <math>
  \left\langle \mathfrak{p} \right\rangle =
  \left\langle\, \mathfrak{I}^{-1}\, S\, | \mathfrak{p} |\, \mathfrak{I}^{-1}\, S\, \right\rangle =
  \left\langle\, S\, | \mathfrak{I}\, \mathfrak{p}\, \mathfrak{I}^{-1} |\, S\, \right\rangle =
  -\left\langle \mathfrak{p} \right\rangle
</math>
it follows that the expectation value changes sign under inversion.  We used here the fact that <math> \mathfrak{I}</math>, being a symmetry operator, is [[unitary operator|unitary]]: <math> \mathfrak{I}^{-1} = \mathfrak{I}^{*}\,</math> and [[Hermitian adjoint#Definition for bounded operators between Hilbert spaces|by definition]] the Hermitian adjoint <math> \mathfrak{I}^*\,</math> may be moved from bra to ket and then becomes <math> \mathfrak{I}^{**} = \mathfrak{I}\,</math>. Since the only quantity that is equal to minus itself is the zero, the expectation  value vanishes,
: <math>\left\langle \mathfrak{p} \right\rangle = 0.</math>


In the case of open-shell atoms with degenerate  energy levels, one could define a dipole moment by the aid of the first-order [[Stark effect]]. This gives a non-vanishing dipole (by definition proportional to a non-vanishing first-order Stark shift) only if some of the wavefunctions belonging to the degenerate energies have opposite [[parity (physics)|parity]]; i.e., have different behavior under inversion. This is a rare occurrence, but happens for the excited H-atom, where 2s and 2p states are "accidentally" degenerate (see article ''[[Laplace–Runge–Lenz vector#Quantum mechanics of the hydrogen atom|Laplace–Runge–Lenz vector]]'' for the origin of this degeneracy) and have opposite parity (2s is even and 2p is odd).
=== Alternative formulation in spherical coordinates ===


== Field of a static magnetic dipole ==
An alternative formulation that simplifies the equations by orienting the z-axis in the direction of the dipole moment is:
{{see also|Magnet#Two models for magnets: magnetic poles and atomic currents}}


=== Magnitude ===
<math display="block">\begin{align}
The far-field strength, ''B'', of a dipole magnetic field is given by
\mathbf{E}_{dip} = \frac{p}{4\pi\varepsilon_0 r^3}\left ( 2 \cos\theta \, \mathbf{\hat{r}} + \sin\theta \, \mathbf{\hat{\theta}} \right ), \\
: <math>B(m, r, \lambda) = \frac{\mu_0}{4\pi} \frac{m}{r^3} \sqrt{1 + 3\sin^2(\lambda)} \, ,</math>
where
: ''B'' is the strength of the field, measured in [[tesla (unit)|tesla]]s
: ''r'' is the distance from the center, measured in metres
: ''λ'' is the magnetic latitude (equal to 90°&nbsp;−&nbsp;''θ'') where ''θ'' is the magnetic colatitude, measured in [[radian]]s or [[degree (angle)|degree]]s from the dipole axis<ref group="note">Magnetic colatitude is 0 along the dipole's axis and 90° in the plane perpendicular to its axis.</ref>
: ''m'' is the dipole moment, measured in [[ampere]]-square metres or [[joule]]s per [[tesla (unit)|tesla]]
: ''μ''<sub>0</sub> is the [[permeability (electromagnetism)|permeability of free space]], measured in [[henry (unit)|henries]] per metre.


Conversion to cylindrical coordinates is achieved using {{nowrap|''r''<sup>2</sup> {{=}} ''z''<sup>2</sup> + ''ρ''<sup>2</sup>}} and
\mathbf{B}_{dip} = \frac{\mu_0 m}{4\pi r^3}\left ( 2 \cos\theta \, \mathbf{\hat{r}} + \sin\theta \, \mathbf{\hat{\theta}} \right ),
: <math>\lambda = \arcsin\left(\frac{z}{\sqrt{z^2 + \rho^2}}\right)</math>
where ''ρ'' is the perpendicular distance from the ''z''-axis. Then,
: <math>B(\rho, z) = \frac{\mu_0 m}{4 \pi \left(z^2 + \rho^2\right)^\frac32} \sqrt{1 + \frac{3 z^2}{z^2 + \rho^2}}</math>


=== Vector form ===
\end{align}</math>
The field itself is a vector quantity:
where {{math|&theta;}} is the angle from the z-axis (the direction of the dipole moment), {{math|r}} is the distance from the dipole to where the field is, <math>\hat r</math> is the direction of {{math|r}}, and <math>\hat \theta</math>  is the direction  perpendicular to <math>\hat r</math> and pointing away from the z-axis.
: <math>\mathbf{B}(\mathbf{m}, \mathbf{r}) =
  \frac{\mu_0}{4\pi} \ \frac{3(\mathbf{m} \cdot \hat{\mathbf{r}}) \hat{\mathbf{r}} - \mathbf{m}}{r^3}
  </math>
where
: '''B''' is the field
: '''r''' is the vector from the position of the dipole to the position where the field is being measured
: ''r'' is the absolute value of '''r''': the distance from the dipole
: '''r̂''' = {{sfrac|'''r'''|''r''}} is the [[unit vector]] parallel to '''r''';
: '''m''' is the (vector) dipole moment
: ''μ''<sub>0</sub> is the permeability of free space


This is ''exactly'' the field of a point dipole, ''exactly'' the dipole term in the multipole expansion of an arbitrary field, and ''approximately'' the field of any dipole-like configuration at large distances.
=== Magnitude ===


=== Magnetic vector potential ===
The magnitude of the dipole field strengths, {{math|'''E'''<sub>dip</sub>}} and {{math|'''E'''<sub>dip</sub>}}, at a distance, {{math|r}}, from the dipoles at the origin are:
The [[vector potential]] '''A''' of a magnetic dipole is
<math display="block">\begin{align}
: <math>\mathbf{A}(\mathbf{r}) = \frac{\mu_0}{4\pi} \frac{\mathbf{m} \times \hat{\mathbf{r}}}{r^2}</math>
E_{dip} \; & = & \frac{1}{4\pi\varepsilon_0} \frac{p}{r^3} \sqrt{1 + 3\cos^2(\theta)} \, ,\\
with the same definitions as above.
B_{dip} \; & = & \frac{\mu_0}{4\pi} \frac{m}{r^3} \sqrt{1 + 3\cos^2(\theta)} \, ,
\end{align}</math>


== Field from an electric dipole ==
where {{mvar|&theta;}} is the angle between the direction that the dipole is pointing (typically chosen as the z-axis) and the direction of {{math|'''r'''}}, {{math|&epsilon;<sub>0</sub>}} is the permittivity of free space, and {{math|''μ''<sub>0</sub>}} is permeability of free space.
<!-- This section is linked from [[Intermolecular force]] -->
The [[electrostatic potential]] at position '''r''' due to an electric dipole at the origin is given by:
: <math> \Phi(\mathbf{r}) = \frac{1}{4\pi\epsilon_0}\,\frac{\mathbf{p}\cdot\hat{\mathbf{r}}}{r^2}</math>
where '''p''' is the (vector) [[Electric dipole moment|dipole moment]], and ''є''<sub>0</sub> is the [[permittivity of free space]].


This term appears as the second term in the [[Multipole expansion#Expansion in Cartesian coordinates|multipole expansion]] of an arbitrary electrostatic potential Φ('''r'''). If the source of Φ('''r''') is a dipole, as it is assumed here, this term is  the only non-vanishing term in the multipole expansion of Φ('''r'''). The [[electric field]] from a dipole can be found from the [[gradient]] of this potential:
Conversion to cylindrical coordinates is achieved using {{math|1=''r''<sup>2</sup> = ''z''<sup>2</sup> + ''ρ''<sup>2</sup>}} and
: <math> \mathbf{E} = - \nabla \Phi =\frac {1} {4\pi\epsilon_0} \ \frac{3(\mathbf{p}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{p}}{r^3} - \delta^3(\mathbf{r})\frac{\mathbf{p}}{3\epsilon_0}.</math>
<math>\theta = \arccos\left(\frac{z}{\sqrt{z^2 + \rho^2}}\right)</math>
where ''ρ'' is the perpendicular distance from the ''z''-axis gives:
<math display="block">\begin{align}
E_{dip} \; & = & \frac{p}{4 \pi \varepsilon_0 \left(z^2 + \rho^2\right)^\frac32} \sqrt{1 + \frac{3 z^2}{z^2 + \rho^2}} \, , \\
B_{dip} \; & = & \frac{\mu_0 m}{4 \pi \left(z^2 + \rho^2\right)^\frac32} \sqrt{1 + \frac{3 z^2}{z^2 + \rho^2}}.


This is of the same form of the expression for the magnetic field of a point magnetic dipole, ignoring the delta function.
\end{align}</math>
In a real electric dipole, however, the charges are physically separate and the electric field diverges or converges at the point charges.
This is different to the magnetic field of a real magnetic dipole which is continuous everywhere. The delta function represents the strong field pointing in the opposite direction between the point charges, which is often omitted since one is rarely interested in the field at the dipole's position.
For further discussions about the internal field of dipoles, see<ref name=":0" /><ref>{{cite book|title=Classical Electrodynamics, 3rd Ed.|last=Jackson|first=John D.|publisher=Wiley|year=1999|isbn=978-0-471-30932-1|pages=148–150}}</ref> or ''{{slink|Magnetic moment#Internal magnetic field of a dipole}}''.


== Torque on a dipole ==
== Torques and forces on static dipoles ==
Since the direction of an [[electric field]] is defined as the direction of the force on a positive charge, electric field lines point away from a positive charge and toward a negative charge.
Since the direction of an [[electric field]] is defined as the direction of the force on a positive charge, electric field lines point away from a positive charge and toward a negative charge.


When placed in a homogeneous [[electric field|electric]] or [[magnetic field]], equal but opposite [[force]]s arise on each side of the dipole creating a [[torque]] {{math|'''&tau;'''}}}:
When placed in a homogeneous [[electric field|electric]] or [[magnetic field]], equal but opposite [[force]]s arise on each side of the dipole creating a [[torque]] {{math|'''''τ'''''}}:
: <math> \boldsymbol{\tau} = \mathbf{p} \times \mathbf{E}</math>
<math display="block"> \boldsymbol{\tau} = \mathbf{p} \times \mathbf{E}</math>
for an [[electrical dipole moment|electric dipole moment]] '''p''' (in coulomb-meters), or
for an [[electrical dipole moment|electric dipole moment]] {{math|'''p'''}} (in coulomb-meters), or
: <math> \boldsymbol{\tau} = \mathbf{m} \times \mathbf{B}</math>
<math display="block"> \boldsymbol{\tau} = \mathbf{m} \times \mathbf{B}</math>
for a [[magnetic dipole moment]] '''m''' (in ampere-square meters).
for a [[magnetic dipole moment]] {{math|'''m'''}} (in ampere-square meters).


The resulting torque will tend to align the dipole with the applied field, which in the case of an electric dipole, yields a potential energy of
The resulting torque will tend to align the dipole with the applied field, which in the case of an electric dipole, yields a potential energy of
: <math> U = -\mathbf{p} \cdot \mathbf{E}</math>.
<math display="block"> U = -\mathbf{p} \cdot \mathbf{E} .</math>


The energy of a magnetic dipole is similarly
The energy of a magnetic dipole is similarly
: <math> U = -\mathbf{m} \cdot \mathbf{B}</math>.
<math display="block"> U = -\mathbf{m} \cdot \mathbf{B}.</math>
 
== Molecular electric dipoles ==
 
<!-- This section is linked from [[Ammonia]] and redirects from [[Molecular dipole]] -->
{{See also|Chemical polarity|Dipole moments of molecules}}
Electric dipole moments are responsible for the behavior of a substance in the presence of external electric fields. The dipoles tend to be aligned to the external field which can be constant or time-dependent. This effect forms the basis of a modern experimental technique called [[dielectric spectroscopy]].
 
Dipole moments can be found in common molecules such as water and also in biomolecules such as proteins due to non-uniform distributions of positive and negative charges on the various atoms.<ref name="ojeda">{{cite journal |author1=Ojeda, P. |author2=Garcia, M. |title=Electric Field-Driven Disruption of a Native beta-Sheet Protein Conformation and Generation of a Helix-Structure  |journal=Biophysical Journal |volume=99 |issue=2 |pages=595–599 |year=2010 |pmid=20643079 |pmc=2905109 |doi= 10.1016/j.bpj.2010.04.040 |bibcode = 2010BpJ....99..595O }}</ref> Therefore, a molecule's dipole is an [[electric dipole]] with an inherent electric field that should not be confused with a [[magnetic dipole]], which generates a magnetic field.
The physical chemist [[Peter Debye|Peter J. W. Debye]] was the first scientist to study molecular dipoles extensively, and, as a consequence, dipole moments are measured in the non-[[SI]] unit named ''[[debye]]'' in his honor.
 
For molecules there are three types of dipoles:
; Permanent dipoles: These occur when two atoms in a molecule have substantially different [[electronegativity]] : One atom attracts electrons more than another, becoming more negative, while the other atom becomes more positive. A molecule with a permanent dipole moment is called a ''polar'' molecule. See ''{{slink|Intermolecular force#Dipole–dipole interactions}}''.
; Instantaneous dipoles : These occur due to chance when [[electron]]s happen to be more concentrated in one place than another in a [[molecule]], creating a temporary dipole. These dipoles are smaller in magnitude than permanent dipoles, but still play a large role in chemistry and biochemistry due to their prevalence. See [[London dispersion force|instantaneous dipole]].
; Induced dipoles : These can occur when one molecule with a permanent dipole repels another molecule's electrons, ''inducing'' a dipole moment in that molecule. A molecule is ''polarized'' when it carries an induced dipole. See [[Intermolecular force#Debye (permanent–induced dipoles) force|induced-dipole attraction]].
 
It is possible to calculate dipole moments from [[Electronic structure|electronic structure theory]], either as a response to constant electric fields or from the [[density matrix]].<ref>{{Cite book|title=Introduction to computational chemistry|last=Frank.|first=Jensen|date=2007|publisher=John Wiley & Sons|isbn=978-0-470-01187-4|edition= 2nd|location=Chichester, England|oclc=70707839}}</ref> Such values however are not directly comparable to experiment due to the potential presence of nuclear quantum effects, which can be substantial for even simple systems like the ammonia molecule.<ref>{{Cite journal|last=Puzzarini|first=Cristina|date=2008-09-01|title=Ab initio characterization of XH3 (X = N,P). Part II. Electric, magnetic and spectroscopic properties of ammonia and phosphine|journal=Theoretical Chemistry Accounts|language=en|volume=121|issue=1–2|pages=1–10|doi=10.1007/s00214-008-0409-8|s2cid=98782005|issn=1432-881X}}</ref> [[Coupled cluster|Coupled cluster theory]] (especially CCSD(T)<ref>{{Cite journal|last1=Raghavachari|first1=Krishnan| last2=Trucks|first2=Gary W.| last3=Pople|first3=John A.|last4=Head-Gordon|first4=Martin|title=A fifth-order perturbation comparison of electron correlation theories|journal=Chemical Physics Letters|volume=157|issue=6|pages=479–483 |doi=10.1016/s0009-2614(89)87395-6 |bibcode=1989CPL...157..479R|year=1989}}</ref>) can give very accurate dipole moments,<ref>{{Cite book|title=Molecular electronic-structure theory|last1=Helgaker|first1=Trygve|last2=Jørgensen|first2=Poul|last3=Olsen|first3=Jeppe|language=en|doi=10.1002/9781119019572|year=2000|isbn=978-1-119-01957-2|url=https://cds.cern.ch/record/1529252|type=Submitted manuscript|publisher=Wiley}}{{Dead link|date=August 2022 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> although it is possible to get reasonable estimates (within about 5%) from [[density functional theory]], especially if [[Hybrid functional|hybrid]] or double hybrid functionals are employed.<ref>{{Cite journal|last1=Hait|first1=Diptarka|last2=Head-Gordon|first2=Martin|date=2018-03-21|title=How Accurate Is Density Functional Theory at Predicting Dipole Moments? An Assessment Using a New Database of 200 Benchmark Values|journal=Journal of Chemical Theory and Computation|language=en|volume=14|issue=4|pages=1969–1981 |doi=10.1021/acs.jctc.7b01252|pmid=29562129 |arxiv=1709.05075|s2cid=4391272}}</ref> The dipole moment of a molecule can also be calculated based on the molecular structure using the concept of group contribution methods.<ref name="mueller">{{cite journal |author1=K. Müller |author2=L. Mokrushina |author3=W. Arlt |title=Second-Order Group Contribution Method for the Determination of the Dipole Moment |journal=J. Chem. Eng. Data |volume=57 |issue=4 |pages=1231–1236 |year=2012 |doi=10.1021/je2013395  }}</ref>
 
By means of the total dipole moment of some material one can compute the [[dielectric constant]]. If <math> \mathcal{M}_{\rm Tot} </math> is the total dipole moment of the sample, then the dielectric constant is given by
<math display="block">\varepsilon = 1 + k \left\langle \mathcal{M}_\text{Tot}^2 \right\rangle</math>
where ''k'' is a constant and <math>\left\langle \mathcal{M}_\text{Tot}^2 \right\rangle = \left\langle \mathcal{M}_\text{Tot} (t = 0) \mathcal{M}_\text{Tot}(t = 0) \right\rangle</math> is the time correlation function of the total dipole moment. In general the total dipole moment have contributions coming from translations and rotations of the molecules in the sample,
<math display="block">\mathcal{M}_\text{Tot} = \mathcal{M}_\text{Trans} + \mathcal{M}_\text{Rot}.</math>
 
Therefore, the dielectric constant has contributions from both terms. This approach can be generalized to compute the frequency dependent dielectric function.<ref name="kim">{{cite journal |author1=Y. Shim |author2=H. Kim |title=Dielectric Relaxation, Ion Conductivity, Solvent Rotation, and Solvation Dynamics in a Room-Temperature Ionic Liquid |journal=J. Phys. Chem. B |volume=112 |issue=35 |pages=11028–11038 |year=2008 |pmid=18693693 |doi=10.1021/jp802595r}}</ref>
 
Alternatively, electric dipole moment values for a given material can be calculated by the measured value of the dielectric constant of that material.
 
== Quantum-mechanical dipole operator ==
Consider a collection of ''N'' particles with charges ''q<sub>i</sub>'' and position vectors '''r'''<sub>''i''</sub>. For instance, this collection may be a molecule consisting of electrons, all with [[electron charge|charge]] −''e'', and nuclei with charge ''eZ<sub>i</sub>'', where ''Z<sub>i</sub>'' is the [[atomic number]] of the ''i''&thinsp;th nucleus.
The dipole observable ([[physical quantity]]) has the quantum mechanical '''dipole operator''':<ref name="matar2025">{{cite journal | vauthors = Matar IK, Matta CF | title = Origin-Dependence of Dipole Moments of Charged Proteins: Theoretical Foundations and Implications, Revisited | journal = Journal of Computational Chemistry | volume = 46 | issue = 25 | article-number = e70207 | date = 2025 | doi = 10.1002/jcc.70207 }}</ref>
<math display="block">\mathfrak{p} = \sum_{i=1}^N \, q_i \, \mathbf{r}_i \, .</math>
 
Notice that this definition is valid only for neutral atoms or molecules, i.e., total charge equal to zero.<ref name="matar2025">{{cite journal | vauthors = Matar IK, Matta CF | title = Origin-Dependence of Dipole Moments of Charged Proteins: Theoretical Foundations and Implications, Revisited | journal = Journal of Computational Chemistry | volume = 46 | issue = 25 | article-number = e70207 | date = 2025 | doi = 10.1002/jcc.70207 }}</ref> In the ionized case, we have
<math display="block">\mathfrak{p} = \sum_{i=1}^N \, q_i \, (\mathbf{r}_i - \mathbf{r}_c) ,</math>
where <math> \mathbf{r}_c</math> is the center of mass of the molecule/group of particles.<ref>{{Cite web|url=http://www.av8n.com/physics/electric-dipole.htm#eq-dipole-ref|title=The Electric Dipole Moment Vector -- Direction, Magnitude, Meaning, et cetera|access-date=2015-08-25|archive-date=2023-07-08|archive-url=https://web.archive.org/web/20230708223415/http://www.av8n.com/physics/electric-dipole.htm#eq-dipole-ref}}</ref>
 
== Atomic dipoles ==
<!-- This section is linked from [[Intermolecular force]] -->
A non-degenerate (''S''-state) atom can have only a zero permanent dipole. This fact follows quantum mechanically from the inversion symmetry of atoms. All 3 components of the dipole operator are antisymmetric under [[Inversion in a point|inversion]] with respect to the nucleus,
<math display="block"> \mathfrak{I} \;\mathfrak{p}\;  \mathfrak{I}^{-1} = -\mathfrak{p}, </math>
where <math>\mathfrak{p}</math> is the dipole operator  and <math>\mathfrak{I}</math> is the inversion operator.
 
The permanent dipole moment  of an atom in a non-degenerate state (see ''[[Degenerate energy level]]'') is given as the expectation (average) value of the dipole operator,
<math display="block">\left\langle \mathfrak{p} \right\rangle = \left\langle S\right| \mathfrak{p} \left| S \right\rangle,</math>
where <math> \left| S \right\rangle </math> is an ''S''-state, non-degenerate, wavefunction, which is symmetric or anti-symmetric under inversion: <math> \mathfrak{I} \left | S \right\rangle = \pm \left| S \right\rangle</math>. Since the product of the wavefunction (in the ket) and its [[complex conjugate]] (in the bra) is always symmetric under inversion and its inverse,
<math display="block">
  \left\langle \mathfrak{p} \right\rangle =
  \left\langle \mathfrak{I}^{-1} S \right| \mathfrak{p} \left| \mathfrak{I}^{-1} S \right\rangle =
  \left\langle S \right | \mathfrak{I}\, \mathfrak{p}\, \mathfrak{I}^{-1} \left| S \right\rangle =
  - \left\langle \mathfrak{p} \right\rangle
</math>
it follows that the expectation value changes sign under inversion.  We used here the fact that <math> \mathfrak{I}</math>, being a symmetry operator, is [[unitary operator|unitary]]: <math> \mathfrak{I}^{-1} =  \mathfrak{I}^{*}\,</math> and [[Hermitian adjoint#Definition for bounded operators between Hilbert spaces|by definition]] the Hermitian adjoint <math> \mathfrak{I}^*\,</math> may be moved from bra to ket and then becomes <math> \mathfrak{I}^{**} = \mathfrak{I}\,</math>. Since the only quantity that is equal to minus itself is the zero, the expectation  value vanishes,
<math display="block">\left\langle \mathfrak{p} \right\rangle = 0.</math>
 
In the case of open-shell atoms with degenerate  energy levels, one could define a dipole moment by the aid of the first-order [[Stark effect]]. This gives a non-vanishing dipole (by definition proportional to a non-vanishing first-order Stark shift) only if some of the wavefunctions belonging to the degenerate energies have opposite [[parity (physics)|parity]]; i.e., have different behavior under inversion. This is a rare occurrence, but happens for the excited H-atom, where 2s and 2p states are "accidentally" degenerate (see article ''[[Laplace–Runge–Lenz vector#Quantum mechanics of the hydrogen atom|Laplace–Runge–Lenz vector]]'' for the origin of this degeneracy) and have opposite parity (2s is even and 2p is odd).
 
== Atomic Dipole Moment in QTAIM ==
In the Quantum Theory of Atoms in Molecules (QTAIM)<ref>[[Richard Bader|Bader RFW]], ''Atoms in Molecules: A Quantum Theory'', Oxford University Press, 1990, ISBN 978-0-19-855865-1.</ref><ref>Popelier PLA, ''Atoms in Molecules: An Introduction'', Prentice Hall, 2000.</ref>, an atomic dipole moment is defined from the electron density by partitioning a molecule or crystal into atomic basins bounded by zero-flux surfaces in the gradient vector field of the electron density. The atomic dipole consists of two contributions: (1) the intrinsic polarization of the atomic basin itself, corresponding to the displacement of the electronic charge centroid relative to the nucleus, and (2) a charge-transfer contribution arising from the transfer of electronic charge across interatomic surfaces shared with neighboring atoms. In this manner, the total molecular or crystalline dipole moment is exactly additive, being equal to the sum of all atomic contributions, thereby providing a real-space decomposition of polarization in molecules and solids<ref>Bader RFW, "Dielectric polarization: a problem in the physics of an open system", ''Molecular Physics'', 100(21), 3333–3344, 2002, doi:10.1080/0026897021000014901.</ref><ref>Bader RFW and Matta CF, "Properties of atoms in crystals: Dielectric polarization", ''International Journal of Quantum Chemistry'', 85, 592–607, 2001, doi:10.1002/qua.1540.</ref>.


== Dipole radiation ==
== Dipole radiation ==
Line 197: Line 207:
In addition to dipoles in electrostatics, it is also common to consider an electric or magnetic dipole that is oscillating in time. It is an extension, or a more physical next-step, to [[spherical wave]] radiation.
In addition to dipoles in electrostatics, it is also common to consider an electric or magnetic dipole that is oscillating in time. It is an extension, or a more physical next-step, to [[spherical wave]] radiation.


In particular, consider a harmonically oscillating electric dipole, with [[angular frequency]] ''ω'' and a dipole moment ''p''<sub>0</sub> along the '''ẑ''' direction of the form
In particular, consider a harmonically oscillating electric dipole, with [[angular frequency]] {{mvar|ω}} and a dipole moment {{math|''p''<sub>0</sub>}} along the {{math|'''ẑ'''}} direction of the form
: <math>\mathbf{p}(\mathbf{r}, t) = \mathbf{p}(\mathbf{r})e^{-i\omega t} = p_0\hat{\mathbf{z}}e^{-i\omega t} .</math>
<math display="block">\mathbf{p}(\mathbf{r}, t) = \mathbf{p}(\mathbf{r})e^{-i\omega t} = p_0\hat{\mathbf{z}}e^{-i\omega t} .</math>


In vacuum, the exact field produced by this oscillating dipole can be derived using the [[retarded potential]] formulation as:
In vacuum, the exact field produced by this oscillating dipole can be derived using the [[retarded potential]] formulation as:
: <math>\begin{align}
<math display="block">\begin{align}
   \mathbf{E} &= \frac{1}{4\pi\varepsilon_0} \left\{
   \mathbf{E} &= \frac{1}{4\pi\varepsilon_0} \left[
       \frac{\omega^2}{c^2 r} \left( \hat{\mathbf{r}} \times \mathbf{p} \right) \times \hat{\mathbf{r}} +
       \frac{\omega^2}{c^2 r} \left( \hat{\mathbf{r}} \times \mathbf{p} \right) \times \hat{\mathbf{r}} +
       \left( \frac{1}{r^3} - \frac{i\omega}{cr^2} \right)
       \left( \frac{1}{r^3} - \frac{i\omega}{cr^2} \right)
         \left( 3\hat{\mathbf{r}} \left[\hat{\mathbf{r}} \cdot \mathbf{p}\right] - \mathbf{p} \right)
         \left( 3\hat{\mathbf{r}} \left[\hat{\mathbf{r}} \cdot \mathbf{p}\right] - \mathbf{p} \right)
     \right\} e^\frac{i\omega r}{c} e^{-i\omega t} \\
     \right] e^{i\omega (r/c-t)} \\[1ex]
   \mathbf{B} &= \frac{\omega^2}{4\pi\varepsilon_0 c^3} (\hat{\mathbf{r}} \times \mathbf{p}) \left( 1 - \frac{c}{i\omega r} \right) \frac{e^{i\omega r/c}}{r} e^{-i\omega t}.
   \mathbf{B} &= \frac{\omega^2}{4\pi\varepsilon_0 c^3} (\hat{\mathbf{r}} \times \mathbf{p}) \left( 1 - \frac{c}{i\omega r} \right) \frac{e^{i\omega (r/c-t)}}{r}.
\end{align}</math>
\end{align}</math>


For {{sfrac|''rω''|''c''}}&nbsp;≫&nbsp;1, the far-field takes the simpler form of a radiating "spherical" wave, but with angular dependence embedded in the cross-product:<ref>[[David J. Griffiths]], Introduction to Electrodynamics, Prentice Hall, 1999, page 447</ref>
For {{math|{{sfrac|''rω''|''c''}} ≫ 1}}, the far-field takes the simpler form of a radiating "spherical" wave, but with angular dependence embedded in the cross-product:<ref>[[David J. Griffiths]], Introduction to Electrodynamics, Prentice Hall, 1999, page 447</ref>
: <math>\begin{align}
<math display="block">\begin{align}
   \mathbf{B}
   \mathbf{B}
     &=  \frac{\omega^2}{4\pi\varepsilon_0 c^3} (\hat{\mathbf{r}} \times \mathbf{p}) \frac{e^{i\omega (r/c - t)}}{r}
     &=  \frac{\omega^2}{4\pi\varepsilon_0 c^3} (\hat{\mathbf{r}} \times \mathbf{p}) \frac{e^{i\omega (r/c - t)}}{r} \\[0.4ex]
    =  \frac{\omega^2 \mu_0 p_0 }{4\pi c} (\hat{\mathbf{r}} \times \hat{\mathbf{z}}) \frac{e^{i\omega (r/c - t)}}{r}
    &=  \frac{\omega^2 \mu_0 p_0 }{4\pi c} (\hat{\mathbf{r}} \times \hat{\mathbf{z}}) \frac{e^{i\omega (r/c - t)}}{r}
     = -\frac{\omega^2 \mu_0 p_0 }{4\pi c} \sin(\theta) \frac{e^{i\omega (r/c - t)}}{r} \mathbf{\hat{\phi}} \\
     = -\frac{\omega^2 \mu_0 p_0 }{4\pi c} \sin(\theta) \frac{e^{i\omega (r/c - t)}}{r} \hat{\boldsymbol\phi} \\[1ex]
   \mathbf{E}
   \mathbf{E}
     &=  c \mathbf{B} \times \hat{\mathbf{r}}
     &=  c \mathbf{B} \times \hat{\mathbf{r}} \\[0.4ex]
    = -\frac{\omega^2 \mu_0 p_0}{4\pi} \sin(\theta) \left(\hat{\phi} \times \mathbf{\hat{r}}\right) \frac{e^{i\omega (r/c - t)}}{r}
    &= -\frac{\omega^2 \mu_0 p_0}{4\pi} \sin(\theta) \left(\hat{\boldsymbol\phi} \times \mathbf{\hat{r}}\right) \frac{e^{i\omega (r/c - t)}}{r}
     = -\frac{\omega^2 \mu_0 p_0}{4\pi} \sin(\theta) \frac{e^{i\omega (r/c - t)}}{r} \hat{\theta}.
     = -\frac{\omega^2 \mu_0 p_0}{4\pi} \sin(\theta) \frac{e^{i\omega (r/c - t)}}{r} \hat{\boldsymbol\theta}.
\end{align}</math>
\end{align}</math>


The time-averaged [[Poynting vector]]
The time-averaged [[Poynting vector]]
: <math>\langle \mathbf{S} \rangle = \left(\frac{\mu_0 p_0^2\omega^4}{32\pi^2 c}\right) \frac{\sin^2(\theta)}{r^2} \mathbf{\hat{r}}</math>
<math display="block">\langle \mathbf{S} \rangle = \frac{\mu_0 p_0^2\omega^4}{32\pi^2 c} \, \frac{\sin^2(\theta)}{r^2} \mathbf{\hat{r}}</math>
is not distributed isotropically, but concentrated around the directions lying perpendicular to the dipole moment, as a result of the non-spherical electric and magnetic waves. In fact, the [[spherical harmonic]] function (sin ''θ'') responsible for such [[torus|toroidal]] angular distribution is precisely the ''l''&nbsp;=&nbsp;1 "p" wave.
is not distributed isotropically, but concentrated around the directions lying perpendicular to the dipole moment, as a result of the non-spherical electric and magnetic waves. In fact, the [[spherical harmonic]] function (sin ''θ'') responsible for such [[torus|toroidal]] angular distribution is precisely the ''l''&nbsp;=&nbsp;1 "p" wave.


The total time-average power radiated by the field can then be derived from the Poynting vector as
The total time-average power radiated by the field can then be derived from the Poynting vector as
: <math>P = \frac{\mu_0 \omega^4 p_0^2}{12\pi c}.</math>
<math display="block">P = \frac{\mu_0 \omega^4 p_0^2}{12\pi c}.</math>


Notice that the dependence of the power on the fourth power of the frequency of the radiation is in accordance with the [[Rayleigh scattering]], and the underlying effects why the sky consists of mainly blue colour.
Notice that the dependence of the power on the fourth power of the frequency of the radiation is in accordance with the [[Rayleigh scattering]], and the underlying effects why the sky consists of mainly blue colour.
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== See also ==
== See also ==
{{div col}}
* [[Polarization density]]
* [[Polarization density]]
* [[Magnet#Two models for magnets: magnetic poles and atomic currents|Magnetic dipole models]]
* [[Magnet#Two models for magnets: magnetic poles and atomic currents|Magnetic dipole models]]
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* [[Laplace expansion (potential)|Laplace expansion]]
* [[Laplace expansion (potential)|Laplace expansion]]
* [[Molecular solid]]
* [[Molecular solid]]
* [[Magnetic moment#Internal magnetic field of a dipole]]
* {{slink|Magnetic moment#Internal magnetic field of a dipole}}
{{div col end}}


== Notes ==
== Notes ==
Retrieved from "https://wiki.tachyony.co.uk/wiki/Dipole"