Dipole
Overview
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, there are several kinds of dipoles:
- An electric dipoleElectric dipole momentIn physics, the electric dipole moment is a measure of the separation of positive and negative electrical charges in a system of charges, that is, a measure of the charge system's overall polarity with SI units of Coulomb-meter...
is a separation of positive and negative charges. The simplest example of this is a pair of electric chargeElectric chargeElectric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
s of equal magnitude but opposite sign, separated by some (usually small) distance. A permanent electric dipole is called an electretElectretElectret is a dielectric material that has a quasi-permanent electric charge or dipole polarisation. An electret generates internal and external electric fields, and is the electrostatic equivalent of a permanent magnet. Oliver Heaviside coined this term in 1885...
. - A magnetic dipole is a closed circulation of electric currentElectric currentElectric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
. A simple example of this is a single loop of wire with some constant current flowing through it. - A flow dipole is a separation of a sink and a source.
Unanswered Questions
Encyclopedia
In physics
, there are several kinds of dipoles:
Dipoles can be characterized by their dipole moment, a vector quantity. For the simple electric dipole given above, 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 e.g. 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.)
For the 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.
In addition to current loops, the electron
, among other fundamental particles, has a magnetic dipole moment. This is because it generates a magnetic field
that is identical to that generated by a very small current loop. However, to the best of our knowledge, the electron's magnetic moment is not due to a current loop, but is instead an intrinsic property of the electron. It is also possible that the electron has an electric dipole moment, although this has not yet been observed (see electron electric dipole moment
for more information).
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 monopoles
), and are labeled "north" and "south." The dipole moment of the bar magnet points from its magnetic south
to its magnetic north pole
. The north pole of a bar magnet in a compass
points north. However, this means that Earth's geomagnetic north pole is the south pole of its dipole moment, and vice versa.
The only known mechanisms for the creation of magnetic dipoles are by current loops or quantum-mechanical spin
since the existence of magnetic monopole
s has never been experimentally demonstrated.
The term comes from the Greek
δίς (dis), "twice" and πόλος (pòlos), "axis".
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
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.
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 1/r3, as compared to 1/r4 for the next (quadrupole) term and higher powers of 1/r for higher terms, or 1/r2 for the monopole term.
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 which should not be confused with a magnetic dipole
which generates a magnetic field.
A molecule with a permanent dipole moment is called a polar molecule. A molecule is polarized when it carries an induced dipole. The physical chemist Peter J. W. Debye
was the first scientist to study molecular dipoles extensively, and, as a consequence, dipole moments are measured in units named debye
in his honor.
With respect to molecules, there are three types of dipoles:
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 is equal to the product of the strength of the
external field and the dipole polarizability
of ρ.
Typical gas phase values of some chemical compounds in debye
units:
These values can be obtained from measurement of the dielectric constant
. When the symmetry of a molecule cancels out a net dipole moment, the value is set at 0. The highest dipole moments are in the range of 10 to 11. From the dipole moment information can be deduced about the molecular geometry
of the molecule. For example the data illustrate that carbon dioxide is a linear molecule but ozone is not.
of the i th nucleus.
The physical quantity (observable) dipole has the quantum mechanical dipole operator:
where is the dipole operator and
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,
where
is an S-state, non-degenerate, wavefunction, which
is symmetric or antisymmetric under inversion:
.
Since the product of the wavefunction (in the ket) and its complex conjugate (in the bra) is always symmetric under inversion and its inverse,
it follows that the expectation value changes sign under inversion. We used here the fact that
, being a symmetry operator, is unitary
:
and by definition
the Hermitian adjoint
may be moved from bra to ket and then becomes 
.
Since the only quantity that is equal to minus itself is the zero, the expectation value vanishes,
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
; 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 this article for the origin of this degeneracy) and have opposite parity (2s is even and 2p is odd).
where
Conversion to cylindrical coordinates is achieved using and
where ρ is the perpendicular distance from the z-axis. Then,
where
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.
A of a magnetic dipole is
with the same definitions as above.
where
is a unit vector in the direction of r, p is the (vector) dipole moment
, and ε0 is the permittivity of free space.
This term appears as the second term in the 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:
where E is the electric field and δ3 is the 3-dimensional delta function
. This is formally identical to the magnetic field of a point magnetic dipole; only a few names have changed.
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 an electric
or magnetic field
, equal but opposite force
s arise on each side of the dipole creating a torque
τ:
for an electric dipole moment p (in coulomb-meters), or
for a magnetic dipole moment 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 energy of a magnetic dipole is similarly
.
In addition to dipoles in electrostatics, it is also common to consider an electric or magnetic dipole that is oscillating in time.
In particular, a harmonically oscillating electric dipole is described by a dipole moment of the form
where ω is the angular frequency
. In vacuum, this produces fields:
Far away (for
), the fields approach the limiting form of a radiating spherical wave:
which produces a total time-average radiated power P given by
This power is not distributed isotropically, but is rather concentrated around the directions lying perpendicular to the dipole moment.
Usually such equations are described by spherical harmonics
, but they look very different.
A circular polarized dipole is described as a superposition of two linear dipoles.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, there are several kinds of dipoles:
- An electric dipoleElectric dipole momentIn physics, the electric dipole moment is a measure of the separation of positive and negative electrical charges in a system of charges, that is, a measure of the charge system's overall polarity with SI units of Coulomb-meter...
is a separation of positive and negative charges. The simplest example of this is a pair of electric chargeElectric chargeElectric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
s of equal magnitude but opposite sign, separated by some (usually small) distance. A permanent electric dipole is called an electretElectretElectret is a dielectric material that has a quasi-permanent electric charge or dipole polarisation. An electret generates internal and external electric fields, and is the electrostatic equivalent of a permanent magnet. Oliver Heaviside coined this term in 1885...
. - A magnetic dipole is a closed circulation of electric currentElectric currentElectric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
. A simple example of this is a single loop of wire with some constant current flowing through it. - A flow dipole is a separation of a sink and a source. In a highly viscous medium, a two-beater kitchen mixer causes a dipole flow field.
- An acoustic dipole is the oscillating version of it. A simple example is a dipole speakerDipole speakerThis article is about loudspeaker systems designed such that one approximate point-source emits sound equally in two opposing directions, 180 degrees out of phase. For discussion of "stereo dipole" speakers, see stereo dipole and ambiophonics....
. - Any scalar or other field may have a dipole momentMultipole expansionA multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original...
.
Dipoles can be characterized by their dipole moment, a vector quantity. For the simple electric dipole given above, the electric dipole moment
Electric dipole moment
In physics, the electric dipole moment is a measure of the separation of positive and negative electrical charges in a system of charges, that is, a measure of the charge system's overall polarity with SI units of Coulomb-meter...
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 e.g. 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.)
For the 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.
In addition to current loops, the electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
, among other fundamental particles, has a magnetic dipole moment. This is because it generates a magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
that is identical to that generated by a very small current loop. However, to the best of our knowledge, the electron's magnetic moment is not due to a current loop, but is instead an intrinsic property of the electron. It is also possible that the electron has an electric dipole moment, although this has not yet been observed (see electron electric dipole moment
Electron electric dipole moment
The electron electric dipole moment de is an intrinsic property of an electron such that the potential energy is linearly related to the strength of the electric field: U=de·E. Within the standard model of elementary particle physics, such a dipole is predicted to be non-zero but very small, at...
for more information).
Magnetic monopole
A magnetic monopole is a hypothetical particle in particle physics that is a magnet with only one magnetic pole . In more technical terms, a magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring...
), and are labeled "north" and "south." The dipole moment of the bar magnet points from its magnetic south
South Pole
The South Pole, also known as the Geographic South Pole or Terrestrial South Pole, is one of the two points where the Earth's axis of rotation intersects its surface. It is the southernmost point on the surface of the Earth and lies on the opposite side of the Earth from the North Pole...
to its magnetic north pole
North Pole
The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is, subject to the caveats explained below, defined as the point in the northern hemisphere where the Earth's axis of rotation meets its surface...
. The north pole of a bar magnet in a compass
Compass
A compass is a navigational instrument that shows directions in a frame of reference that is stationary relative to the surface of the earth. The frame of reference defines the four cardinal directions – north, south, east, and west. Intermediate directions are also defined...
points north. However, this means that Earth's geomagnetic north pole is the south pole of its dipole moment, and vice versa.
The only known mechanisms for the creation of magnetic dipoles are by current loops or quantum-mechanical spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
since the existence of magnetic monopole
Magnetic monopole
A magnetic monopole is a hypothetical particle in particle physics that is a magnet with only one magnetic pole . In more technical terms, a magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring...
s has never been experimentally demonstrated.
The term comes from the Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
δίς (dis), "twice" and πόλος (pòlos), "axis".
Classification
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 expansionMultipole expansion
A multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original...
is precisely the point dipole field.
Although there are no known magnetic monopole
Magnetic monopole
A magnetic monopole is a hypothetical particle in particle physics that is a magnet with only one magnetic pole . In more technical terms, a magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring...
s in nature, there are magnetic dipoles in the form of the quantum-mechanical spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
associated with particles such as electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
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.
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 1/r3, as compared to 1/r4 for the next (quadrupole) term and higher powers of 1/r for higher terms, or 1/r2 for the monopole term.
Molecular dipoles
Many moleculeMolecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
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
Chemical polarity
In chemistry, polarity refers to a separation of electric charge leading to a molecule or its chemical groups having an electric dipole or multipole moment. Polar molecules interact through dipole–dipole intermolecular forces and hydrogen bonds. Molecular polarity is dependent on the difference in...
compounds like hydrogen fluoride
Hydrogen fluoride
Hydrogen fluoride is a chemical compound with the formula HF. This colorless gas is the principal industrial source of fluorine, often in the aqueous form as hydrofluoric acid, and thus is the precursor to many important compounds including pharmaceuticals and polymers . HF is widely used in the...
(HF), where electron density
Electron density
Electron density is the measure of the probability of an electron being present at a specific location.In molecules, regions of electron density are usually found around the atom, and its bonds...
is shared unequally between atoms. Therefore, a molecule's dipole is an electric dipole with an inherent electric field which should not be confused with a magnetic dipole
Magnetic dipole
A magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the dimensions of the source are reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric dipole, but the analogy is not complete. In particular, a magnetic...
which generates a magnetic field.
A molecule with a permanent dipole moment is called a polar molecule. A molecule is polarized when it carries an induced dipole. The physical chemist Peter J. W. Debye
Peter Debye
Peter Joseph William Debye FRS was a Dutch physicist and physical chemist, and Nobel laureate in Chemistry.-Early life:...
was the first scientist to study molecular dipoles extensively, and, as a consequence, dipole moments are measured in units named debye
Debye
The debye is a CGS unit of electric dipole momentElectric dipole moment is defined as charge times displacement: Historically the debye was defined as the dipole moment resulting from two charges of opposite sign but an equal magnitude of 10-10 statcoulomb10-10 statcoulomb is approximately 0.2083...
in his honor.
With respect to molecules, there are three types of dipoles:
- Permanent dipoles: These occur when two atoms in a molecule have substantially different electronegativityElectronegativityElectronegativity, symbol χ , is a chemical property that describes the tendency of an atom or a functional group to attract electrons towards itself. An atom's electronegativity is affected by both its atomic number and the distance that its valence electrons reside from the charged nucleus...
: One atom attracts electrons more than another, becoming more negative, while the other atom becomes more positive. See dipole-dipole attractions. - Instantaneous dipoles: These occur due to chance when electronElectronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s happen to be more concentrated in one place than another in a moleculeMoleculeA molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
, creating a temporary dipole. See instantaneous dipoleLondon dispersion forceLondon dispersion forces is a type of force acting between atoms and molecules. They are part of the van der Waals forces...
. - Induced dipoles: These can occur when one molecule with a permanent dipole repels another molecule's electrons, "inducing" a dipole moment in that molecule. See 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
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
). The size of the induced dipole is equal to the product of the strength of the
external field and the dipole polarizability
Polarizability
Polarizability is the measure of the change in a molecule's electron distribution in response to an applied electric field, which can also be induced by electric interactions with solvents or ionic reagents. It is a property of matter...
of ρ.
Typical gas phase values of some chemical compounds in debye
Debye
The debye is a CGS unit of electric dipole momentElectric dipole moment is defined as charge times displacement: Historically the debye was defined as the dipole moment resulting from two charges of opposite sign but an equal magnitude of 10-10 statcoulomb10-10 statcoulomb is approximately 0.2083...
units:
- carbon dioxideCarbon dioxideCarbon dioxide is a naturally occurring chemical compound composed of two oxygen atoms covalently bonded to a single carbon atom...
: 0 - carbon monoxideCarbon monoxideCarbon monoxide , also called carbonous oxide, is a colorless, odorless, and tasteless gas that is slightly lighter than air. It is highly toxic to humans and animals in higher quantities, although it is also produced in normal animal metabolism in low quantities, and is thought to have some normal...
: 0.112 - ozoneOzoneOzone , or trioxygen, is a triatomic molecule, consisting of three oxygen atoms. It is an allotrope of oxygen that is much less stable than the diatomic allotrope...
: 0.53 - phosgenePhosgenePhosgene is the chemical compound with the formula COCl2. This colorless gas gained infamy as a chemical weapon during World War I. It is also a valued industrial reagent and building block in synthesis of pharmaceuticals and other organic compounds. In low concentrations, its odor resembles...
: 1.17 - water vaporWater vaporWater vapor or water vapour , also aqueous vapor, is the gas phase of water. It is one state of water within the hydrosphere. Water vapor can be produced from the evaporation or boiling of liquid water or from the sublimation of ice. Under typical atmospheric conditions, water vapor is continuously...
: 1.85 - hydrogen cyanide: 2.98
- cyanamideCyanamideCyanamide is an organic compound with the formula CN2H2. This white solid is widely used in agriculture and the production of pharmaceuticals and other organic compounds. It is also used as an alcohol deterrent drug in Canada, Europe and Japan. The molecule features a nitrile group attached to an...
: 4.27 - potassium bromidePotassium bromidePotassium bromide is a salt, widely used as an anticonvulsant and a sedative in the late 19th and early 20th centuries, with over-the-counter use extending to 1975 in the United States. Its action is due to the bromide ion...
: 10.41
These values can be obtained from measurement of the dielectric constant
Dielectric constant
The relative permittivity of a material under given conditions reflects the extent to which it concentrates electrostatic lines of flux. In technical terms, it is the ratio of the amount of electrical energy stored in a material by an applied voltage, relative to that stored in a vacuum...
. When the symmetry of a molecule cancels out a net dipole moment, the value is set at 0. The highest dipole moments are in the range of 10 to 11. From the dipole moment information can be deduced about the molecular geometry
Molecular geometry
Molecular geometry or molecular structure is the three-dimensional arrangement of the atoms that constitute a molecule. It determines several properties of a substance including its reactivity, polarity, phase of matter, color, magnetism, and biological activity.- Molecular geometry determination...
of the molecule. For example the data illustrate that carbon dioxide is a linear molecule but ozone is not.
Quantum mechanical dipole operator
Consider a collection of N particles with charges qi and position vectors ri. For instance, this collection may be a molecule consisting of electrons, all with charge −e, and nuclei with charge eZi, where Zi is the atomic numberAtomic number
In chemistry and physics, the atomic number is the number of protons found in the nucleus of an atom and therefore identical to the charge number of the nucleus. It is conventionally represented by the symbol Z. The atomic number uniquely identifies a chemical element...
of the i th nucleus.
The physical quantity (observable) dipole has the quantum mechanical dipole operator:
Atomic dipoles
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 with respect to the nucleus,where is the dipole operator and
is the inversion operator.The permanent dipole moment of an atom in a non-degenerate state (see degenerate energy level
Degenerate energy level
In physics, two or more different quantum states are said to be degenerate if they are all at the same energy level. Statistically this means that they are all equally probable of being filled, and in Quantum Mechanics it is represented mathematically by the Hamiltonian for the system having more...
) is given as the expectation (average) value of the dipole operator,
where
is an S-state, non-degenerate, wavefunction, whichis symmetric or antisymmetric under inversion:
.Since the product of the wavefunction (in the ket) and its complex conjugate (in the bra) is always symmetric under inversion and its inverse,
it follows that the expectation value changes sign under inversion. We used here the fact that
, being a symmetry operator, is unitaryUnitary operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...
:
and by definitionthe Hermitian adjoint
may be moved from bra to ket and then becomes .Since the only quantity that is equal to minus itself is the zero, the expectation value vanishes,
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
Stark effect
The Stark effect is the shifting and splitting of spectral lines of atoms and molecules due to presence of an external static electric field. The amount of splitting and or shifting is called the Stark splitting or Stark shift. In general one distinguishes first- and second-order Stark effects...
. 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
Parity (physics)
In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...
; 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 this article for the origin of this degeneracy) and have opposite parity (2s is even and 2p is odd).
Field of a static magnetic dipole
Magnitude
The far-field strength, B, of a dipole magnetic field is given bywhere
- B is the strength of the field, measured in teslaTesla (unit)The tesla is the SI derived unit of magnetic field B . One tesla is equal to one weber per square meter, and it was defined in 1960 in honour of the inventor, physicist, and electrical engineer Nikola Tesla...
s - r is the distance from the center, measured in metres
- λ is the magnetic latitude (equal to 90° − θ) where θ is the magnetic colatitude, measured in radianRadianRadian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s or degreeDegree (angle)A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...
s from the dipole axisMagnetic colatitude is 0 along the dipole's axis and 90° in the plane perpendicular to its axis. - m is the dipole moment (VADM=virtual axial dipole moment), measured in ampere square-metres (A·m2), which equals jouleJouleThe joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...
s per teslaTesla (unit)The tesla is the SI derived unit of magnetic field B . One tesla is equal to one weber per square meter, and it was defined in 1960 in honour of the inventor, physicist, and electrical engineer Nikola Tesla... - μ0 is the permeability of free spacePermeability (electromagnetism)In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...
, measured in henries per metre.
Conversion to cylindrical coordinates is achieved using and
where ρ is the perpendicular distance from the z-axis. Then,
Vector form
The field itself is a vector quantity: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
is the unit vector parallel to r; - m is the (vector) dipole moment
- μ0 is the permeability of free space
- δ3 is the three-dimensional delta functionDirac delta functionThe Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
.δ3(r) = 0 except at , so this term is ignored in multipole expansion.
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.
Magnetic vector potential
The vector potentialVector potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field....
A of a magnetic dipole is
with the same definitions as above.
Field from an electric dipole
The electrostatic potential at position r due to an electric dipole at the origin is given by:where
is a unit vector in the direction of r, p is the (vector) dipole momentElectric dipole moment
In physics, the electric dipole moment is a measure of the separation of positive and negative electrical charges in a system of charges, that is, a measure of the charge system's overall polarity with SI units of Coulomb-meter...
, and ε0 is the permittivity of free space.
This term appears as the second term in the 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
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
from a dipole can be found from the gradient
Gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
of this potential:
where E is the electric field and δ3 is the 3-dimensional delta function
Dirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
. This is formally identical to the magnetic field of a point magnetic dipole; only a few names have changed.
Torque on a dipole
Since the direction of an electric fieldElectric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
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 an electric
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
or magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
, equal but opposite force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
s arise on each side of the dipole creating a torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
τ:
for an electric dipole moment p (in coulomb-meters), or
for a magnetic dipole moment 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 energy of a magnetic dipole is similarly
.Dipole radiation
In addition to dipoles in electrostatics, it is also common to consider an electric or magnetic dipole that is oscillating in time.
In particular, a harmonically oscillating electric dipole is described by a dipole moment of the form
where ω is the angular frequency
Angular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
. In vacuum, this produces fields:
Far away (for
), the fields approach the limiting form of a radiating spherical wave:which produces a total time-average radiated power P given by
This power is not distributed isotropically, but is rather concentrated around the directions lying perpendicular to the dipole moment.
Usually such equations are described by spherical harmonics
Spherical harmonics
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...
, but they look very different.
A circular polarized dipole is described as a superposition of two linear dipoles.
See also
- Charge polarization
- Magnetic dipole models
- Dipole model of the Earth's magnetic fieldDipole model of the Earth's magnetic fieldThe dipole model of the Earth's magnetic field is a first order approximation of the rather complex true Earth's magnetic field. Due to effects of the interplanetary magnetic field, and the solar wind, the dipole model is particularly inaccurate at high L-shells , but may be a good approximation...
- ElectretElectretElectret is a dielectric material that has a quasi-permanent electric charge or dipole polarisation. An electret generates internal and external electric fields, and is the electrostatic equivalent of a permanent magnet. Oliver Heaviside coined this term in 1885...
- Indian Ocean DipoleIndian Ocean DipoleThe Indian Ocean Dipole is an irregular oscillation of sea-surface temperatures in which the western Indian Ocean becomes alternately warmer and then colder than the eastern part of the ocean.-The phenomenon:...
(an oceanographic phenomenon) - Magnetic dipole-dipole interactionMagnetic dipole-dipole interactionMagnetic dipole–dipole interaction, also called dipolar coupling, refers to the direct interaction between two magnetic dipoles. The potential energy of the interaction is as follows:...
- Spin magnetic momentSpin magnetic moment-Basis for spin magnetic moments:A spin magnetic moment is induced by all charged particles. The electron is an example of one such charged particle. A spin magnetic moment is created because a particle has physical properties known as spin and electric charge. The spin within classical physics...
- MonopoleMonopoleMonopole may refer to:*Magnetic monopole, or Dirac monopole, a hypothetical particle that may be loosely described as a magnet with only one pole, or related concepts in physics and mathematics:...
- Solid harmonics
- Axial multipole momentsAxial multipole momentsAxial multipole moments are a series expansionof the electric potential of acharge distribution localized close tothe origin along oneCartesian axis,denoted here as the z-axis...
- Cylindrical multipole momentsCylindrical multipole momentsCylindrical multipole moments are the coefficients in a series expansion of a potential that varies logarithmically with the distance to a source, i.e., as \ln \ R...
- Spherical multipole momentsSpherical multipole momentsSpherical multipole moments are the coefficients in a series expansionof a potential that varies inversely with the distance R to a source, i.e., as 1/R...
- Laplace expansionLaplace expansion (potential)In physics, the Laplace expansion of a 1/r - type potential is applied to expand Newton's gravitational potential or Coulomb's electrostatic potential. In quantum mechanical calculations on atoms the expansion is used in the evaluation of integrals of the interelectronic repulsion.The Laplace...
- Molecular solidMolecular solidMolecular solid is a solid composed of molecules held together by the van der Waals forces. Because these dipole forces are weaker than covalent or ionic bonds, molecular solids are soft and have relatively low melting temperature. Pure molecular solids are electrical insulators but they can be...
External links
- An interactive JAVA applet displaying the behavior of two-dimensional dipoles.
- USGS Geomagnetism Program
- Fields of Force: a chapter from an online textbook
- Electric Dipole Potential by Stephen WolframStephen WolframStephen Wolfram is a British scientist and the chief designer of the Mathematica software application and the Wolfram Alpha computational knowledge engine.- Biography :...
and Energy Density of a Magnetic Dipole by Franz Krafft. Wolfram Demonstrations ProjectWolfram Demonstrations ProjectThe Wolfram Demonstrations Project is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. It consists of an organized, open-source collection of small interactive programs called Demonstrations, which are meant to visually and...
.