Dipole

This article is about the electromagnetic phenomenon. For other uses, seedipole (disambiguation).

Inphysics,adipole(fromAncient Greekδίς(dís)'twice', andπόλος(pólos)'axis')is anelectromagneticphenomenon which occurs in two ways:

• Anelectric dipoledeals with the separation of the positive and negativeelectric chargesfound 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 anelectret.)
• Amagnetic dipoleis the closed circulation of anelectric currentsystem. A simple example is a single loop of wire with constant current through it. Abar magnetis an example of a magnet with a permanentmagnetic dipole moment.

The magnetic field of a sphere with a north magnetic pole at the top and a south magnetic pole at the bottom. By comparison,Earthhas asouthmagnetic pole near its north geographic pole and anorthmagnetic pole near its South Pole.

Dipoles, whether electric or magnetic, can be characterized by their dipole moment, a vector quantity. For the simple electric dipole, theelectric dipole momentpoints 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 shouldconvergeto 0 while simultaneously, the charge strength shoulddivergeto infinity in such a way that the product remains a positive constant.)

For the magnetic (dipole) current loop, themagnetic dipole momentpoints through the loop (according to theright hand grip rule), with a magnitude equal to the current in the loop times the area of the loop.

Similar to magnetic current loops, theelectronparticle and some otherfundamental particleshave magnetic dipole moments, as an electron generates amagnetic fieldidentical 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 anintrinsicproperty of the electron.The electron may also have anelectricdipole moment though such has yet to be observed (seeelectron electric dipole moment).

Contour plot of theelectrostatic potentialof a horizontally oriented electrical dipole of infinitesimal size. Strong colors indicate highest and lowest potential (where the opposing charges of the dipole are located).

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 withmonopoles,seeClassificationbelow) 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 magneticsouthto its magneticnorth pole.In a magneticcompass,the north pole of a bar magnet points north. However, that means that Earth's geomagnetic north pole is thesouthpole (south-seeking pole) of its dipole moment and vice versa.

The only known mechanisms for the creation of magnetic dipoles are by current loops or quantum-mechanicalspinsince the existence ofmagnetic monopoleshas never been experimentally demonstrated.

Classification

Electric field lines of two opposing charges separated by a finite distance.

Magnetic field lines of a ring current of finite diameter.

Field lines of a point dipole of any type, electric, magnetic, acoustic, etc.

Aphysical dipoleconsists 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. Apoint (electric) dipoleis 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 themultipole expansionis precisely the point dipole field.

Although there are no knownmagnetic monopolesin nature, there are magnetic dipoles in the form of the quantum-mechanicalspinassociated with particles such aselectrons(although the accurate description of such effects falls outside of classical electromagnetism). A theoretical magneticpoint dipolehas 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 italwaysis 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 to⁠1/r³⁠,as compared to⁠1/r⁴⁠for the next (quadrupole) term and higher powers of⁠1/r⁠for higher terms, or⁠1/r²⁠for the monopole term.

Molecular dipoles

See also:Chemical polarityandDipole moments of molecules

Manymoleculeshave such dipole moments due to non-uniform distributions of positive and negative charges on the various atoms. Such is the case with polar compounds likehydrogen fluoride(HF), whereelectron densityis shared unequally between atoms. Therefore, a molecule's dipole is anelectric dipolewith an inherent electric field that should not be confused with amagnetic dipole,which generates a magnetic field.

The physical chemistPeter J. W. Debyewas the first scientist to study molecular dipoles extensively, and, as a consequence, dipole moments are measured in the non-SIunit nameddebyein his honor.

For molecules there are three types of dipoles:

Permanent dipoles

These occur when two atoms in a molecule have substantially differentelectronegativity: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 apolarmolecule. Seedipole–dipole attractions.

Instantaneous dipoles

These occur due to chance whenelectronshappen to be more concentrated in one place than another in amolecule,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. Seeinstantaneous dipole.

Induced dipoles

These can occur when one molecule with a permanent dipole repels another molecule's electrons,inducinga dipole moment in that molecule. A molecule ispolarizedwhen it carries an induced dipole. Seeinduced-dipole attraction.

More generally, an induced dipole ofanypolarizable charge distributionρ(remember that a molecule has a charge distribution) is caused by an electric field external toρ.This field may, for instance, originate from an ion or polar molecule in the vicinity ofρor may be macroscopic (e.g., a molecule between the plates of a chargedcapacitor). The size of the induced dipole moment is equal to the product of the strength of the external field and the dipolepolarizabilityofρ.

Dipole moment values can be obtained from measurement of thedielectric constant.Some typical gas phase values indebyeunits are:

• carbon dioxide:0
• carbon monoxide:0.112 D
• ozone:0.53 D
• phosgene:1.17 D
• ammonia:1.42 D
• water vapor:1.85 D
• hydrogen cyanide:2.98 D
• cyanamide:4.27 D
• potassium bromide:10.41 D

The linear molecule CO₂has a zero dipole as the two bond dipoles cancel.

Potassium bromide (KBr) has one of the highest dipole moments because it is anionic compoundthat exists as a molecule in the gas phase.

The bent molecule H₂O has a net dipole. The two bond dipoles do not cancel.

The overall dipole moment of a molecule may be approximated as avector sumofbond dipole moments.As a vector sum it depends on the relative orientation of the bonds, so that from the dipole moment information can be deduced about themolecular geometry.

For example, the zero dipole of CO₂implies that the two C=O bond dipole moments cancel so that the molecule must be linear. For H₂O the O−H bond moments do not cancel because the molecule is bent. For ozone (O₃) 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.

Resonance Lewis structures of the ozone molecule

Cisisomer, dipole moment 1.90 D

Transisomer, dipole moment zero

An example in organic chemistry of the role of geometry in determining dipole moment is thecisandtransisomersof1,2-dichloroethene.In thecisisomer 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 D. In thetransisomer, 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 isboron trifluoride,which has three polar bonds with a difference inelectronegativitygreater than the traditionally cited threshold of 1.7 forionic 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.

Quantum mechanical dipole operator

Consider a collection ofNparticles with chargesq_iand position vectorsr_i.For instance, this collection may be a molecule consisting of electrons, all withcharge−e,and nuclei with chargeeZ_i,whereZ_iis theatomic numberof thei th nucleus. The dipole observable (physical quantity) has the quantum mechanicaldipole operator:

${\mathfrak {p}}=\sum _{i=1}^{N}\,q_{i}\,\mathbf {r} _{i}\,.$

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

${\mathfrak {p}}=\sum _{i=1}^{N}\,q_{i}\,(\mathbf {r} _{i}-\mathbf {r} _{c}),$

where$\mathbf {r} _{c}$is the center of mass of the molecule/group of particles.

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 underinversionwith respect to the nucleus,

${\mathfrak {I}}\;{\mathfrak {p}}\;{\mathfrak {I}}^{-1}=-{\mathfrak {p}},$

where${\mathfrak {p}}$is the dipole operator and${\mathfrak {I}}$is the inversion operator.

The permanent dipole moment of an atom in a non-degenerate state (seedegenerate energy level) is given as the expectation (average) value of the dipole operator,

$\left\langle {\mathfrak {p}}\right\rangle =\left\langle \,S\,|{\mathfrak {p}}|\,S\,\right\rangle,$

where$|\,S\,\rangle$is anS-state, non-degenerate, wavefunction, which is symmetric or antisymmetric under inversion:${\mathfrak {I}}\,|\,S\,\rangle =\pm |\,S\,\rangle$.Since the product of the wavefunction (in the ket) and its complex conjugate (in the bra) is always symmetric under inversion and its inverse,

$\left\langle {\mathfrak {p}}\right\rangle =\left\langle \,{\mathfrak {I}}^{-1}\,S\,|{\mathfrak {p}}|\,{\mathfrak {I}}^{-1}\,S\,\right\rangle =\left\langle \,S\,|{\mathfrak {I}}\,{\mathfrak {p}}\,{\mathfrak {I}}^{-1}|\,S\,\right\rangle =-\left\langle {\mathfrak {p}}\right\rangle$

it follows that the expectation value changes sign under inversion. We used here the fact that${\mathfrak {I}}$,being a symmetry operator, isunitary:${\mathfrak {I}}^{-1}={\mathfrak {I}}^{*}\,$andby definitionthe Hermitian adjoint${\mathfrak {I}}^{*}\,$may be moved from bra to ket and then becomes${\mathfrak {I}}^{**}={\mathfrak {I}}\,$.Since the only quantity that is equal to minus itself is the zero, the expectation value vanishes,

$\left\langle {\mathfrak {p}}\right\rangle =0.$

In the case of open-shell atoms with degenerate energy levels, one could define a dipole moment by the aid of the first-orderStark 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 oppositeparity;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 articleLaplace–Runge–Lenz vectorfor the origin of this degeneracy) and have opposite parity (2s is even and 2p is odd).

Field of a static magnetic dipole

See also:Magnet § Two models for magnets: magnetic poles and atomic currents

Magnitude

The far-field strength,B,of a dipole magnetic field is given by

$B(m,r,\lambda )={\frac {\mu _{0}}{4\pi }}{\frac {m}{r^{3}}}{\sqrt {1+3\sin ^{2}(\lambda )}}\,,$

where

Bis the strength of the field, measured inteslas

ris the distance from the center, measured in metres

λis the magnetic latitude (equal to 90° −θ) whereθis the magnetic colatitude, measured inradiansordegreesfrom the dipole axis

mis the dipole moment, measured inampere-square metres orjoulespertesla

μ₀is thepermeability of free space,measured inhenriesper metre.

Conversion to cylindrical coordinates is achieved usingr²=z²+ρ²and

$\lambda =\arcsin \left({\frac {z}{\sqrt {z^{2}+\rho ^{2}}}}\right)$

whereρis the perpendicular distance from thez-axis. Then,

$B(\rho,z)={\frac {\mu _{0}m}{4\pi \left(z^{2}+\rho ^{2}\right)^{\frac {3}{2}}}}{\sqrt {1+{\frac {3z^{2}}{z^{2}+\rho ^{2}}}}}$

Vector form

The field itself is a vector quantity:

$\mathbf {B} (\mathbf {m},\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\ {\frac {3(\mathbf {m} \cdot {\hat {\mathbf {r} }}){\hat {\mathbf {r} }}-\mathbf {m} }{r^{3}}}$

where

Bis the field

ris the vector from the position of the dipole to the position where the field is being measured

ris the absolute value ofr:the distance from the dipole

r̂=⁠r/r⁠is the unit vector parallel tor;

mis the (vector) dipole moment

μ₀is the permeability of free space

This isexactlythe field of a point dipole,exactlythe dipole term in the multipole expansion of an arbitrary field, andapproximatelythe field of any dipole-like configuration at large distances.

Magnetic vector potential

Thevector potentialAof a magnetic dipole is

$\mathbf {A} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}{\frac {\mathbf {m} \times {\hat {\mathbf {r} }}}{r^{2}}}$

with the same definitions as above.

Field from an electric dipole

Theelectrostatic potentialat positionrdue to an electric dipole at the origin is given by:

$\Phi (\mathbf {r} )={\frac {1}{4\pi \epsilon _{0}}}\,{\frac {\mathbf {p} \cdot {\hat {\mathbf {r} }}}{r^{2}}}$

wherepis the (vector)dipole moment,andє₀is thepermittivity of free space.

This term appears as the second term in themultipole expansionof 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). Theelectric fieldfrom a dipole can be found from thegradientof this potential:

$\mathbf {E} =-\nabla \Phi ={\frac {1}{4\pi \epsilon _{0}}}\ {\frac {3(\mathbf {p} \cdot {\hat {\mathbf {r} }}){\hat {\mathbf {r} }}-\mathbf {p} }{r^{3}}}-\delta ^{3}(\mathbf {r} ){\frac {\mathbf {p} }{3\epsilon _{0}}}.$

This is of the same form of the expression for the magnetic field of a point magnetic dipole, ignoring the delta function. 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, seeorMagnetic moment#Internal magnetic field of a dipole.

Torque on a dipole

Since the direction of anelectric fieldis 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 homogeneouselectricormagnetic field,equal but oppositeforcesarise on each side of the dipole creating atorqueτ}:

${\boldsymbol {\tau }}=\mathbf {p} \times \mathbf {E}$

for anelectric dipole momentp(in coulomb-meters), or

${\boldsymbol {\tau }}=\mathbf {m} \times \mathbf {B}$

for amagnetic dipole momentm(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

$U=-\mathbf {p} \cdot \mathbf {E}$.

The energy of a magnetic dipole is similarly

$U=-\mathbf {m} \cdot \mathbf {B}$.

Dipole radiation

Modulus of the Poynting vector for an oscillating electric dipole (exact solution). The two charges are shown as two small black dots.

See also:Dipole antenna

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, tospherical waveradiation.

In particular, consider a harmonically oscillating electric dipole, withangular frequencyωand a dipole momentp₀along theẑdirection of the form

$\mathbf {p} (\mathbf {r},t)=\mathbf {p} (\mathbf {r} )e^{-i\omega t}=p_{0}{\hat {\mathbf {z} }}e^{-i\omega t}.$

In vacuum, the exact field produced by this oscillating dipole can be derived using theretarded potentialformulation as:

${\begin{aligned}\mathbf {E} &={\frac {1}{4\pi \varepsilon _{0}}}\left\{{\frac {\omega ^{2}}{c^{2}r}}\left({\hat {\mathbf {r} }}\times \mathbf {p} \right)\times {\hat {\mathbf {r} }}+\left({\frac {1}{r^{3}}}-{\frac {i\omega }{cr^{2}}}\right)\left(3{\hat {\mathbf {r} }}\left[{\hat {\mathbf {r} }}\cdot \mathbf {p} \right]-\mathbf {p} \right)\right\}e^{\frac {i\omega r}{c}}e^{-i\omega t}\\\mathbf {B} &={\frac {\omega ^{2}}{4\pi \varepsilon _{0}c^{3}}}({\hat {\mathbf {r} }}\times \mathbf {p} )\left(1-{\frac {c}{i\omega r}}\right){\frac {e^{i\omega r/c}}{r}}e^{-i\omega t}.\end{aligned}}$

For⁠rω/c⁠≫ 1, the far-field takes the simpler form of a radiating "spherical" wave, but with angular dependence embedded in the cross-product:

${\begin{aligned}\mathbf {B} &={\frac {\omega ^{2}}{4\pi \varepsilon _{0}c^{3}}}({\hat {\mathbf {r} }}\times \mathbf {p} ){\frac {e^{i\omega (r/c-t)}}{r}}={\frac {\omega ^{2}\mu _{0}p_{0}}{4\pi c}}({\hat {\mathbf {r} }}\times {\hat {\mathbf {z} }}){\frac {e^{i\omega (r/c-t)}}{r}}=-{\frac {\omega ^{2}\mu _{0}p_{0}}{4\pi c}}\sin(\theta ){\frac {e^{i\omega (r/c-t)}}{r}}\mathbf {\hat {\phi }} \\\mathbf {E} &=c\mathbf {B} \times {\hat {\mathbf {r} }}=-{\frac {\omega ^{2}\mu _{0}p_{0}}{4\pi }}\sin(\theta )\left({\hat {\phi }}\times \mathbf {\hat {r}} \right){\frac {e^{i\omega (r/c-t)}}{r}}=-{\frac {\omega ^{2}\mu _{0}p_{0}}{4\pi }}\sin(\theta ){\frac {e^{i\omega (r/c-t)}}{r}}{\hat {\theta }}.\end{aligned}}$

The time-averagedPoynting vector

$\langle \mathbf {S} \rangle =\left({\frac {\mu _{0}p_{0}^{2}\omega ^{4}}{32\pi ^{2}c}}\right){\frac {\sin ^{2}(\theta )}{r^{2}}}\mathbf {\hat {r}}$

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, thespherical harmonicfunction (sinθ) responsible for suchtoroidalangular distribution is precisely thel= 1 "p" wave.

The total time-average power radiated by the field can then be derived from the Poynting vector as

$P={\frac {\mu _{0}\omega ^{4}p_{0}^{2}}{12\pi c}}.$

Notice that the dependence of the power on the fourth power of the frequency of the radiation is in accordance with theRayleigh scattering,and the underlying effects why the sky consists of mainly blue colour.

A circular polarized dipole is described as a superposition of two linear dipoles.

See also

• Polarization density
• Magnetic dipole models
• Dipole model of the Earth's magnetic field
• Electret
• Indian Ocean DipoleandSubtropical Indian Ocean Dipole,two oceanographic phenomena
• Magnetic dipole-dipole interaction
• Spin magnetic moment
• Monopole
• Solid harmonics
• Axial multipole moments
• Cylindrical multipole moments
• Spherical multipole moments
• Laplace expansion
• Molecular solid
• Magnetic moment#Internal magnetic field of a dipole

External links

• USGS Geomagnetism Program
• Fields of ForceArchived2010-12-14 at theWayback Machine:a chapter from an online textbook
• Electric Dipole PotentialbyStephen WolframandEnergy Density of a Magnetic Dipoleby Franz Krafft.Wolfram Demonstrations Project.