Dipole

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.

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.

Although there are no known magnetic monopoles in nature, there are magnetic dipoles in the form of the quantum-mechanical spin associated with particles such as electrons (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.

Many molecules 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 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.

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 dipole–dipole attractions.

Instantaneous dipoles

These occur due to chance when electrons 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 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 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.

Dipole moment values can be obtained from measurement of the dielectric constant. Some typical gas phase values in debye units are:[7]

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

The linear molecule CO₂ has a zero dipole as the two bond dipoles cancel.

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.

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 a vector sum of bond 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 the molecular geometry.

For example, the zero dipole of CO₂ implies 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.

Cis isomer, dipole moment 1.90 D

Trans isomer, dipole moment zero

An example in organic chemistry of the role of geometry in determining dipole moment is the 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 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 about the boron cation center, the molecule as a whole does not exhibit any identifiable pole: one cannot construct a plane that divides the molecule into a net negative part and a net positive part.

Consider a collection of N particles with charges q_i and position vectors r_i. For instance, this collection may be a molecule consisting of electrons, all with charge −e, and nuclei with charge eZ_i, where Z_i is the atomic number of the i th nucleus. The dipole observable (physical quantity) has the quantum mechanical dipole operator:

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

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

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

where ${\displaystyle \mathbf {r} _{c}}$ is the center of mass of the molecule/group of particles.[8]

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,

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

where ${\displaystyle {\mathfrak {p}}}$ is the dipole operator and ${\displaystyle {\mathfrak {I}}}$ 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,

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

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

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

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

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

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

The electrostatic potential at position r due to an electric dipole at the origin is given by:

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

where p is the (vector) dipole moment, and є₀ 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:

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

This is formally identical to the expression for the magnetic field of a point magnetic dipole with only a few names changed. In a real dipole, however, where the charges are physically separate, the "internal" field lines are different, since the magnetic field lines are continuous, while those of the electric field diverge or converge from the point charges. For further discussions about the internal field of dipoles, see[2][9] or Magnetic moment#Internal magnetic field of a dipole.

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 or magnetic field, equal but opposite forces arise on each side of the dipole creating a torque τ}:

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

for an electric dipole moment p (in coulomb-meters), or

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

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

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

The energy of a magnetic dipole is similarly

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

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

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 p₀ along the ẑ direction of the form

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

In vacuum, the exact field produced by this oscillating dipole can be derived using the retarded potential formulation as:

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

For rω/c ≫ 1, the far-field takes the simpler form of a radiating "spherical" wave, but with angular dependence embedded in the cross-product:[10]

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

The time-averaged Poynting vector

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

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, the spherical harmonic function (sin θ) responsible for such toroidal angular distribution is precisely the l = 1 "p" wave.

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

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

Notice that the dependence of the power on the fourth power of the frequency of the radiation is in accordance with the Rayleigh 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.

• Polarization density
• Magnetic dipole models
• Dipole model of the Earth's magnetic field
• Electret
• Indian Ocean Dipole and Subtropical 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

• USGS Geomagnetism Program
• Fields of Force: a chapter from an online textbook
• Electric Dipole Potential by Stephen Wolfram and Energy Density of a Magnetic Dipole by Franz Krafft. Wolfram Demonstrations Project.