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This article is about the electromagnetic phenomenon. For the radio antenna, see dipole antenna. For the permanent magnet, see dipole magnet.

    Dipole
        Definition
        Torque on a dipole
        Physical dipoles, point dipoles, and approximate dipoles
        Molecular dipoles
        Atomic dipoles
            Magnitude
            Vector form
            Magnetic vector potential
        Field from an electric dipole
        Dipole radiation
        See also

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Definition

In physics, there are two kinds of dipoles (Greek: di(s) = double and polos = pivot). An electric dipole is a separation of positive and negative charge. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some, usually small, distance. By contrast, a magnetic dipole is a closed circulation of electric current. A simple example of this is a single loop of wire with some constant current flowing through it. •
•

Dipoles can be characterized by their dipole moment, a vector quantity. For the simple electric dipole given above, the electric dipole moment would point from the negative charge towards the positive charge, and have a magnitude equal to the strength of each charge times the separation between the charges. For the current loop, the magnetic dipole moment would point through the loop (according to the right hand 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, is said to have a magnetic dipole moment. This is because it generates a magnetic field which 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 magnetic monopole), and are labeled "north" and "south." The dipole moment of the bar magnet points from its magnetic south to its magnetic north pole — confusingly, the "north" and "south" convention for magnetic dipoles is the opposite of that used to describe the Earth's geographic and magnetic poles, so that the Earth's geomagnetic north pole is the south pole of its dipole moment.

The only known mechanisms for the creation of magnetic dipoles are by current loops or quantum-mechanical spin since the existence of magnetic monopoles has never been experimentally demonstrated.

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

$oldsymbol = mathbf imes mathbf$

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

$oldsymbol = mathbf imes mathbf$

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

$U = -mathbf cdot mathbf$ .

The energy of a magnetic dipole is similarly

$U = -mathbf cdot mathbf$ .

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Physical dipoles, point dipoles, and approximate dipoles

A physical dipole consists of two equal and opposite point charges: literally, 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 the exact 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 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 / r^3

, as compared to

1 / r^4

for the next (quadrupole) term and higher powers of

1 / r

for higher terms, or

1 / r^2

for the monopole term.

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Molecular dipoles
Many molecules have such dipole moments due to non-uniform distributions of positive and negative charges on the various atoms. For example:

(positive) H-Cl (negative)

A molecule with a permanent dipole moment is called a polar molecule. The physical chemist Peter J. W. Debye was the first scientist to study molecular dipoles extensively, and dipole moments are consequently measured in units named debye in his honor.

Note added by other author: A molecule is polarized when it carries an induced dipole.

With respect to molecules there are three types of dipoles:

Permanent dipoles: These occur when 2 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. 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. See Instantaneous dipole attraction.

Induced dipoles These occur when one molecule with a permanent dipole repels another molecule's electrons, "inducing" a dipole moment in that molecule. See induced-dipole attraction.

Notes added by other author:

(i) The definition of an induced dipole given in the previous sentence is too restrictive and misleading. An induced dipole of any polarizable charge distribution

ho

(remember that a molecule is a charge distribution) is caused by an electric field external to

ho

. This field may, for instance, originate from an ion or polar molecule in the vicinity of

ho

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

ho

.

(ii) Whereas permanent and induced dipoles have
quantitative definitions, the instantaneous dipole is fictitious. It is
only used in a handwaving argument to "explain" the London dispersion force, see intermolecular forces. While permanent and induced dipoles can be measured and computed by quantum chemical methods, (see quantum chemistry), the instantaneous dipole can neither be measured nor computed.

Typical gas phase values of some chemical compounds in Debye units •:

carbon dioxide: 0

carbon monoxide: 0.112

ozone: 0.53

phosgene: 1.17

water vapor: 1.85

hydrogen cyanide: 2.98

cyanamide: 4.27

potassium bromide: 10.41

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.

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Atomic dipoles
A closed-shell, non-degenerate, (S-state) atom can only have 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. 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. Since the product of the wavefunction and its complex conjugate appearing in the expectation value is always symmetric under inversion, it follows that the expectation value changes sign under inversion. Since, on the other hand, inversion is a symmetry of the atom, the expectation value must be invariant (stay the same) under inversion. The only quantity equal to minus itself is zero and hence 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 only gives a non-vanishing dipole (by definition proportional to a non-vanishing first-order Stark shift) 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 and have opposite parity (2s is even and 2p is odd).

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Magnitude
The strength, B, of a dipole magnetic field is given by:

$B(mathbf, lambda) = rac rac sqrt$

where:

B is the strength of the field, measured in teslas

r is the distance from the center, measured in metres

λ is the magnetic latitude (90°-θ) where θ = magnetic colatitude, measured in radians or degrees from the dipole axis (magnetic colatitude is 0 along the dipole's axis and 90° in the plane perpendicular to its axis)

M is the dipole moment, measured in ampere square-metres, which equals joules per tesla.

μ₀ is the permeability of free space, measured in henrys per metre.

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Vector form
The field itself is a vector quantity:

$mathbf(mathbf) = rac left(3(mathbfcdothat)hat-mathbf$

ight)

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

$hat = mathbf/r$ is the unit vector parallel to r

m is the (vector) dipole moment

μ₀ 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.

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Magnetic vector potential
The vector potential A of a magnetic dipole is

$mathbf(mathbf) = rac (mathbf imeshat)$

with the same definitions as above.

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Field from an electric dipole
The electrostatic potential of an electric dipole is

$Phi (mathbf) = rac (mathbfcdothat).$

And the electric field from a dipole can be found from the gradient of this potential:

where

E is the electric field

r, r, $hat$ are as above

p is the (vector) dipole moment

ε₀ is the permittivity of free space.

Notice that this is formally identical to the magnetic field of a point magnetic dipole; only a few names have changed.

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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

mathbf=mathbfe^

where ω is the angular frequency. In vacuum, this produces fields:

$mathbf = rac left e^$

$mathbf = rac hat imes mathbf left( 1 - rac$

ight) rac

Far away (for

romega/c gg 1

), the fields approach the limiting form of a radiating spherical wave:

$mathbf = rac (hat imes mathbf) rac$

$mathbf = sqrt mathbf imes hat$

which produces a total time-average radiated power P given by:

$P = sqrt rac |mathbf|^2$

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.

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