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The Earth's magnetic field, which is approximately a dipole. However, the "N" and "S" (north and south) poles are labeled here geographically, which is the opposite of the convention for labeling the poles of a magnetic dipole moment.
A dipole (Greek: di(s) = double and polos = pivot) is a pair of electric charges or magnetic poles of equal magnitude but opposite polarity (opposite electronic charges), separated by some (usually small) distance. Dipoles can be characterized by their dipole moment, a vector quantity with a magnitude equal to the product of the charge or magnetic strength of one of the poles and the distance separating the two poles. The direction of the dipole moment corresponds, for electric dipoles, to the direction from the negative to the positive charge. For magnetic dipoles, the dipole moment points from the magnetic south to the 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. (Because of the absence of magnetic monopoles, magnetic dipoles are actually created by current loops and/or by quantum-mechanical spin.)
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 (E) or magnetic (B) field, equal but opposite forces 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.
Physical dipoles, point dipoles, and approximate dipoles
Diagram of a physical dipole, with equipotential surfaces and field lines indicated
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/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 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 and is polarized. 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.
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.
- 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.
Field from a magnetic dipole
Magnitude
The strength, B, of a dipole magnetic field is given by:
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
- μ0 is the permeability of free space, measured in henrys per metre.
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
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 potential A of a magnetic dipole is
with the same definitions as above.
Field from an electric dipole
The electrostatic potential of an electric dipole is
And the electric field from a dipole can be found from the gradient of this potential:
-
where
- E is the electric field
- r, r,
 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.
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. In vacuum, this produces fields:
![\mathbf{E} = \frac{1}{4\pi\varepsilon_0} \left\{ \frac{\omega^2}{c^2 r} \hat{\mathbf{r}} \times \mathbf{p} \times \hat{\mathbf{r}} + \left( \frac{1}{r^3} - \frac{i\omega}{cr^2} \right) \left[ 3 \hat{\mathbf{r}} (\hat{\mathbf{r}} \cdot \mathbf{p}) - \mathbf{p} \right] \right\} e^{i\omega r/c}](/math/1/c/b/1cb95474b78138db05d3763ac2fcd75a.png) 
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.
See also
References
- Brau, Charles A. (2004). Modern Problems in Classical Electrodynamics, Oxford University Press. ISBN 0195146654.
- Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.), Prentice Hall. ISBN 013805326X.
External links
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