Dipole

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

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. 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.

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:

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

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

\mathbf{\tau} = \mathbf{m} \times \mathbf{B}    =    \mu \mathbf{m} \times \mathbf{H}

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{p} \cdot \mathbf{E}

The energy of a magnetic dipole is similarly

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

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.

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.
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.

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.

Field from a magnetic dipole

Magnitude

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

B(\mathbf{r}, \lambda) = \frac {\mu_0} {4\pi} \frac {\mathbf{M}} {r^3} \sqrt {1+3\sin^2\lambda}

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.

Vector form

The field itself is a vector quantity:

\mathbf{B}(\mathbf{r}) = \frac {\mu_0} {4\pi r^3} \left(3(\mathbf{m}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{m}\right)

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}} = \mathbf{r}/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.

Magnetic vector potential

The vector potential A of a magnetic dipole is

\mathbf{A}(\mathbf{r}) = \frac {\mu_0} {4\pi r^2} (\mathbf{m}\times\hat{\mathbf{r}})

with the same definitions as above.

Field from an electric dipole

The electrostatic potential of an electric dipole is

\Phi  (\mathbf{r}) = \frac {1} {4\pi\epsilon_0 r^2} (\mathbf{p}\cdot\hat{\mathbf{r}}).

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

\mathbf{E} \,

= - \nabla \Phi \,

=\frac {1} {4\pi\epsilon_0 r^3} \left(3(\mathbf{p}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{p}\right)

where

E is the electric field

r, r,

\hat{\mathbf{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 $\mathbf{p}=\mathbf{p'(\mathbf r)}e^{-i\omega t}$ 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) \left3 \hat{\mathbf{r}} (\hat{\mathbf{r}} \cdot \mathbf{p}) - \mathbf{p} \right \right\} e^{i\omega r/c}

\mathbf{H} = \frac{\omega^2}{4\pi c} \hat{\mathbf{r}} \times \mathbf{p} \left( 1 - \frac{c}{i\omega r} \right) \frac{e^{i\omega r/c}}{r}

Far away (for $r\omega/c \gg 1$ ), the fields approach the limiting form of a radiating spherical wave:

\mathbf{H} = \frac{\omega^2}{4\pi c} (\hat{\mathbf{r}} \times \mathbf{p}) \frac{e^{i\omega r/c}}{r}

\mathbf{E} = \sqrt{\frac{\mu_0}{\epsilon_0}} \mathbf{H} \times \hat{\mathbf{r}}

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

P = \sqrt{\frac{\mu_0}{\epsilon_0}} \frac{\omega^4}{12\pi c^2} |\mathbf{p}|^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.

References

External links

USGS Geomagnetism Program

Electromagnetism | chemical properties

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