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Dipole
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(Redirected from Magnetic dipole)
In physics, there are two kinds of dipoles:
An electric dipole is a separation of positive and
negative charges. The simplest example of this is a
pair of electric charges of equal magnitude but
opposite sign, separated by some (usually small)
distance. A permanent electric dipole is called an
electret.
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.[1][2]
The Earth's magnetic field, which is
approximately a magnetic 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
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 grip 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 that 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).
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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 monopoles), and are
labeled "north" and "south." The dipole moment of the
bar magnet points from its magnetic south to its
magnetic north pole. What can be confusing is that the
"north" and "south" convention for magnetic dipoles is
the opposite of that used to describe Earth's geographic
and magnetic poles, so that Earth's geomagnetic north
pole is the south pole of its dipole moment. (This should
not be difficult to remember; it simply means that the
north pole of a bar magnet is the one that points north if
used as a compass.)
Contour plot of an electrical dipole, with
equipotential surfaces indicated
The only known mechanisms for the creation of
magnetic dipoles are by current loops or quantummechanical spin since the existence of magnetic monopoles has never been experimentally
demonstrated.
The term comes from the Greek di(s)- = "two" and pòla "pivot, hinge".
Contents
1 Classification
2 Molecular dipoles
3 Quantum mechanical dipole operator
4 Atomic dipoles
5 Field from a magnetic dipole
5.1 Magnitude
5.2 Vector form
5.3 Magnetic vector potential
6 Field from an electric dipole
7 Torque on a dipole
8 Dipole radiation
9 See also
10 Notes
11 References
12 External links
Classification
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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
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.
Electric dipole field lines
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,
Magnetic dipole field lines
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. Such is the case with polar compounds like hydroxide (OH−),
where electron density is shared unequally between atoms.
A molecule with a permanent dipole moment is called a polar molecule. A molecule is polarized
when it carries an induced dipole. 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.
With respect to molecules, there are three types of dipoles:
Permanent dipoles: These occur when two atoms in a molecule have substantially different
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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.
Induced dipoles: These can occur when one molecule with a permanent dipole repels another
molecule's electrons, "inducing" a dipole moment in that molecule. 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 is equal to the
product of the strength of the external field and the dipole polarizability of ρ.
Typical gas phase values of some chemical compounds in debye units:[3]
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.
Quantum mechanical dipole operator
Consider a collection of N particles with charges qi and position vectors ri. For instance, this
collection may be a molecule consisting of electrons, all with charge −e, and nuclei with charge eZi,
where Zi is the atomic number of the i th nucleus. The physical quantity (observable) dipole has the
quantum mechanical operator:
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 under inversion with respect to the nucleus,
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where is the dipole operator and 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,
where
is an S-state, non-degenerate, wavefunction, which is symmetric or antisymmetric
under inversion:
. Since the product of the wavefunction (in the ket) and its
complex conjugate (in the bra) is always symmetric under inversion and its inverse,
it follows that the expectation value changes sign under inversion. We used here the fact that ,
being a symmetry operator, is unitary:
and by definition the Hermitian adjoint
may
be moved from bra to ket and then becomes
. Since the only quantity that is equal to minus
itself is the zero, 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 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 this
article for the origin of this degeneracy) and have opposite parity (2s is even and 2p is odd).
Field from a 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
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 (equal to 90° − θ) where θ is the magnetic colatitude, measured in
radians or degrees from the dipole axis[note 1]
m is the dipole moment (VADM=virtual axial dipole moment), measured in ampere squaremetres (A·m2), which equals joules per tesla
μ0 is the permeability of free space, measured in henries per metre.
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Conversion to cylindrical coordinates is achieved using r2 = z2 + ρ2 and
where ρ is the perpendicular distance from the z-axis. Then,
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
δ3 is the three-dimensional delta function.[note 2]
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 at position r due to an electric dipole at the origin is given by:
where
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is a unit vector in the direction of r', p is the (vector) dipole moment, and ε0 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:
where E is the electric field and δ3 is the 3-dimensional delta function.[note 2] This is formally
identical to the magnetic field of a point magnetic dipole; only a few names have changed.
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:
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, which in the case of an
electric dipole, yields a potential energy of
.
The energy of a magnetic dipole is similarly
.
Dipole radiation
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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.
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:
Far away (for
Real-time evolution of
the electric field of an
oscillating electric
dipole. The dipole is
located at (60,60) in the
graph, oscillating at 1 Hz
in the vertical direction
), 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. 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.
See also
Magnetic dipole models
Dipole model of the Earth's magnetic field
Electret
Indian Ocean Dipole (an oceanographic phenomenon)
Magnetic dipole-dipole interaction
Magnetic dipole moment
Spin magnetic moment
Monopole
Solid harmonics
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Axial multipole moments
Cylindrical multipole moments
Spherical multipole moments
Laplace expansion
Molecular solid
Notes
1. ^ Magnetic colatitude is 0 along the dipole's axis and 90° in the plane perpendicular to its axis.
2. ^ a b δ3(r) = 0 except at r = (0,0,0), so this term is ignored in multipole expansion.
References
1. ^ Brau, Charles A. (2004). Modern Problems in Classical Electrodynamics. Oxford University
Press. ISBN 0-19-514665-4.
2. ^ Griffiths, David J. (1999). Introduction to Electrodynamics (3rd ed. ed.). Prentice Hall.
ISBN 0-13-805326-X.
3. ^ Weast, Robert C. (1984). CRC Handbook of Chemistry and Physics (65rd ed. ed.). CRC
Press. ISBN 0-8493-0465-2.
External links
An interactive JAVA applet displaying the behavior of two-dimensional dipoles
(http://www.princeton.edu/~achremos/Applet3-page.htm) .
USGS Geomagnetism Program (http://geomag.usgs.gov)
Fields of Force (http://lightandmatter.com/html_books/4em/ch05/ch05.html) : a chapter from
an online textbook
Electric Dipole Potential (http://demonstrations.wolfram.com/ElectricDipolePotential/) by
Stephen Wolfram and Energy Density of a Magnetic Dipole
(http://demonstrations.wolfram.com/EnergyDensityOfAMagneticDipole/) by Franz Krafft.
Wolfram Demonstrations Project.
The inverse cube law (http://blazelabs.com/inversecubelaw.pdf) The inverse cube law for
dipoles (PDF file) by Eng. Xavier Borg
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Categories: Electromagnetism | Chemical properties | Fundamental physics concepts
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