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.
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).
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.)
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.
The term comes from the
Greek
di(
s)- = "two" and
pòla "pivot,
hinge".
Classification
Electric dipole field lines
Magnetic dipole field lines
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.
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. 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 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 occur when one molecule
with a permanent dipole repels another molecule's electrons,
"inducing" a dipole moment in that molecule. See induced-dipole attraction.
The definition of an induced dipole given in the previous sentence
is too restrictive and misleading. 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
theexternal field and the dipole
polarizability of
ρ.
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.
Quantum mechanical dipole operator
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 physical quantity (observable)
dipole has the quantum mechanical operator:
- \mathfrak{p} = \sum_{i=1}^N \, q_i \, \mathbf{r}_i \, .
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,
- \mathfrak{I} \;\mathfrak{p}\; \mathfrak{I}^{-1} = -
\mathfrak{p},
where \stackrel{\mathfrak{p}}{} is the dipole operator and
\stackrel{\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,\langle
\mathfrak{p} \rangle = \langle\, S\, | \mathfrak{p} |\, S
\,\rangle,where |\, S\, \rangle is an S-state, non-degenerate,
wavefunction, whichis symmetric or antisymmetric under inversion:
\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,\langle \mathfrak{p} \rangle = \langle\,
\mathfrak{I}^{-1}\, S\, | \mathfrak{p} |\, \mathfrak{I}^{-1}\, S
\,\rangle
= \langle\, S\, | \mathfrak{I}\, \mathfrak{p} \, \mathfrak{I}^{-1}| \, S \,\rangle = -\langle \mathfrak{p} \rangle
it follows that the expectation value changes sign under inversion.
We used here the fact that \mathfrak{I}\,, being a symmetry
operator, is
unitary:
\mathfrak{I}^{-1} = \mathfrak{I}^{*}\, and
by
definitionthe Hermitian adjoint \mathfrak{I}^*\, may be moved
from bra to ket and then becomes \mathfrak{I}^{**} =
\mathfrak{I}\,.Since the only quantity that is equal to minus
itself is the zero, the expectation value vanishes,\langle
\mathfrak{p}\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 this
article for the origin of this degeneracy) and have opposite
parity (2s is even and 2p is odd).
Field from a magnetic dipole
Magnitude
The far-field strength,
B, of a dipole magnetic field is
given by
- B(m, r, \lambda) = \frac {\mu_0} {4\pi} \frac {m} {r^3} \sqrt
{1+3\sin^2\lambda} \, ,
where
- B is the strength of the field, measured in tesla
- 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 degree from
the dipole axis
- m is the dipole moment (VADM=virtual axial dipole
moment), measured in ampere square-metres (A·m^{2}), which
equals joules per tesla
- μ_{0} is the permeability of free space,
measured in henries per metre.
Conversion to cylindrical coordinates is achieved using and
- \lambda = \arcsin\left(\frac{z}{\sqrt{z^2+\rho^2}}\right)
where
ρ is the perpendicular distance from the
z-axis. Then,
- B(\rho,z) = \frac{\mu_0 m}{4 \pi (z^2+\rho^2)^{3/2}}
\sqrt{1+\frac{3 z^2}{z^2 + \rho^2}}
Vector form
The field itself is a vector quantity:
- \mathbf{B}(\mathbf{m}, \mathbf{r}) = \frac {\mu_0} {4\pi r^3}
\left(3(\mathbf{m}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{m}\right)
+ \frac{2\mu_0}{3}\mathbf{m}\delta^3(\mathbf{r})
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
- μ_{0} is the permeability of free space
- δ^{3} is the three-dimensional delta
function.δ^{3}(r) = 0 except
at , so this term is ignored in multipole expansion.
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
at position
r due to an electric dipole at the
origin is given by:
- \Phi(\mathbf{r}) =
\frac{1}{4\pi\varepsilon_0}\,\frac{\mathbf{p}\cdot\hat{\mathbf{r}}}{r^2}
where
- \hat{\mathbf{r}} is a unit vector in the direction of
r', p is the (vector) dipole moment, and ε_{0} 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:
- \mathbf{E} = - \nabla \Phi =\frac {1} {4\pi\epsilon_0}
\left(\frac{3(\mathbf{p}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{p}}{r^3}\right)
- \frac{1}{3\epsilon_0}\mathbf{p}\delta^3(\mathbf{r})
where
E is the electric field and
δ^{3} is the 3-dimensional
delta function. 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:
- \boldsymbol{\tau} = \mathbf{p} \times \mathbf{E}
for an
electric dipole
moment p (in coulomb-meters), or
- \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
- U = -\mathbf{p} \cdot \mathbf{E}.
The energy of a magnetic dipole is similarly
- U = -\mathbf{m} \cdot \mathbf{B}.
Dipole radiation
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
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) \left[ 3
\hat{\mathbf{r}} (\hat{\mathbf{r}} \cdot \mathbf{p}) - \mathbf{p}
\right] \right\} e^{i\omega r/c}
- \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}.
Far away (for \scriptstyle r \omega /c \gg 1), the fields approach
the limiting form of a radiating spherical wave:
- \mathbf{B} = \frac{\omega^2}{4\pi\varepsilon_0 c^3}
(\hat{\mathbf{r}} \times \mathbf{p}) \frac{e^{i\omega r/c}}{r}
- \mathbf{E} = c \mathbf{B} \times \hat{\mathbf{r}}
which produces a total time-average radiated power
P given
by
- P = \frac{\omega^4}{12\pi\varepsilon_0 c^3}
|\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.
See also
Notes
- Magnetic colatitude is 0 along the dipole's axis and 90° in the
plane perpendicular to its axis.
References
External links