A magnet is a material or object that produces
a magnetic field. This magnetic field is invisible
but is responsible for the most notable property
of a magnet: a force that pulls on other ferromagnetic
materials, such as iron, and attracts or repels
other magnets.
A permanent magnet is an object made from
a material that is magnetized and creates
its own persistent magnetic field. An everyday
example is a refrigerator magnet used to hold
notes on a refrigerator door. Materials that
can be magnetized, which are also the ones
that are strongly attracted to a magnet, are
called ferromagnetic (or ferrimagnetic). These
include the elements iron, nickel and cobalt,
some alloys of rare-earth metals, and some
naturally occurring minerals such as lodestone.
Although ferromagnetic (and ferrimagnetic)
materials are the only ones attracted to a
magnet strongly enough to be commonly considered
magnetic, all other substances respond weakly
to a magnetic field, by one of several other
types of magnetism.
Ferromagnetic materials can be divided into
magnetically "soft" materials like annealed
iron, which can be magnetized but do not tend
to stay magnetized, and magnetically "hard"
materials, which do. Permanent magnets are
made from "hard" ferromagnetic materials such
as alnico and ferrite that are subjected to
special processing in a strong magnetic field
during manufacture to align their internal
microcrystalline structure, making them very
hard to demagnetize. To demagnetize a saturated
magnet, a certain magnetic field must be applied,
and this threshold depends on coercivity of
the respective material. "Hard" materials
have high coercivity, whereas "soft" materials
have low coercivity. The overall strength
of a magnet is measured by its magnetic moment
or, alternatively, the total magnetic flux
it produces. The local strength of magnetism
in a material is measured by its magnetization.
An electromagnet is made from a coil of wire
that acts as a magnet when an electric current
passes through it but stops being a magnet
when the current stops. Often, the coil is
wrapped around a core of "soft" ferromagnetic
material such as mild steel, which greatly
enhances the magnetic field produced by the
coil.
== Discovery and development ==
Ancient people learned about magnetism from
lodestones (or magnetite) which are naturally
magnetized pieces of iron ore. The word magnet
was adopted in Middle English from Latin magnetum
"lodestone", ultimately from Greek μαγνῆτις
[λίθος] (magnētis [lithos]) meaning
"[stone] from Magnesia", a part of ancient
Greece where lodestones were found. Lodestones,
suspended so they could turn, were the first
magnetic compasses. The earliest known surviving
descriptions of magnets and their properties
are from Greece, India, and China around 2500
years ago. The properties of lodestones and
their affinity for iron were written of by
Pliny the Elder in his encyclopedia Naturalis
Historia.By the 12th to 13th centuries AD,
magnetic compasses were used in navigation
in China, Europe, the Arabian Peninsula and
elsewhere.
== Physics ==
=== Magnetic field ===
The magnetic flux density (also called magnetic
B field or just magnetic field, usually denoted
B) is a vector field. The magnetic B field
vector at a given point in space is specified
by two properties:
Its direction, which is along the orientation
of a compass needle.
Its magnitude (also called strength), which
is proportional to how strongly the compass
needle orients along that direction.In SI
units, the strength of the magnetic B field
is given in teslas.
=== Magnetic moment ===
A magnet's magnetic moment (also called magnetic
dipole moment and usually denoted μ) is a
vector that characterizes the magnet's overall
magnetic properties. For a bar magnet, the
direction of the magnetic moment points from
the magnet's south pole to its north pole,
and the magnitude relates to how strong and
how far apart these poles are. In SI units,
the magnetic moment is specified in terms
of A·m2 (amperes times meters squared).
A magnet both produces its own magnetic field
and responds to magnetic fields. The strength
of the magnetic field it produces is at any
given point proportional to the magnitude
of its magnetic moment. In addition, when
the magnet is put into an external magnetic
field, produced by a different source, it
is subject to a torque tending to orient the
magnetic moment parallel to the field. The
amount of this torque is proportional both
to the magnetic moment and the external field.
A magnet may also be subject to a force driving
it in one direction or another, according
to the positions and orientations of the magnet
and source. If the field is uniform in space,
the magnet is subject to no net force, although
it is subject to a torque.A wire in the shape
of a circle with area A and carrying current
I has a magnetic moment of magnitude equal
to IA.
=== Magnetization ===
The magnetization of a magnetized material
is the local value of its magnetic moment
per unit volume, usually denoted M, with units
A/m. It is a vector field, rather than just
a vector (like the magnetic moment), because
different areas in a magnet can be magnetized
with different directions and strengths (for
example, because of domains, see below). A
good bar magnet may have a magnetic moment
of magnitude 0.1 A•m2 and a volume of 1
cm3, or 1×10−6 m3, and therefore an average
magnetization magnitude is 100,000 A/m. Iron
can have a magnetization of around a million
amperes per meter. Such a large value explains
why iron magnets are so effective at producing
magnetic fields.
=== Modelling magnets ===
Two different models exist for magnets: magnetic
poles and atomic currents.
Although for many purposes it is convenient
to think of a magnet as having distinct north
and south magnetic poles, the concept of poles
should not be taken literally: it is merely
a way of referring to the two different ends
of a magnet. The magnet does not have distinct
north or south particles on opposing sides.
If a bar magnet is broken into two pieces,
in an attempt to separate the north and south
poles, the result will be two bar magnets,
each of which has both a north and south pole.
However, a version of the magnetic-pole approach
is used by professional magneticians to design
permanent magnets.In this approach, the divergence
of the magnetization ∇·M inside a magnet
and the surface normal component M·n are
treated as a distribution of magnetic monopoles.
This is a mathematical convenience and does
not imply that there are actually monopoles
in the magnet. If the magnetic-pole distribution
is known, then the pole model gives the magnetic
field H. Outside the magnet, the field B is
proportional to H, while inside the magnetization
must be added to H. An extension of this method
that allows for internal magnetic charges
is used in theories of ferromagnetism.
Another model is the Ampère model, where
all magnetization is due to the effect of
microscopic, or atomic, circular bound currents,
also called Ampèrian currents, throughout
the material. For a uniformly magnetized cylindrical
bar magnet, the net effect of the microscopic
bound currents is to make the magnet behave
as if there is a macroscopic sheet of electric
current flowing around the surface, with local
flow direction normal to the cylinder axis.
Microscopic currents in atoms inside the material
are generally canceled by currents in neighboring
atoms, so only the surface makes a net contribution;
shaving off the outer layer of a magnet will
not destroy its magnetic field, but will leave
a new surface of uncancelled currents from
the circular currents throughout the material.
The right-hand rule tells which direction
positively-charged current flows. However,
current due to negatively-charged electricity
is far more prevalent in practice.
=== Polarity ===
The north pole of a magnet is defined as the
pole that, when the magnet is freely suspended,
points towards the Earth's North Magnetic
Pole in the Arctic (the magnetic and geographic
poles do not coincide, see magnetic declination).
Since opposite poles (north and south) attract,
the North Magnetic Pole is actually the south
pole of the Earth's magnetic field. As a practical
matter, to tell which pole of a magnet is
north and which is south, it is not necessary
to use the Earth's magnetic field at all.
For example, one method would be to compare
it to an electromagnet, whose poles can be
identified by the right-hand rule. The magnetic
field lines of a magnet are considered by
convention to emerge from the magnet's north
pole and reenter at the south pole.
=== Magnetic materials ===
The term magnet is typically reserved for
objects that produce their own persistent
magnetic field even in the absence of an applied
magnetic field. Only certain classes of materials
can do this. Most materials, however, produce
a magnetic field in response to an applied
magnetic field – a phenomenon known as magnetism.
There are several types of magnetism, and
all materials exhibit at least one of them.
The overall magnetic behavior of a material
can vary widely, depending on the structure
of the material, particularly on its electron
configuration. Several forms of magnetic behavior
have been observed in different materials,
including:
Ferromagnetic and ferrimagnetic materials
are the ones normally thought of as magnetic;
they are attracted to a magnet strongly enough
that the attraction can be felt. These materials
are the only ones that can retain magnetization
and become magnets; a common example is a
traditional refrigerator magnet. Ferrimagnetic
materials, which include ferrites and the
oldest magnetic materials magnetite and lodestone,
are similar to but weaker than ferromagnetics.
The difference between ferro- and ferrimagnetic
materials is related to their microscopic
structure, as explained in Magnetism.
Paramagnetic substances, such as platinum,
aluminum, and oxygen, are weakly attracted
to either pole of a magnet. This attraction
is hundreds of thousands of times weaker than
that of ferromagnetic materials, so it can
only be detected by using sensitive instruments
or using extremely strong magnets. Magnetic
ferrofluids, although they are made of tiny
ferromagnetic particles suspended in liquid,
are sometimes considered paramagnetic since
they cannot be magnetized.
Diamagnetic means repelled by both poles.
Compared to paramagnetic and ferromagnetic
substances, diamagnetic substances, such as
carbon, copper, water, and plastic, are even
more weakly repelled by a magnet. The permeability
of diamagnetic materials is less than the
permeability of a vacuum. All substances not
possessing one of the other types of magnetism
are diamagnetic; this includes most substances.
Although force on a diamagnetic object from
an ordinary magnet is far too weak to be felt,
using extremely strong superconducting magnets,
diamagnetic objects such as pieces of lead
and even mice can be levitated, so they float
in mid-air. Superconductors repel magnetic
fields from their interior and are strongly
diamagnetic.There are various other types
of magnetism, such as spin glass, superparamagnetism,
superdiamagnetism, and metamagnetism.
== Common uses ==
Magnetic recording media: VHS tapes contain
a reel of magnetic tape. The information that
makes up the video and sound is encoded on
the magnetic coating on the tape. Common audio
cassettes also rely on magnetic tape. Similarly,
in computers, floppy disks and hard disks
record data on a thin magnetic coating.
Credit, debit, and automatic teller machine
cards: All of these cards have a magnetic
strip on one side. This strip encodes the
information to contact an individual's financial
institution and connect with their account(s).
Older types of televisions (non flat screen)
and older large computer monitors: TV and
computer screens containing a cathode ray
tube employ an electromagnet to guide electrons
to the screen.
Speakers and microphones: Most speakers employ
a permanent magnet and a current-carrying
coil to convert electric energy (the signal)
into mechanical energy (movement that creates
the sound). The coil is wrapped around a bobbin
attached to the speaker cone and carries the
signal as changing current that interacts
with the field of the permanent magnet. The
voice coil feels a magnetic force and in response,
moves the cone and pressurizes the neighboring
air, thus generating sound. Dynamic microphones
employ the same concept, but in reverse. A
microphone has a diaphragm or membrane attached
to a coil of wire. The coil rests inside a
specially shaped magnet. When sound vibrates
the membrane, the coil is vibrated as well.
As the coil moves through the magnetic field,
a voltage is induced across the coil. This
voltage drives a current in the wire that
is characteristic of the original sound.
Electric guitars use magnetic pickups to transduce
the vibration of guitar strings into electric
current that can then be amplified. This is
different from the principle behind the speaker
and dynamic microphone because the vibrations
are sensed directly by the magnet, and a diaphragm
is not employed. The Hammond organ used a
similar principle, with rotating tonewheels
instead of strings.
Electric motors and generators: Some electric
motors rely upon a combination of an electromagnet
and a permanent magnet, and, much like loudspeakers,
they convert electric energy into mechanical
energy. A generator is the reverse: it converts
mechanical energy into electric energy by
moving a conductor through a magnetic field.
Medicine: Hospitals use magnetic resonance
imaging to spot problems in a patient's organs
without invasive surgery.
Chemistry: Chemists use nuclear magnetic resonance
to characterize synthesized compounds.
Chucks are used in the metalworking field
to hold objects. Magnets are also used in
other types of fastening devices, such as
the magnetic base, the magnetic clamp and
the refrigerator magnet.
Compasses: A compass (or mariner's compass)
is a magnetized pointer free to align itself
with a magnetic field, most commonly Earth's
magnetic field.
Art: Vinyl magnet sheets may be attached to
paintings, photographs, and other ornamental
articles, allowing them to be attached to
refrigerators and other metal surfaces. Objects
and paint can be applied directly to the magnet
surface to create collage pieces of art. Magnetic
art is portable, inexpensive and easy to create.
Vinyl magnetic art is not for the refrigerator
anymore. Colorful metal magnetic boards, strips,
doors, microwave ovens, dishwashers, cars,
metal I beams, and any metal surface can be
receptive of magnetic vinyl art. Being a relatively
new media for art, the creative uses for this
material is just beginning.
Science projects: Many topic questions are
based on magnets, including the repulsion
of current-carrying wires, the effect of temperature,
and motors involving magnets.
Toys: Given their ability to counteract the
force of gravity at close range, magnets are
often employed in children's toys, such as
the Magnet Space Wheel and Levitron, to amusing
effect.
Refrigerator magnets are used to adorn kitchens,
as a souvenir, or simply to hold a note or
photo to the refrigerator door.
Magnets can be used to make jewelry. Necklaces
and bracelets can have a magnetic clasp, or
may be constructed entirely from a linked
series of magnets and ferrous beads.
Magnets can pick up magnetic items (iron nails,
staples, tacks, paper clips) that are either
too small, too hard to reach, or too thin
for fingers to hold. Some screwdrivers are
magnetized for this purpose.
Magnets can be used in scrap and salvage operations
to separate magnetic metals (iron, cobalt,
and nickel) from non-magnetic metals (aluminum,
non-ferrous alloys, etc.). The same idea can
be used in the so-called "magnet test", in
which an auto body is inspected with a magnet
to detect areas repaired using fiberglass
or plastic putty.
Magnets are found in process industries, food
manufacturing especially, in order to remove
metal foreign bodies from materials entering
the process (raw materials) or to detect a
possible contamination at the end of the process
and prior to packaging. They constitute an
important layer of protection for the process
equipment and for the final consumer.
Magnetic levitation transport, or maglev,
is a form of transportation that suspends,
guides and propels vehicles (especially trains)
through electromagnetic force. Eliminating
rolling resistance increases efficiency. The
maximum recorded speed of a maglev train is
581 kilometers per hour (361 mph).
Magnets may be used to serve as a fail-safe
device for some cable connections. For example,
the power cords of some laptops are magnetic
to prevent accidental damage to the port when
tripped over. The MagSafe power connection
to the Apple MacBook is one such example.
== Medical issues and safety ==
Because human tissues have a very low level
of susceptibility to static magnetic fields,
there is little mainstream scientific evidence
showing a health effect associated with exposure
to static fields. Dynamic magnetic fields
may be a different issue, however; correlations
between electromagnetic radiation and cancer
rates have been postulated due to demographic
correlations (see Electromagnetic radiation
and health).
If a ferromagnetic foreign body is present
in human tissue, an external magnetic field
interacting with it can pose a serious safety
risk.A different type of indirect magnetic
health risk exists involving pacemakers. If
a pacemaker has been embedded in a patient's
chest (usually for the purpose of monitoring
and regulating the heart for steady electrically
induced beats), care should be taken to keep
it away from magnetic fields. It is for this
reason that a patient with the device installed
cannot be tested with the use of a magnetic
resonance imaging device.
Children sometimes swallow small magnets from
toys, and this can be hazardous if two or
more magnets are swallowed, as the magnets
can pinch or puncture internal tissues.Magnetic
imaging devices (e.g. MRIs) generate enormous
magnetic fields, and therefore rooms intended
to hold them exclude ferrous metals. Bringing
objects made of ferrous metals (such as oxygen
canisters) into such a room creates a severe
safety risk, as those objects may be powerfully
thrown about by the intense magnetic fields.
== Magnetizing ferromagnets ==
Ferromagnetic materials can be magnetized
in the following ways:
Heating the object higher than its Curie temperature,
allowing it to cool in a magnetic field and
hammering it as it cools. This is the most
effective method and is similar to the industrial
processes used to create permanent magnets.
Placing the item in an external magnetic field
will result in the item retaining some of
the magnetism on removal. Vibration has been
shown to increase the effect. Ferrous materials
aligned with the Earth's magnetic field that
are subject to vibration (e.g., frame of a
conveyor) have been shown to acquire significant
residual magnetism. Likewise, striking a steel
nail held by fingers in a N-S direction with
a hammer will temporarily magnetize the nail.
Stroking: An existing magnet is moved from
one end of the item to the other repeatedly
in the same direction (single touch method)
or two magnets are moved outwards from the
center of a third (double touch method).
Electric Current: The magnetic field produced
by passing an electric current through a coil
can get domains to line up. Once all of the
domains are lined up, increasing the current
will not increase the magnetization.
== Demagnetizing ferromagnets ==
Magnetized ferromagnetic materials can be
demagnetized (or degaussed) in the following
ways:
Heating a magnet past its Curie temperature;
the molecular motion destroys the alignment
of the magnetic domains. This always removes
all magnetization.
Placing the magnet in an alternating magnetic
field with intensity above the material's
coercivity and then either slowly drawing
the magnet out or slowly decreasing the magnetic
field to zero. This is the principle used
in commercial demagnetizers to demagnetize
tools, erase credit cards, hard disks, and
degaussing coils used to demagnetize CRTs.
Some demagnetization or reverse magnetization
will occur if any part of the magnet is subjected
to a reverse field above the magnetic material's
coercivity.
Demagnetization progressively occurs if the
magnet is subjected to cyclic fields sufficient
to move the magnet away from the linear part
on the second quadrant of the B-H curve of
the magnetic material (the demagnetization
curve).
Hammering or jarring: mechanical disturbance
tends to randomize the magnetic domains and
reduce magnetization of an object, but may
cause unacceptable damage.
== Types of permanent magnets ==
=== Magnetic metallic elements ===
Many materials have unpaired electron spins,
and the majority of these materials are paramagnetic.
When the spins interact with each other in
such a way that the spins align spontaneously,
the materials are called ferromagnetic (what
is often loosely termed as magnetic). Because
of the way their regular crystalline atomic
structure causes their spins to interact,
some metals are ferromagnetic when found in
their natural states, as ores. These include
iron ore (magnetite or lodestone), cobalt
and nickel, as well as the rare earth metals
gadolinium and dysprosium (when at a very
low temperature). Such naturally occurring
ferromagnets were used in the first experiments
with magnetism. Technology has since expanded
the availability of magnetic materials to
include various man-made products, all based,
however, on naturally magnetic elements.
=== Composites ===
Ceramic, or ferrite, magnets are made of a
sintered composite of powdered iron oxide
and barium/strontium carbonate ceramic. Given
the low cost of the materials and manufacturing
methods, inexpensive magnets (or non-magnetized
ferromagnetic cores, for use in electronic
components such as portable AM radio antennas)
of various shapes can be easily mass-produced.
The resulting magnets are non-corroding but
brittle and must be treated like other ceramics.
Alnico magnets are made by casting or sintering
a combination of aluminium, nickel and cobalt
with iron and small amounts of other elements
added to enhance the properties of the magnet.
Sintering offers superior mechanical characteristics,
whereas casting delivers higher magnetic fields
and allows for the design of intricate shapes.
Alnico magnets resist corrosion and have physical
properties more forgiving than ferrite, but
not quite as desirable as a metal. Trade names
for alloys in this family include: Alni, Alcomax,
Hycomax, Columax, and Ticonal.Injection-molded
magnets are a composite of various types of
resin and magnetic powders, allowing parts
of complex shapes to be manufactured by injection
molding. The physical and magnetic properties
of the product depend on the raw materials,
but are generally lower in magnetic strength
and resemble plastics in their physical properties.
Flexible magnets are composed of a high-coercivity
ferromagnetic compound (usually ferric oxide)
mixed with a plastic binder. This is extruded
as a sheet and passed over a line of powerful
cylindrical permanent magnets. These magnets
are arranged in a stack with alternating magnetic
poles facing up (N, S, N, S...) on a rotating
shaft. This impresses the plastic sheet with
the magnetic poles in an alternating line
format. No electromagnetism is used to generate
the magnets. The pole-to-pole distance is
on the order of 5 mm, but varies with manufacturer.
These magnets are lower in magnetic strength
but can be very flexible, depending on the
binder used.
=== Rare-earth magnets ===
Rare earth (lanthanoid) elements have a partially
occupied f electron shell (which can accommodate
up to 14 electrons). The spin of these electrons
can be aligned, resulting in very strong magnetic
fields, and therefore, these elements are
used in compact high-strength magnets where
their higher price is not a concern. The most
common types of rare-earth magnets are samarium-cobalt
and neodymium-iron-boron (NIB) magnets.
=== Single-molecule magnets (SMMs) and single-chain
magnets (SCMs) ===
In the 1990s, it was discovered that certain
molecules containing paramagnetic metal ions
are capable of storing a magnetic moment at
very low temperatures. These are very different
from conventional magnets that store information
at a magnetic domain level and theoretically
could provide a far denser storage medium
than conventional magnets. In this direction,
research on monolayers of SMMs is currently
under way. Very briefly, the two main attributes
of an SMM are:
a large ground state spin value (S), which
is provided by ferromagnetic or ferrimagnetic
coupling between the paramagnetic metal centres
a negative value of the anisotropy of the
zero field splitting (D)Most SMMs contain
manganese but can also be found with vanadium,
iron, nickel and cobalt clusters. More recently,
it has been found that some chain systems
can also display a magnetization that persists
for long times at higher temperatures. These
systems have been called single-chain magnets.
=== Nano-structured magnets ===
Some nano-structured materials exhibit energy
waves, called magnons, that coalesce into
a common ground state in the manner of a Bose–Einstein
condensate.
=== Rare-earth-free permanent magnets ===
The United States Department of Energy has
identified a need to find substitutes for
rare-earth metals in permanent-magnet technology,
and has begun funding such research. The Advanced
Research Projects Agency-Energy (ARPA-E) has
sponsored a Rare Earth Alternatives in Critical
Technologies (REACT) program to develop alternative
materials. In 2011, ARPA-E awarded 31.6 million
dollars to fund Rare-Earth Substitute projects.
=== Costs ===
The current cheapest permanent magnets, allowing
for field strengths, are flexible and ceramic
magnets, but these are also among the weakest
types. The ferrite magnets are mainly low-cost
magnets since they are made from cheap raw
materials: iron oxide and Ba- or Sr-carbonate.
However, a new low cost magnet, Mn-Al alloy,
has been developed and is now dominating the
low-cost magnets field. It has a higher saturation
magnetization than the ferrite magnets. It
also has more favorable temperature coefficients,
although it can be thermally unstable.
Neodymium-iron-boron (NIB) magnets are among
the strongest. These cost more per kilogram
than most other magnetic materials but, owing
to their intense field, are smaller and cheaper
in many applications.
=== Temperature ===
Temperature sensitivity varies, but when a
magnet is heated to a temperature known as
the Curie point, it loses all of its magnetism,
even after cooling below that temperature.
The magnets can often be remagnetized, however.
Additionally, some magnets are brittle and
can fracture at high temperatures.
The maximum usable temperature is highest
for alnico magnets at over 540 °C (1,000
°F), around 300 °C (570 °F) for ferrite
and SmCo, about 140 °C (280 °F) for NIB
and lower for flexible ceramics, but the exact
numbers depend on the grade of material.
== Electromagnets ==
An electromagnet, in its simplest form, is
a wire that has been coiled into one or more
loops, known as a solenoid. When electric
current flows through the wire, a magnetic
field is generated. It is concentrated near
(and especially inside) the coil, and its
field lines are very similar to those of a
magnet. The orientation of this effective
magnet is determined by the right hand rule.
The magnetic moment and the magnetic field
of the electromagnet are proportional to the
number of loops of wire, to the cross-section
of each loop, and to the current passing through
the wire.If the coil of wire is wrapped around
a material with no special magnetic properties
(e.g., cardboard), it will tend to generate
a very weak field. However, if it is wrapped
around a soft ferromagnetic material, such
as an iron nail, then the net field produced
can result in a several hundred- to thousandfold
increase of field strength.
Uses for electromagnets include particle accelerators,
electric motors, junkyard cranes, and magnetic
resonance imaging machines. Some applications
involve configurations more than a simple
magnetic dipole; for example, quadrupole and
sextupole magnets are used to focus particle
beams.
== Units and calculations ==
For most engineering applications, MKS (rationalized)
or SI (Système International) units are commonly
used. Two other sets of units, Gaussian and
CGS-EMU, are the same for magnetic properties
and are commonly used in physics.In all units,
it is convenient to employ two types of magnetic
field, B and H, as well as the magnetization
M, defined as the magnetic moment per unit
volume.
The magnetic induction field B is given in
SI units of teslas (T). B is the magnetic
field whose time variation produces, by Faraday's
Law, circulating electric fields (which the
power companies sell). B also produces a deflection
force on moving charged particles (as in TV
tubes). The tesla is equivalent to the magnetic
flux (in webers) per unit area (in meters
squared), thus giving B the unit of a flux
density. In CGS, the unit of B is the gauss
(G). One tesla equals 104 G.
The magnetic field H is given in SI units
of ampere-turns per meter (A-turn/m). The
turns appear because when H is produced by
a current-carrying wire, its value is proportional
to the number of turns of that wire. In CGS,
the unit of H is the oersted (Oe). One A-turn/m
equals 4π×10−3 Oe.
The magnetization M is given in SI units of
amperes per meter (A/m). In CGS, the unit
of M is the oersted (Oe). One A/m equals 10−3
emu/cm3. A good permanent magnet can have
a magnetization as large as a million amperes
per meter.
In SI units, the relation B = μ0(H + M) holds,
where μ0 is the permeability of space, which
equals 4π×10−7 T•m/A. In CGS, it is
written as B = H + 4πM. (The pole approach
gives μ0H in SI units. A μ0M term in SI
must then supplement this μ0H to give the
correct field within B, the magnet. It will
agree with the field B calculated using Ampèrian
currents).Materials that are not permanent
magnets usually satisfy the relation M = χH
in SI, where χ is the (dimensionless) magnetic
susceptibility. Most non-magnetic materials
have a relatively small χ (on the order of
a millionth), but soft magnets can have χ
on the order of hundreds or thousands. For
materials satisfying M = χH, we can also
write B = μ0(1 + χ)H = μ0μrH = μH, where
μr = 1 + χ is the (dimensionless) relative
permeability and μ =μ0μr is the magnetic
permeability. Both hard and soft magnets have
a more complex, history-dependent, behavior
described by what are called hysteresis loops,
which give either B vs. H or M vs. H. In CGS,
M = χH, but χSI = 4πχCGS, and μ = μr.
Caution: in part because there are not enough
Roman and Greek symbols, there is no commonly
agreed-upon symbol for magnetic pole strength
and magnetic moment. The symbol m has been
used for both pole strength (unit A•m, where
here the upright m is for meter) and for magnetic
moment (unit A•m2). The symbol μ has been
used in some texts for magnetic permeability
and in other texts for magnetic moment. We
will use μ for magnetic permeability and
m for magnetic moment. For pole strength,
we will employ qm. For a bar magnet of cross-section
A with uniform magnetization M along its axis,
the pole strength is given by qm = MA, so
that M can be thought of as a pole strength
per unit area.
=== Fields of a magnet ===
Far away from a magnet, the magnetic field
created by that magnet is almost always described
(to a good approximation) by a dipole field
characterized by its total magnetic moment.
This is true regardless of the shape of the
magnet, so long as the magnetic moment is
non-zero. One characteristic of a dipole field
is that the strength of the field falls off
inversely with the cube of the distance from
the magnet's center.
Closer to the magnet, the magnetic field becomes
more complicated and more dependent on the
detailed shape and magnetization of the magnet.
Formally, the field can be expressed as a
multipole expansion: A dipole field, plus
a quadrupole field, plus an octupole field,
etc.
At close range, many different fields are
possible. For example, for a long, skinny
bar magnet with its north pole at one end
and south pole at the other, the magnetic
field near either end falls off inversely
with the square of the distance from that
pole.
=== Calculating the magnetic force ===
==== 
Pull force of a single magnet ====
The strength of a given magnet is sometimes
given in terms of its pull force— its ability
to move (push/ pull) other objects. The pull
force exerted by either an electromagnet or
a permanent magnet at the "air gap" (i.e.,
the point in space where the magnet ends)
is given by the Maxwell equation:
F
=
B
2
A
2
μ
0
{\displaystyle F={{B^{2}A} \over {2\mu _{0}}}}
,where
F is force (SI unit: newton)
A is the cross section of the area of the
pole in square meters
B is the magnetic induction exerted by the
magnetTherefore, if a magnet is acting vertically,
it can lift a mass m in kilograms given by
the simple equation:
m
=
B
2
A
2
μ
0
g
n
{\displaystyle m={{B^{2}A} \over {2\mu _{0}g_{n}}}}
.
==== Force between two magnetic poles ====
Classically, the force between two magnetic
poles is given by:
F
=
μ
q
m
1
q
m
2
4
π
r
2
{\displaystyle F={{\mu q_{m1}q_{m2}} \over
{4\pi r^{2}}}}
where
F is force (SI unit: newton)
qm1 and qm2 are the magnitudes of magnetic
poles (SI unit: ampere-meter)
μ is the permeability of the intervening
medium (SI unit: tesla meter per ampere, henry
per meter or newton per ampere squared)
r is the separation (SI unit: meter).The pole
description is useful to the engineers designing
real-world magnets, but real magnets have
a pole distribution more complex than a single
north and south. Therefore, implementation
of the pole idea is not simple. In some cases,
one of the more complex formulae given below
will be more useful.
==== Force between two nearby magnetized surfaces
of area A ====
The mechanical force between two nearby magnetized
surfaces can be calculated with the following
equation. The equation is valid only for cases
in which the effect of fringing is negligible
and the volume of the air gap is much smaller
than that of the magnetized material:
F
=
μ
0
H
2
A
2
=
B
2
A
2
μ
0
{\displaystyle F={\frac {\mu _{0}H^{2}A}{2}}={\frac
{B^{2}A}{2\mu _{0}}}}
where:
A is the area of each surface, in m2
H is their magnetizing field, in A/m
μ0 is the permeability of space, which equals
4π×10−7 T•m/A
B is the flux density, in T.
==== 
Force between two bar magnets ====
The force between two identical cylindrical
bar magnets placed end to end at large distance
z
≫
R
{\displaystyle z\gg R}
is approximately:,
F
≃
[
B
0
2
A
2
(
L
2
+
R
2
)
π
μ
0
L
2
]
[
1
z
2
+
1
(
z
+
2
L
)
2
−
2
(
z
+
L
)
2
]
{\displaystyle F\simeq \left[{\frac {B_{0}^{2}A^{2}\left(L^{2}+R^{2}\right)}{\pi
\mu _{0}L^{2}}}\right]\left[{\frac {1}{z^{2}}}+{\frac
{1}{(z+2L)^{2}}}-{\frac {2}{(z+L)^{2}}}\right]}
where:
B0 is the magnetic flux density very close
to each pole, in T,
A is the area of each pole, in m2,
L is the length of each magnet, in m,
R is the radius of each magnet, in m, and
z is the separation between the two magnets,
in m.
B
0
=
μ
0
2
M
{\displaystyle B_{0}\,=\,{\frac {\mu _{0}}{2}}M}
relates the flux density at the pole to the
magnetization of the magnet.Note that all
these formulations are based on Gilbert's
model, which is usable in relatively great
distances. In other models (e.g., Ampère's
model), a more complicated formulation is
used that sometimes cannot be solved analytically.
In these cases, numerical methods must be
used.
==== Force between two cylindrical magnets
====
For two cylindrical magnets with radius
R
{\displaystyle R}
and length
L
{\displaystyle L}
, with their magnetic dipole aligned, the
force can be asymptotically approximated at
large distance
z
≫
R
{\displaystyle z\gg R}
by,
F
(
z
)
≃
π
μ
0
4
M
2
R
4
[
1
z
2
+
1
(
z
+
2
L
)
2
−
2
(
z
+
L
)
2
]
{\displaystyle F(z)\simeq {\frac {\pi \mu
_{0}}{4}}M^{2}R^{4}\left[{\frac {1}{z^{2}}}+{\frac
{1}{(z+2L)^{2}}}-{\frac {2}{(z+L)^{2}}}\right]}
where
M
{\displaystyle M}
is the magnetization of the magnets and
z
{\displaystyle z}
is the gap between the magnets.
A measurement of the magnetic flux density
very close to the magnet
B
0
{\displaystyle B_{0}}
is related to
M
{\displaystyle M}
approximately by the formula
B
0
=
μ
0
2
M
{\displaystyle B_{0}={\frac {\mu _{0}}{2}}M}
The effective magnetic dipole can be written
as
m
=
M
V
{\displaystyle m=MV}
Where
V
{\displaystyle V}
is the volume of the magnet. For a cylinder,
this is
V
=
π
R
2
L
{\displaystyle V=\pi R^{2}L}
.
When
z
≫
L
{\displaystyle z\gg L}
, the point dipole approximation is obtained,
F
(
x
)
=
3
π
μ
0
2
M
2
R
4
L
2
1
z
4
=
3
μ
0
2
π
M
2
V
2
1
z
4
=
3
μ
0
2
π
m
1
m
2
1
z
4
{\displaystyle F(x)={\frac {3\pi \mu _{0}}{2}}M^{2}R^{4}L^{2}{\frac
{1}{z^{4}}}={\frac {3\mu _{0}}{2\pi }}M^{2}V^{2}{\frac
{1}{z^{4}}}={\frac {3\mu _{0}}{2\pi }}m_{1}m_{2}{\frac
{1}{z^{4}}}}
which matches the expression of the force
between two magnetic dipoles.
== See also ==
== 
Notes
