# Magnet

A "horseshoe magnet" made of alnico, an iron alloy. The magnet is made in the shape of a horseshoe to bring the two magnetic poles close to each other, to create a strong magnetic field there that can pick up heavy pieces of iron.
Iron filings that have oriented in the magnetic field produced by a bar magnet
Magnetic field lines of a solenoid electromagnet, which are similar to a bar magnet as illustrated above with the iron filings

A magnet (from Greek , "Magnesian stone") 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 iron, nickel, 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 powerful 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.

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, an electromagnet is wrapped around a core of ferromagnetic material like steel, which enhances the magnetic field produced by the coil.

The overall strength of a magnet is measured by its magnetic moment or, alternately, the total magnetic flux it produces. The local strength of the magnetism in a material is measured by its magnetization.

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## History

Ancient people learned about magnetism from lodestones, naturally magnetized pieces of iron ore. They are naturally created magnets, which attract pieces of iron. The word magnet in Greek meant "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.[1][2][3] The properties of lodestones and their affinity for iron were written of by Pliny the Elder in his encyclopedia Naturalis Historia.[4]

By the 12th to 13th centuries AD, magnetic compasses were used in navigation in China, Europe, and elsewhere.[5] For more details, see the main Magnetism article.

## Background on the physics of magnetism and magnets

An ovoid-shaped rare earth magnet hanging from another

### Magnetic field

The magnetic field (usually denoted B) is a vector field. The magnetic field vector at a given point in space is specified by two properties:

1. Its direction, which is along the orientation of a compass needle.
2. 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 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,[6] 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.

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 is a magnet, with 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.

### Two models for magnets: magnetic poles and atomic currents

Field of a cylindrical bar magnet calculated with Ampère's model

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.

Gilbert model: However, a version of the magnetic-pole approach is used by professional magneticians to design permanent magnets. In this approach, the pole surfaces of a permanent magnet are imagined to be covered with so-called magnetic charge, north-pole particles on the north pole and south-pole particles on the south pole that are the source of the magnetic field lines. If the magnetic-pole distribution is known, then outside the magnet, the pole model gives the magnetic field exactly. In the interior of the magnet, this model fails to give the correct field (see #Units and calculations, below). This pole model is also called the Gilbert model of a magnetic dipole.[7] Griffiths suggests (p. 258): "My advice is to use the Gilbert model, if you like, to get an intuitive 'feel' for a problem, but never rely on it for quantitative results."

Ampère model: 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. (Since scraping off the outer layer of a magnet will not destroy its magnetic field, it can be seen that this is just a model, and the tiny currents are actually distributed throughout the material). The right-hand rule tells which direction the current flows. The Ampère model gives the exact magnetic field both inside and outside the magnet. It is usually difficult to calculate the Ampèrian currents on the surface of a magnet, whereas it is often easier to find the effective poles for the same magnet.

### Pole naming conventions

The north pole of the magnet is the pole that, when the magnet is freely suspended, points towards the Earth's north magnetic pole in northern Canada. Since opposite poles (north and south) attract, whereas like poles (north and north, or south and south) repel, the Earth's present geographic north pole is thus actually its magnetic south pole. Confounding the situation further, the Earth's magnetic field has reversed itself many times in the distant past. As a practical matter, in order 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 calibration method would be to compare it to an electromagnet, whose poles can be identified by the right-hand rule.

### 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 below.
• Paramagnetic substances, such as platinum, aluminium, and oxygen, are weakly attracted to a magnet. This effect is hundreds of thousands of times weaker than the attraction 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 [8] 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 of magnets

Hard disks record data on a thin magnetic coating.
• 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 ATM 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).
• Common televisions and computer monitors: TV and computer screens containing a cathode ray tube employ an electromagnet to guide electrons to the screen. Plasma screens and LCDs use different technologies.
• 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.
Magnetic hand separator for heavy minerals
• 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.
• Transformers: Transformers are devices that transfer electric energy between two windings of wire that are electrically isolated but are coupled magnetically.
• Chucks: 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. For example: how is the strength of a magnet affected by glass, plastic, and cardboard?
Magnets have many uses in toys. M-tic uses magnetic rods connected to metal spheres for construction. Note the geodesic pyramid.
• 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.
• Magnets can be used to make jewellery. 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, steel, and nickel) from non-magnetic metals (aluminium, 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.
• Magnetic levitation transport, or maglev, is a form of transportation that suspends, guides and propels vehicles (especially trains) through electromagnetic force. 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 hazard 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.[9]

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 an MRI, which is a magnetic 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; one death has been reported.[10]

## Magnetizing ferromagnets

Ferromagnetic materials can be magnetized in the following ways:

• Heating the object above 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. A magnetic field much stronger than the Earth's can be generated inside a solenoid by passing direct current through it.
• Stroking: An existing magnet is moved from one end of the item to the other repeatedly in the same direction.

## 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 an 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 and erase credit cards and 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.
• Demagnetisation 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 demagnetisation curve).
• Hammering or jarring: the mechanical disturbance tends to randomize the magnetic domains. Will leave some residual magnetization.

## Types of permanent magnets

A stack of ferrite 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 (ferro)magnetic 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 (ferro)magnets 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

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 radio antennas, for example) of various shapes can be easily mass-produced. The resulting magnets are non-corroding but brittle and must be treated like other ceramics.

#### Alnico

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.[11]

#### Injection-molded

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

Flexible magnets are similar to injection-molded magnets, using a flexible resin or binder such as vinyl, and produced in flat strips, shapes or sheets. These magnets are lower in magnetic strength but can be very flexible, depending on the binder used. Flexible magnets can be used in industrial printers.

### 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:

1. a large ground state spin value (S), which is provided by ferromagnetic or ferrimagnetic coupling between the paramagnetic metal centres
2. 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.[12][13]

### Costs

The current cheapest permanent magnets, allowing for field strengths, are flexible and ceramic magnets, but these are also among the weakest types. 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.[14]

### 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.

## Microscopic origin of magnetism

Every electron, on account of its spin, is a small magnet (see electron magnetic dipole moment). In most materials, the countless electrons have randomly oriented spins, leaving no magnetic effect on average. However, in a bar magnet, many of the electron spins are aligned in the same direction, so they act cooperatively, creating a net magnetic field.

In addition to the electron's intrinsic magnetic field, there is sometimes an additional magnetic field that results from the electron's orbital motion around the nucleus. This effect is analogous to how a current-carrying loop of wire generates a magnetic field (see Magnetic dipole). Again, ordinarily, the motion of the electrons is such that there is no average field from the material, but in certain conditions, the motion can line up so as to produce a measurable total field.

### Ferromagnetism

A ferromagnet, like a paramagnetic substance, has unpaired electrons. However, in addition to the electrons' intrinsic magnetic moments wanting to be parallel to an applied field, there is also in these materials a tendency for these magnetic moments to want to be parallel to each other. Thus, even when the applied field is removed, the electrons in the material can keep each other continually pointed in the same direction.

Every ferromagnetic substance has its own individual temperature, called the Curie temperature, or Curie point, above which it loses its ferromagnetic properties. This is because the thermal tendency to disorder overwhelms the energy lowering due to ferromagnetic order.

### Magnetic domains

Magnetic domains in ferromagnetic material

The magnetic moment of atoms in a ferromagnetic material causes them to behave something like tiny permanent magnets. They stick together and align themselves into small regions of more or less uniform alignment called magnetic domains or Weiss domains. Magnetic domains can be observed with a magnetic force microscope to reveal magnetic-domain boundaries that resemble white lines in the sketch. There are many scientific experiments that can physically show magnetic fields.

Effect of a magnet on the domains

When a domain contains too many molecules, it becomes unstable and divides into two domains aligned in opposite directions, so that they stick together more stably as shown at the right.

When exposed to a magnetic field, the domain boundaries move so that the domains aligned with the magnetic field grow and dominate the structure as shown at the left. When the magnetizing field is removed, the domains may not return to an unmagnetized state. This results in the ferromagnetic material being magnetized, forming a permanent magnet.

When magnetized strongly enough that the prevailing domain overruns all others to result in only one single domain, the material is magnetically saturated. When a magnetized ferromagnetic material is heated to the Curie point temperature, the molecules are agitated to the point that the magnetic domains lose the organization and the magnetic properties they cause cease. When the material is cooled, this domain alignment structure spontaneously returns, in a manner roughly analogous to how a liquid can freeze into a crystalline solid.

### Ferrimagnetism

Ferrimagnetic ordering

Like ferromagnetism, ferrimagnets retain their magnetization in the absence of a field. However, like antiferromagnets, neighboring pairs of electron spins like to point in opposite directions. These two properties are not contradictory, because in the optimal geometrical arrangement, there is more magnetic moment from the sublattice of electrons that point in one direction than from the sublattice that points in the opposite direction.

The first discovered magnetic substance, magnetite, was originally believed to be a ferromagnet; Louis Néel disproved this, however, with the discovery of ferrimagnetism.

## 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.

1. The magnetic induction field B is given in SI units of teslas (T). B is the true 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.
2. The magnetic field H is given in SI units of ampere-turns per meter (A-turn/m). The turns appears 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.
3. 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.
4. 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

#### Force between two magnetic poles

Classically, the force between two magnetic poles is given by:[15]

$F={{\mu q_{m1} q_{m2}}\over{4\pi r^2}}$

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 practicing magneticians who design 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:[16][17]

$F=\frac{\mu_0 H^2 A}{2} = \frac{B^2 A}{2 \mu_0}$

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 is given by:[16]

$F=\left[\frac {B_0^2 A^2 \left( L^2+R^2 \right)} {\pi\mu_0L^2}\right] \left[{\frac 1 {x^2}} + {\frac 1 {(x+2L)^2}} - {\frac 2 {(x+L)^2}} \right]$

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
x is the separation between the two magnets, in m.
$B_0 \,=\, \frac{\mu_0}{2}M$ 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 and height t, with their magnetic dipole aligned, the force can be well approximated (even at distances of the order of t) by,[18]

$F(x) = \frac{\pi\mu_0}{4} M^2 R^4 \left[\frac{1}{x^2} + \frac{1}{(x+2t)^2} - \frac{2}{(x + t)^2}\right]$

where M is the magnetization of the magnets and x is the distance between them. In disagreement to the statement in the previous section, a measurement of the magnetic flux density very close to the magnet B0 is related to M by the formula

B0 = μ0M

The effective magnetic dipole can be written as

m = MV

Where V is the volume of the magnet. For a cylinder, this is V = πR2t.

When t < < x, the point dipole approximation is obtained,

$F(x) = \frac{3\pi\mu_0}{2} M^2 R^4 t^2\frac{1}{x^4} = \frac{3\mu_0}{2\pi} M^2 V^2\frac{1}{x^4} = \frac{3\mu_0}{2\pi} m_1 m_2\frac{1}{x^4}$

which matches the expression of the force between two magnetic dipoles.

## Footnotes and in-line references

1. ^ Fowler, Michael (1997). "Historical Beginnings of Theories of Electricity and Magnetism". Retrieved 2008-04-02.
2. ^ Vowles, Hugh P. (1932). "Early Evolution of Power Engineering". Isis (University of Chicago Press) 17 (2): 412–420 [419–20]. doi:10.1086/346662.
3. ^ Li Shu-hua, “Origine de la Boussole 11. Aimant et Boussole,” Isis, Vol. 45, No. 2. (Jul., 1954), p.175
4. ^ http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.02.0137:book=34:chapter=42&highlight=magnet
5. ^ Schmidl, Petra G. (1996-1997). "Two Early Arabic Sources On The Magnetic Compass". Journal of Arabic and Islamic Studies 1: 81–132.
6. ^ Knight, Jones, & Field, "College Physics" (2007) p. 815
7. ^ Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X. , section 6.1.
8. ^ Mice levitated in NASA lab
9. ^ Schenck JF (2000). "Safety of strong, static magnetic fields". J Magn Reson Imaging 12 (1): 2–19. doi:10.1002/1522-2586(200007)12:1<2::AID-JMRI2>3.0.CO;2-V. PMID 10931560.
10. ^ Oestreich AE (2008). "Worldwide survey of damage from swallowing multiple magnets". Pediatr Radiol 39 (2): 142. doi:10.1007/s00247-008-1059-7. PMID 19020871.
11. ^ Brady, George Stuart; Henry R. Clauser & John A. Vaccari (2002). Materials Handbook: An Encyclopedia for Managers. McGraw-Hill Professional. pp. 577. ISBN 007136076X.
12. ^ "Nanomagnets Bend The Rules". Retrieved November 14, 2005.
13. ^ Della Torre, E.; Bennett, L.; Watson, R. (2005). "Extension of the Bloch T3/2 Law to Magnetic Nanostructures: Bose-Einstein Condensation". Physical Review Letters 94 (14): 147210. doi:10.1103/PhysRevLett.94.147210.
14. ^ Magnet sales- Frequently Asked Questions
15. ^ "Basic Relationships". Geophysics.ou.edu. Retrieved 2009-10-19.
16. ^ a b "Magnetic Fields and Forces". Retrieved 2009-12-24.
17. ^
18. ^ David Vokoun, Marco Beleggia, Ludek Heller, Petr Sittner, Magnetostatic interactions and forces between cylindrical permanent magnets, Journal of Magnetism and Magnetic Materials, Volume 321, Issue 22, November 2009, Pages 3758-3763, DOI:10.1016/j.jmmm.2009.07.030. [Article http://www.sciencedirect.com/science/article/B6TJJ-4WSRF7C-2/2/5ede3141fb91e35e83abf6edab5abb94]. Retrieved 02.2009

## Printed references

• Wayne M. Saslow, Electricity, Magnetism, and Light, Academic (2002). ISBN 0-12-619455-6. Chapter 9 discusses magnets and their magnetic fields using the concept of magnetic poles, but it also gives evidence that magnetic poles do not really exist in ordinary matter. Chapters 10 and 11, following what appears to be a 19th-century approach, use the pole concept to obtain the laws describing the magnetism of electric currents.
• Edward P. Furlani, Permanent Magnet and Electromechanical Devices: Materials, Analysis and Applications, Academic Press Series in Electromagnetism (2001). ISBN 0-12-269951-3.