# Magnet

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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
Magnetic field lines of a solenoid electromagnet, which are similar to a bar magnet as illustrated above with the iron filings

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A magnet (from Greek {{#invoke:Category handler|main}} Template:Transl, "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, the coil is wrapped around a core of "soft" ferromagnetic material such as steel, which greatly enhances the magnetic field produced by the coil.

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.

## Discovery and development

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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",[1] 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.[2][3][4] The properties of lodestones and their affinity for iron were written of by Pliny the Elder in his encyclopedia Naturalis Historia.[5]

By the 12th to 13th centuries AD, magnetic compasses were used in navigation in China, Europe, and elsewhere.[6]

## Physics

### Magnetic field

Iron filings that have oriented in the magnetic field produced by a bar magnet

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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:

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 B field is given in teslas.[7]

### Magnetic moment

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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,[8] 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.[9] 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.[10]

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

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The magnetization of a magnetized material is the local value of its magnetic moment per unit volume, usually denoted M, with units A/m.[11] 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

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

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Two different models exist for magnets: magnetic poles and atomic currents.

### 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 Template:Convert, around Template:Convert for ferrite and SmCo, about Template:Convert for NIB and lower for flexible ceramics, but the exact numbers depend on the grade of material.

## Electromagnets

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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.[30]

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

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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 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

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#### 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:[31]

${\displaystyle F={{B^{2}A} \over {2\mu _{0}}}}$,

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 magnet

Therefore, if a magnet is acting vertically, it can lift a mass m in kilograms given by the simple equation:

${\displaystyle m={{B^{2}A} \over {2\mu _{0}g_{n}}}}$.

#### Force between two magnetic poles

Classically, the force between two magnetic poles is given by:[32]

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

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:[33][34]

${\displaystyle F={\frac {\mu _{0}H^{2}A}{2}}={\frac {B^{2}A}{2\mu _{0}}}}$

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 is given by:[33]

${\displaystyle F=\left[{\frac {B_{0}^{2}A^{2}\left(L^{2}+R^{2}\right)}{\pi \mu _{0}L^{2}}}\right]\left[{\frac {1}{x^{2}}}+{\frac {1}{(x+2L)^{2}}}-{\frac {2}{(x+L)^{2}}}\right]}$

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.
${\displaystyle B_{0}\,=\,{\frac {\mu _{0}}{2}}M}$ 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 ${\displaystyle R}$ and height ${\displaystyle t}$, with their magnetic dipole aligned, the force can be well approximated (even at distances of the order of ${\displaystyle t}$) by,[35]

${\displaystyle F(x)={\frac {\pi \mu _{0}}{4}}M^{2}R^{4}\left[{\frac {1}{x^{2}}}+{\frac {1}{(x+2t)^{2}}}-{\frac {2}{(x+t)^{2}}}\right]}$

where ${\displaystyle M}$ is the magnetization of the magnets and ${\displaystyle x}$ is the gap between the magnets. In disagreement to the statement in the previous section, a measurement of the magnetic flux density very close to the magnet ${\displaystyle B_{0}}$ is related to ${\displaystyle M}$ by the formula

${\displaystyle B_{0}=\mu _{0}M}$

The effective magnetic dipole can be written as

${\displaystyle m=MV}$

Where ${\displaystyle V}$ is the volume of the magnet. For a cylinder, this is ${\displaystyle V=\pi R^{2}t}$.

When ${\displaystyle t<, the point dipole approximation is obtained,

${\displaystyle F(x)={\frac {3\pi \mu _{0}}{2}}M^{2}R^{4}t^{2}{\frac {1}{x^{4}}}={\frac {3\mu _{0}}{2\pi }}M^{2}V^{2}{\frac {1}{x^{4}}}={\frac {3\mu _{0}}{2\pi }}m_{1}m_{2}{\frac {1}{x^{4}}}}$

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

## Notes

1. The location of Magnesia is debated; it could be the regional unit or Magnesia ad Sipylum. See, for example, Template:Cite web
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18. Mice levitated in NASA lab. Livescience.com (2009-09-09). Retrieved on 2011-10-08.
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29. Frequently Asked Questions. Magnet sales. Retrieved on 2011-10-08.
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## References

• "The Early History of the Permanent Magnet". Edward Neville Da Costa Andrade, Endeavour, Volume 17, Number 65, January 1958. Contains an excellent description of early methods of producing permanent magnets.
• "positive pole n". The Concise Oxford English Dictionary. Catherine Soanes and Angus Stevenson. Oxford University Press, 2004. Oxford Reference Online. Oxford University Press.
• 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.