A magnet (from Greek wiktionary:µa???t?? ''[vague]. One A/m equals 10-3 emu. A good permanent magnet can have a magnetization as large as a million amperes per meter. Magnetic fields produced by current-carrying wires would require comparably huge currents per unit length, one reason we employ permanent magnets and electromagnets. # In SI units, the relation B  = µ 0(H  + M ) holds, where µ 0 is the permeability of space, which equals $4\backslash pi$ x 10-7 tesla meters per ampere. In CGS it is written as B  = H  + 4pM . [The pole approach gives µ 0H in SI units. A µ 0M term in SI must then supplement this µ 0H to give the correct field within B the magnet. It will agree with the field B calculated using Amperian currents.] Materials that are not permanent magnets usually satisfy the relation M  = ''?H in SI, where ? is the (dimensionless) magnetic susceptibility. Most non-magnetic materials have a relatively small ? (on the order of a millionth), but soft magnets can have ? on the order of hundreds or thousands. For materials satisfying M  = ''?H , we can also write B  = µ 0(1 + ? )H  = µ 0µ rH  = ''µH , where µ r = 1 + ? is the (dimensionless) relative permeability and $\backslash mu=\backslash mu\_0\backslash mu\_r$ 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  = ''?H , but ? SI = 4p? CGS, and $\backslash mu=\backslash mu\_r$. 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 q m . For a bar magnet of cross-section A with uniform magnetization M along its axis, the pole strength is given by q m  = MA , so that M can be thought of as a pole strength per unit area.

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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 nonzero. 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

Calculating the attractive or repulsive force between two magnets is, in the general case, an extremely complex operation, as it depends on the shape, magnetization, orientation and separation of the magnets.

Force between two magnetic poles

The force between two magnetic poles is given by: 2

where

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 attracting surfaces of area A and equal but opposite magnetizations M

where

Force between two bar magnets

The force between two identical cylindrical bar magnets placed end to end is given by: where B ₀ =$\backslash frac\{\backslash mu\_0\}\{2\}$M relates the flux density at the pole to the magnetization of the magnet.

Footnotes and in-line references

1. Knight, Jones, & Field, "College Physics" (2007) p. 815
2. Introduction to Electrodynamics (3rd ed.)
3. Safety of strong, static magnetic fields
4. Worldwide survey of damage from swallowing multiple magnets
5. Nanomagnets Bend The Rules
6. Extension of the Bloch T^3/2 Law to Magnetic Nanostructures: Bose-Einstein Condensation
7. Magnet sales- Frequently Asked Questions

Online references

• , good complete tree diagram of electromagnetic relationships with magnets
• Maxwell's Equations and some history...
• or a Coil Gun

Printed references

1. "positive pole n." The Concise Oxford English Dictionary . Ed. Catherine Soanes and Angus Stevenson. Oxford University Press, 2004. Oxford Reference Online. Oxford University Press.

2. 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.

3. Edward P. Furlani, "Permanent Magnet and Electromechanical Devices: Materials, Analysis and Applications", Academic Press Series in Electromagnetism (2001). ISBN 0-12-269951-3.

See also

B-H Analyzer - a mechanism for measuring the response of materials to an applied field Dipole magnet - a magnet constructed to create a homogeneous magnetic field over some distance Earnshaw's theorem - static magnetic levitation under gravity is impossible except for diamagnets or with control systems Electromagnet - magnets produced by electric current Electromagnetism - the branch of physics related to magnetic and electric fields Electromagnetic field Diamagnetism - materials that are repelled by magnetic fields Halbach Array - a configuration of magnets that focuses the field Halbach cylinder - a cylindrical very strong (~5T) magnetic configuration that has little external field

Magnetic dipole Magnetic levitation - magnetism used to levitate objects Magnetic monopole - a theoretical source of magnetism that has never been found in real life Magnetism - the general study of magnetic phenomena Magneto Molecular magnet Paramagnetism Quadrupole magnet - group of four magnets laid out so that the dipole terms cancel Reversible temperature coefficient Sextupole magnet - group of six magnets laid in alternating arrangement of poles Supermagnets - Neodymium magnets

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