Editing Magnetic field (section)

===Force between magnets===
{{Main|Force between magnets}}

Specifying the [[magnetic moment#Forces between two magnetic dipoles|force between two small magnets]] is quite complicated because it depends on the strength and [[Orientation (geometry)|orientation]] of both magnets and their distance and direction relative to each other. The force is particularly sensitive to rotations of the magnets due to magnetic torque. The force on each magnet depends on its magnetic moment and the magnetic field<ref group="note" name="ex04">Either {{math|'''B'''}} or {{math|'''H'''}} may be used for the magnetic field outside the magnet.</ref> of the other.

To understand the force between magnets, it is useful to examine the ''magnetic pole model'' given above. In this model, the ''{{math|'''H'''}}-field'' of one magnet pushes and pulls on ''both'' poles of a second magnet. If this {{math|'''H'''}}-field is the same at both poles of the second magnet then there is no net force on that magnet since the force is opposite for opposite poles. If, however, the magnetic field of the first magnet is ''nonuniform'' (such as the {{math|'''H'''}} near one of its poles), each pole of the second magnet sees a different field and is subject to a different force. This difference in the two forces moves the magnet in the direction of increasing magnetic field and may also cause a net torque.

This is a specific example of a general rule that magnets are attracted (or repulsed depending on the orientation of the magnet) into regions of higher magnetic field. Any non-uniform magnetic field, whether caused by permanent magnets or electric currents, exerts a force on a small magnet in this way.

The details of the Amperian loop model are different and more complicated but yield the same result: that magnetic dipoles are attracted/repelled into regions of higher magnetic field. Mathematically, the force on a small magnet having a magnetic moment {{math|'''m'''}} due to a magnetic field {{math|'''B'''}} is:<ref name=Trigg>{{cite book |title=AIP physics desk reference |author1=E. Richard Cohen |author2=David R. Lide |author3=George L. Trigg |url=https://books.google.com/books?id=JStYf6WlXpgC&pg=PA381 |page=381 |isbn=978-0-387-98973-0 |publisher=Birkhäuser |year=2003 |edition=3}}</ref>{{rp|at=Eq. 11.42}}

<math display="block">\mathbf{F} = \boldsymbol{\nabla} \left(\mathbf{m}\cdot\mathbf{B}\right),</math>

where the [[gradient]] {{math|'''∇'''}} is the change of the quantity {{math|'''m''' · '''B'''}} per unit distance and the direction is that of maximum increase of {{math|'''m''' · '''B'''}}. The [[dot product]] {{math|1='''m''' · '''B''' = ''mB''cos(''θ'')}}, where {{math|''m''}} and {{math|''B''}} represent the [[magnitude (vector)|magnitude]] of the {{math|'''m'''}} and {{math|'''B'''}} vectors and {{math|''θ''}} is the angle between them. If {{math|'''m'''}} is in the same direction as {{math|'''B'''}} then the dot product is positive and the gradient points "uphill" pulling the magnet into regions of higher {{math|'''B'''}}-field (more strictly larger {{math|'''m''' · '''B'''}}). This equation is strictly only valid for magnets of zero size, but is often a good approximation for not too large magnets. The magnetic force on larger magnets is determined by dividing them into smaller regions each having their own {{math|'''m'''}} then [[Integral|summing up the forces on each of these very small regions]].