Editing Magnetic field (section)

==Interactions with magnets==
===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]].

===Magnetic torque on permanent magnets===
{{Main|Magnetic torque}}
If two like poles of two separate magnets are brought near each other, and one of the magnets is allowed to turn, it promptly rotates to align itself with the first. In this example, the magnetic field of the stationary magnet creates a ''magnetic torque'' on the magnet that is free to rotate. This magnetic torque {{math|'''τ'''}} tends to align a magnet's poles with the magnetic field lines. A compass, therefore, turns to align itself with Earth's magnetic field.

{{Multiple image|header=Torque on a dipole
| align = right | total_width =320
|image1=Dipole in uniform H field.svg | alt1 = |caption1=In the pole model of a dipole, an {{math|'''H'''}} field (to right) causes equal but opposite forces on a N pole ({{math|+''q''}}) and a S pole ({{math|−''q''}}) creating a torque.
|image2=Torque-current-loop.svg| alt2 = |caption2=Equivalently, a {{math|'''B'''}} field induces the same torque on a current loop with the same magnetic dipole moment.
|footer=}} In terms of the pole model, two equal and opposite magnetic charges experiencing the same {{math|'''H'''}} also experience equal and opposite forces. Since these equal and opposite forces are in different locations, this produces a torque proportional to the distance (perpendicular to the force) between them. With the definition of {{math|'''m'''}} as the pole strength times the distance between the poles, this leads to {{math|1=''τ'' = ''μ''<sub>0</sub> ''m H'' sin&thinsp;''θ''}}, where {{math|''μ''<sub>0</sub>}} is a constant called the [[vacuum permeability]], measuring {{val|4|end= π|e=-7}} [[Volt|V]]·[[Second|s]]/([[Ampere|A]]·[[meter|m]]) and {{mvar|θ}} is the angle between {{math|'''H'''}} and {{math|'''m'''}}.

Mathematically, the torque {{math|'''τ'''}} on a small magnet is proportional both to the applied magnetic field and to the magnetic moment {{math|'''m'''}} of the magnet:

<math display="block">\boldsymbol{\tau}=\mathbf{m}\times\mathbf{B} = \mu_0\mathbf{m}\times\mathbf{H}, \,</math>

where × represents the vector [[cross product]]. This equation includes all of the qualitative information included above. There is no torque on a magnet if {{math|'''m'''}} is in the same direction as the magnetic field, since the cross product is zero for two vectors that are in the same direction. Further, all other orientations feel a torque that twists them toward the direction of magnetic field.