Magnet
A magnet 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, steel, nickel, ...
s exert
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
s and
torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
s on each other due to the rules of
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
. The forces of attraction field of magnets are due to microscopic currents of electrically charged
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
s orbiting nuclei and the intrinsic magnetism of fundamental particles (such as electrons) that make up the material. Both of these are modeled quite well as tiny loops of current called
magnetic dipoles that produce their own
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
and are affected by external magnetic fields. The most elementary force between magnets is the
magnetic dipole–dipole interaction. If all of the magnetic dipoles that make up two magnets are known then the net force on both magnets can be determined by summing up all these interactions between the dipoles of the first magnet and that of the second.
It is often more convenient to model the force between two magnets as being due to forces between magnetic poles having
magnetic charges spread over them. Positive and negative magnetic charge is always connected by a string of magnetized material; isolated magnetic charge does not exist. This model works well in predicting the forces between simple magnets where good models of how the magnetic charge is distributed are available.
Magnetic poles vs. atomic currents
The field of a magnet is the sum of fields from all
magnetized
Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...
volume elements, which consist of small
magnetic dipoles on an atomic level. The direct summation of all those dipole fields would require three-dimensional
integration
Integration may refer to:
Biology
*Multisensory integration
*Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome into a host genome
*DNA integration, by means of site-specific recombinase technology, ...
just to obtain the field of one magnet, which may be intricate.
In case of a homogeneous magnetization, the problem can be simplified at least in two different ways, using
Stokes' theorem
Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
. Upon integration along the direction of magnetization, all dipoles along the line of integration cancel each other, except at the magnet's end surface. The field then emerges only from those (mathematical) magnetic charges spread over the magnet's end facets. On the contrary, when integrating over a magnetized area orthogonal to the direction of magnetization, the dipoles within this area
cancel each other, except at the magnet's outer surface, where they (mathematically) sum up to a ring current. This is called Ampère model. In both models, only two-dimensional distributions over the magnet's surface have to be considered, which is simpler than the original three-dimensional problem.
Magnetic-charge model: In the magnetic-charge model, 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. The field due to magnetic charges is obtained through
Coulomb's law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
with magnetic instead of electric charges. If the magnetic pole distribution is known, then the pole model gives the exact distribution of the
magnetic field intensity
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
H both inside and outside the magnet. The surface charge distribution is uniform, if the magnet is homogeneously magnetized and has flat end facets (such as a cylinder or prism).
Ampère model: In the
Ampère model, all magnetization is due to the effect of microscopic, or atomic, circular ''
bound currents'', also called ''Ampèrian currents'' throughout the material. The net effect of these microscopic bound currents is to make the magnet behave as if there is a macroscopic
electric current
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...
flowing in loops in the magnet with the magnetic field normal to the loops. The field due to such currents is then obtained through the
Biot–Savart law. The Ampère model gives the correct magnetic flux density B both inside and outside the magnet. It is sometimes difficult to calculate the Ampèrian currents on the surface of a magnet.
Magnetic dipole moment
Far away from a magnet, its magnetic field is almost always described (to a good approximation) by a
dipole field characterized by its total
magnetic dipole moment
In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnets ...
, ''m''. 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.
The magnetic moment of a
magnet
A magnet 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, steel, nickel, ...
is therefore a measure of its strength and orientation. A loop of
electric current
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...
, a bar
magnet
A magnet 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, steel, nickel, ...
, an
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
, a
molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
, and a
planet
A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
all have magnetic moments. More precisely, the term ''magnetic moment'' normally refers to a system's magnetic dipole moment, which produces the first term in the
multipole expansion
A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similarly ...
[The magnetic dipole portion of the magnetic field can be understood as being due to one pair of north/south poles. Higher order terms such as the quadrupole can be considered as due to 2 or more north/south poles ordered such that they have no lower order contribution. For example the quadrupole configuration has no net dipole moment.] of a general magnetic field.
Both the torque and force exerted on a magnet by an external magnetic field are proportional to that magnet's magnetic moment. The magnetic moment is a
vector: it has both a magnitude and direction. The direction of the magnetic moment points from the south to north pole of a magnet (inside the magnet). For example, the direction of the magnetic moment of a bar magnet, such as the one in a
compass
A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with ...
is the direction that the north poles points toward.
In the physically correct Ampère model, magnetic dipole moments are due to infinitesimally small loops of current. For a sufficiently small loop of current, ''I'', and area, ''A'', the magnetic dipole moment is:
where the direction of ''m'' is
normal to the area in a direction determined using the current and the
right-hand rule
In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors.
Most of th ...
. As such, the
SI unit of magnetic dipole moment is
ampere
The ampere (, ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to elect ...
meter
2. More precisely, to account for solenoids with many turns the unit of magnetic dipole moment is
Ampere-turn
The ampere-turn (A⋅t) is the MKS (metre–kilogram–second) unit of magnetomotive force (MMF), represented by a direct current of one ampere flowing in a single-turn loop in a vacuum. " Turns" refers to the winding number of an electrical con ...
meter
2.
In the magnetic-charge model, the magnetic dipole moment is due to two equal and opposite magnetic charges that are separated by a distance, ''d''. In this model, ''m'' is similar to the electric dipole moment ''p'' due to electrical charges:
where ''q''
m is the ‘magnetic charge’. The direction of the magnetic dipole moment points from the negative south pole to the positive north pole of this tiny magnet.
Magnetic force due to non-uniform magnetic field
Magnets are drawn along the magnetic field gradient. The simplest example of this is the attraction of opposite poles of two magnets. Every magnet produces a magnetic field that is stronger near its poles. If opposite poles of two separate magnets are facing each other, each of the magnets is drawn into the stronger magnetic field near the pole of the other. If like poles are facing each other though, they are repulsed from the larger magnetic field.
The magnetic-charge model predicts a correct mathematical form for this force and is easier to understand qualitatively. For if a magnet is placed in a uniform magnetic field then both poles will feel the same magnetic force but in opposite directions, since they have opposite magnetic charge. But, when a magnet is placed in the non-uniform field, such as that due to another magnet, the pole experiencing the large magnetic field will experience the large force and there will be a net force on the magnet. If the magnet is aligned with the magnetic field, corresponding to two magnets oriented in the same direction near the poles, then it will be drawn into the larger magnetic field. If it is oppositely aligned, such as the case of two magnets with like poles facing each other, then the magnet will be repelled from the region of higher magnetic field.
In the Ampère model, there is also a force on a magnetic dipole due to a non-uniform magnetic field, but this is due to
Lorentz force
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
s on the current loop that makes up the magnetic dipole. The force obtained in the case of a current loop model is
where the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
∇ is the change of the quantity per unit distance, and the direction is that of maximum increase of . To understand this equation, note that the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
, where ''m'' and ''B'' represent the
magnitude of the m and B vectors and ''θ'' is the angle between them. If m is in the same direction as B then the dot product is positive and the gradient points 'uphill' pulling the magnet into regions of higher B-field (more strictly larger m · B). B represents the strength and direction of the magnetic field. 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 having their own m then summing up the forces on each of these regions.
Magnetic-charge model
The magnetic-charge model assumes that the magnetic forces between magnets are due to
magnetic charges near the poles. This model works even close to the magnet when the magnetic field becomes more complicated, and more dependent on the detailed shape and magnetization of the magnet than just the magnetic dipole contribution. Formally, the field can be expressed as a
multipole expansion
A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similarly ...
: A dipole field, plus a
quadrupole field, plus an octopole field, etc. in the Ampère model, but this can be very cumbersome mathematically.
Calculating the magnetic force
Calculating the attractive or repulsive force between two magnets is, in the general case, a very complex operation, as it depends on the shape, magnetization, orientation and separation of the magnets. The magnetic-charge model does depend on some knowledge of how the ‘magnetic charge’ is distributed over the magnetic poles. It is only truly useful for simple configurations even then. Fortunately, this restriction covers many useful cases.
Force between two magnetic poles
If both poles are small enough to be represented as single points then they can be considered to be point magnetic charges.
Classically, the force between two magnetic poles is given by:
where
*''F'' is force (SI unit:
newton
Newton most commonly refers to:
* Isaac Newton (1642–1726/1727), English scientist
* Newton (unit), SI unit of force named after Isaac Newton
Newton may also refer to:
Arts and entertainment
* ''Newton'' (film), a 2017 Indian film
* Newton ( ...
)
*''q''
''m''1 and ''q''
''m''2 are the magnitudes of
magnetic charge
In particle physics, a magnetic monopole is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magneti ...
on magnetic poles (SI unit:
ampere
The ampere (, ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to elect ...
-
meter)
*''μ'' is the
permeability of the intervening medium (SI unit:
tesla meter per
ampere
The ampere (, ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to elect ...
, 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 formulas 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, the force for each magnetized surface is:
where:
*''A'' is the area of each surface, in m
2
*''H'' is their
magnetizing field, in A/m.
*''μ''
0 is the
permeability of space, which equals 4π×10
−7 T·m/A
*''B'' is the
flux density
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
, in T
The derivation of this equation is analogous to the force between two nearby electrically charged surfaces, which assumes that the field in between the plates is uniform.
Force between two bar magnets
The force between two identical cylindrical bar magnets placed end to end at great distance
is approximately: