Force between magnets
Encyclopedia
Magnets exert forces and torques on each other due to the complex rules of electromagnetism
. The magnetic field of magnets are due to microscopic currents of electrically charged electron
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 dipole
s that produce their own magnetic field and are affected by external magnetic fields. The most elementary force between magnets, therefore, 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 interaction 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 'smeared' over them. Such a model fails to account for many important properties of magnetism such as the relationship between angular momentum and magnetic dipoles. Further, magnetic charge does not exist. This model works quite well, though, in predicting the forces between simple magnets where good models of how the 'magnetic charge' is distributed is available.
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
flowing in loops in the magnet with the magnetic field normal to the loops. The Ampère model gives the exact magnetic field both inside and outside the magnet. It is usually difficult to calculate the Ampèrian currents on the surface of a magnet, though, whereas it is often easier to find the effective poles for the same magnet.
Gilbert model: However, a version of the magnetic pole approach is used by professional magneticians to design permanent magnets. In this approach, 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. If the magnetic pole distribution is known, then outside the magnet the pole model gives the magnetic field exactly. In the interior of the magnet this model fails to give the correct field. This pole model is also called the Gilbert model of a magnetic dipole
. Griffiths suggests (p. 258): "My advice is to use the Gilbert model, if you like, to get an intuitive 'feel' for a problem, but never rely on it for quantitative results."
characterized by its total magnetic dipole moment, 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, therefore, of a magnet
is a measure of its strength and orientation. A loop of electric current
, a bar magnet
, an electron
, a molecule
, and a planet
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
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, like the magnetic field it produces, is a vector
field; it has both a magnitude and direction. The direction of the magnetic moment points from the south to north pole of a magnet. For example the direction of the magnetic moment of a bar magnet, such as the one in a compass
it 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
. As such, the SI
unit of magnetic dipole moment is ampere
metre2. More precisely, to account for solenoids with many turns the unit of magnetic dipole moment is Ampere-turn
metre2.
In the Gilbert 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 qm 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.
The Gilbert model almost predicts the 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 physically correct 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
s on the current loop that makes up the magnetic dipole. The force obtained in the case of a current loop model is
\mathbf{F}=\nabla \left(\mathbf{m}\cdot\mathbf{B}\right)
,
where the gradient
∇ is the change of the quantity m · B per unit distance and the direction is that of maximum increase of m · B. To understand this equation, note that the dot product
m · B = mBcos(θ), 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). 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.
: A dipole field, plus a quadrupole field
, plus an octopole field, etc. in the Ampère model, but this can be very cumbersome mathematically.
, the force between two magnetic poles is given by:
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 formulas given below will be more useful.
where:
where
relates the flux density at the pole to the magnetization of the magnet.
Note that all these formulations are based on the Gilbert's model, which is usable in relatively great distances. In other models, (e.g., Ampère's model) use a more complicated formulation that sometimes cannot be solved analytically. In these cases, numerical methods must be used.
, and height 
, with their magnetic dipole aligned, the force can be well approximated (even at distances of the order of 
) by ,
Where
is the magnetization of the magnets and 
is the distance between them.
In disagreement to the statement in the previous section, a measurement of the magnetic flux density very close to the magnet
is related to 
by the formula
The effective magnetic dipole can be written as
Where
is the volume of the magnet. For a cylinder this is 
.
When
the point dipole approximation is obtained,
Which matches the expression of the force between two magnetic dipoles.
s, and even neutron
s (despite the electrical neutrality of neutrons). Of the elementary particles, the magnetism of the electrons dominates for most cases, which can be owed to its lighter mass, and thus, higher mobility in response to magnetic and electrical forces.
s having a magnetic moment
s m1 and m2.
The magnetic field of a magnetic dipole in vector notation
is:
where
This is exactly the field of a point dipole, exactly the dipole term in the multipole expansion of an arbitrary field, and approximately the field of any dipole-like configuration at large distances.
If the coordinate system is shifted to center it on m1 and rotated such that the z-axis points in the direction of m1 then the previous equation simplifies to
B_{z}(\mathbf{r}) = \frac{\mu_0}{4 \pi} m_1 \left(\frac{3\cos^2\theta-1}{r^3}\right)
B_{x}(\mathbf{r}) = \frac{\mu_0}{4 \pi} m_1 \left(\frac{3\cos\theta\sin\theta}{r^3}\right)
,
where the variables r and θ are measured in a frame of reference with origin in m1 and oriented such that m1 is at the origin pointing in the z-direction. This frame is called Local coordinates and is shown in the Figure on the right.
The force of one magnetic dipole on another is determined by using the magnetic field of the first dipole given above and determining the force due to the magnetic field on the second dipole using the force equation given above. Using vector notation, the force of a magnetic dipole m1 on the magnetic dipole m2 is:
where r is the distance-vector from dipole moment m1 to dipole moment m2, with r=||r||. The force acting on m1 is in opposite direction. As an example the magnetic field for two magnets pointing in the z-direction and aligned on the z-axis and separated by the distance z is:
, z-direction.
The final formulas are shown next. They are expressed in the global coordinate system,
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
. The magnetic field of magnets are due to microscopic currents of electrically charged electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
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 dipole
Magnetic dipole
A magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the dimensions of the source are reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric dipole, but the analogy is not complete. In particular, a magnetic...
s that produce their own magnetic field and are affected by external magnetic fields. The most elementary force between magnets, therefore, 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 interaction 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 'smeared' over them. Such a model fails to account for many important properties of magnetism such as the relationship between angular momentum and magnetic dipoles. Further, magnetic charge does not exist. This model works quite well, though, in predicting the forces between simple magnets where good models of how the 'magnetic charge' is distributed is available.
Magnetic poles vs. atomic currents
Two models are used to calculate the magnetic fields of and the forces between magnets. The physically correct method is called the Ampère model while the easier model to use is often the Gilbert model.Ampère model: In the Ampère
André-Marie Ampère
André-Marie Ampère was a French physicist and mathematician who is generally regarded as one of the main discoverers of electromagnetism. The SI unit of measurement of electric current, the ampere, is named after him....
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
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
flowing in loops in the magnet with the magnetic field normal to the loops. The Ampère model gives the exact magnetic field both inside and outside the magnet. It is usually difficult to calculate the Ampèrian currents on the surface of a magnet, though, whereas it is often easier to find the effective poles for the same magnet.
Gilbert model: However, a version of the magnetic pole approach is used by professional magneticians to design permanent magnets. In this approach, 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. If the magnetic pole distribution is known, then outside the magnet the pole model gives the magnetic field exactly. In the interior of the magnet this model fails to give the correct field. This pole model is also called the Gilbert model of a magnetic dipole
Magnetic dipole
A magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the dimensions of the source are reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric dipole, but the analogy is not complete. In particular, a magnetic...
. Griffiths suggests (p. 258): "My advice is to use the Gilbert model, if you like, to get an intuitive 'feel' for a problem, but never rely on it for quantitative results."
Magnetic dipole moment
Far away from a magnet, the magnetic field created by that magnet is almost always described (to a good approximation) by a dipole fieldDipole
In physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...
characterized by its total magnetic dipole moment, 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, therefore, of a magnet
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, and attracts or repels other magnets.A permanent magnet is an object...
is a measure of its strength and orientation. A loop of electric current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
, a bar magnet
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, and attracts or repels other magnets.A permanent magnet is an object...
, an electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
, a molecule
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...
, and a planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
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
Multipole expansion
A multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original...
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
Quadrupole
A quadrupole or quadrapole is one of a sequence of configurations of—for example—electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure reflecting various orders of complexity.-Mathematical...
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, like the magnetic field it produces, is a vector
Vector (mathematics and physics)
In mathematics and physics, a vector is an element of a vector space. If n is a non negative integer and K is either the field of the real numbers or the field of the complex number, then K^n is naturally endowed with a structure of vector space, where K^n is the set of the ordered sequences of n...
field; it has both a magnitude and direction. The direction of the magnetic moment points from the south to north pole of a magnet. For example the direction of the magnetic moment of a bar magnet, such as the one in a compass
Compass
A compass is a navigational instrument that shows directions in a frame of reference that is stationary relative to the surface of the earth. The frame of reference defines the four cardinal directions – north, south, east, and west. Intermediate directions are also defined...
it 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
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...
to the area in a direction determined using the current and the right-hand rule
Right-hand rule
In mathematics and physics, the right-hand rule is a common mnemonic for understanding notation conventions for vectors in 3 dimensions. It was invented for use in electromagnetism by British physicist John Ambrose Fleming in the late 19th century....
. As such, the SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
unit of magnetic dipole moment is ampere
Ampere
The ampere , often shortened to amp, is the SI unit of electric current and is one of the seven SI base units. It is named after André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...
metre2. More precisely, to account for solenoids with many turns the unit of magnetic dipole moment is Ampere-turn
Ampere-turn
The ampere-turn was the MKS unit of magnetomotive force , represented by a direct current of one ampere flowing in a single-turn loop in a vacuum...
metre2.
In the Gilbert 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 qm 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 into regions of higher magnetic field. 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 are 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 Gilbert model almost predicts the 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 physically correct 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
Lorentz force
In physics, the Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the following equation in terms of the electric and magnetic fields:...
s on the current loop that makes up the magnetic dipole. The force obtained in the case of a current loop model is
\mathbf{F}=\nabla \left(\mathbf{m}\cdot\mathbf{B}\right)
,
where the gradient
Gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
∇ is the change of the quantity m · B per unit distance and the direction is that of maximum increase of m · B. To understand this equation, note that the dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
m · B = mBcos(θ), 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). 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.
Gilbert Model
The Gilbert model assumes that the magnetic forces between magnets are due to magnetic charges near the poles. While physically incorrect, this model produces good approximations that work 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 expansionMultipole expansion
A multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original...
: A dipole field, plus a quadrupole field
Quadrupole
A quadrupole or quadrapole is one of a sequence of configurations of—for example—electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure reflecting various orders of complexity.-Mathematical...
, 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, an extremely complex operation, as it depends on the shape, magnetization, orientation and separation of the magnets. The Gilbert 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 a single points then they can be considered to be point magnetic charges. ClassicallyClassical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, the force between two magnetic poles is given by:
where
- F is force (SI unit: newton)
- qm1 and qm2 are the magnitudes of magnetic poles (SI unit: ampere-meterAmpere-meterThe ampere-metre which has the symbol A m, A-m, or A·m is the SI unit for pole strength in a magnet.-Derivation:Einstein proved that a magnetic field is the relativistic part of an electric field...
) - μ is the permeabilityPermeability (electromagnetism)In electromagnetism, permeability is the measure of the ability of a material to support the formation of a magnetic field within itself. In other words, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically...
of the intervening medium (SI unit: teslaTesla (unit)The tesla is the SI derived unit of magnetic field B . One tesla is equal to one weber per square meter, and it was defined in 1960 in honour of the inventor, physicist, and electrical engineer Nikola Tesla...
meter per ampereAmpereThe ampere , often shortened to amp, is the SI unit of electric current and is one of the seven SI base units. It is named after André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...
, 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: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 approximately: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
relates the flux density at the pole to the magnetization of the magnet.Note that all these formulations are based on the Gilbert's model, which is usable in relatively great distances. In other models, (e.g., Ampère's model) use a more complicated formulation 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, and height , with their magnetic dipole aligned, the force can be well approximated (even at distances of the order of ) by ,Where
is the magnetization of the magnets and is the distance between them.In disagreement to the statement in the previous section, a measurement of the magnetic flux density very close to the magnet
is related to by the formulaThe effective magnetic dipole can be written as
Where
is the volume of the magnet. For a cylinder this is .When
the point dipole approximation is obtained,Which matches the expression of the force between two magnetic dipoles.
Ampère model
A more precise physical model of magnetism considers the microscopic movement of charge and/or the intrinsic magnetism of elementary particles. Examples of movement include electrons moving around atomic nuclei or through a solid medium, such as copper. Intrinsic magnetism exists in elementary particles such as electrons, protonProton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
s, and even neutron
Neutron
The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...
s (despite the electrical neutrality of neutrons). Of the elementary particles, the magnetism of the electrons dominates for most cases, which can be owed to its lighter mass, and thus, higher mobility in response to magnetic and electrical forces.
Magnetic dipole-dipole interaction
If two or more magnets are small enough or sufficiently distant that their shape and size is not important then both magnets can be modeled as being magnetic dipoleMagnetic dipole
A magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the dimensions of the source are reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric dipole, but the analogy is not complete. In particular, a magnetic...
s having a magnetic moment
Magnetic moment
The magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...
s m1 and m2.
The magnetic field of a magnetic dipole in vector notation
Vector notation
This page is an overview of the common notations used when working with vectors, which may be spatial or more abstract members of vector spaces....
is:
where
- B is the field
- r is the vector from the position of the dipole to the position where the field is being measured
- r is the absolute value of r: the distance from the dipole
is the unit vector parallel to r; - m is the (vector) dipole moment
- μ0 is the permeability of free space
- δ3 is the three-dimensional delta functionDirac delta functionThe Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
.δ3(r) = 0 except at , so this term is ignored in multipole expansion.
This is exactly the field of a point dipole, exactly the dipole term in the multipole expansion of an arbitrary field, and approximately the field of any dipole-like configuration at large distances.
B_{z}(\mathbf{r}) = \frac{\mu_0}{4 \pi} m_1 \left(\frac{3\cos^2\theta-1}{r^3}\right)
B_{x}(\mathbf{r}) = \frac{\mu_0}{4 \pi} m_1 \left(\frac{3\cos\theta\sin\theta}{r^3}\right)
,
where the variables r and θ are measured in a frame of reference with origin in m1 and oriented such that m1 is at the origin pointing in the z-direction. This frame is called Local coordinates and is shown in the Figure on the right.
The force of one magnetic dipole on another is determined by using the magnetic field of the first dipole given above and determining the force due to the magnetic field on the second dipole using the force equation given above. Using vector notation, the force of a magnetic dipole m1 on the magnetic dipole m2 is:
where r is the distance-vector from dipole moment m1 to dipole moment m2, with r=||r||. The force acting on m1 is in opposite direction. As an example the magnetic field for two magnets pointing in the z-direction and aligned on the z-axis and separated by the distance z is:
, z-direction.The final formulas are shown next. They are expressed in the global coordinate system,