Force between magnets

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, 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 is therefore 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[note 1] 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 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:

${\displaystyle \mathbf {m} =I\mathbf {A} }$,

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 meter². More precisely, to account for solenoids with many turns the unit of magnetic dipole moment is Ampere-turn meter².

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:

${\displaystyle m=q_{m}d\,}$,

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.

Top: ${\displaystyle \mathbf {B} }$, the force on magnetic north-poles.
Bottom: ${\displaystyle \mathbf {\nabla } |\mathbf {B} |}$, the force on aligned dipoles, such as iron particles.

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 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 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 forces on the current loop that makes up the magnetic dipole. The force obtained in the case of a current loop model is

${\displaystyle \mathbf {F} =\nabla \left(\mathbf {m} \cdot \mathbf {B} \right)}$,

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). 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.

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.

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:[1]

${\displaystyle F={{\mu q_{m1}q_{m2}} \over {4\pi r^{2}}}}$

where

F is force (SI unit: newton)

q_m1 and q_m2 are the magnitudes of magnetic charge on magnetic poles (SI unit: ampere-meter)

μ is the permeability of the intervening medium (SI unit: tesla meter per ampere, 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.

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:[2][3][4]

${\displaystyle F={\frac {\mu _{0}H^{2}A}{2}}={\frac {B^{2}A}{2\mu _{0}}}}$

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

The derivation of this equation is analogous to the force between two nearby electrically charged surfaces,[5] which assumes that the field in between the plates is uniform.

Field of two attracting cylindrical bar magnets

Field of two repelling cylindrical bar magnets

The force between two identical cylindrical bar magnets placed end to end at great distance ${\displaystyle x\gg R}$ is approximately:[2]

${\displaystyle F\simeq \left[{\frac {B_{0}^{2}A^{2}\left(L^{2}+R^{2}\right)}{\pi \mu _{0}L^{2}}}\right]\left[{\frac {1}{x^{2}}}+{\frac {1}{(x+2L)^{2}}}-{\frac {2}{(x+L)^{2}}}\right]}$

where

B₀ is the 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

${\displaystyle B_{0}\,=\,{\frac {\mu _{0}}{2}}M}$ relates the flux density at the pole to the magnetization of the magnet.

Note that these formulations assume point-like magnetic-charge distributions instead of a uniform distribution over the end facets, which is a good approximation only at relatively great distances. For intermediate distances, numerical methods must be used.

Exact force between two coaxial cylindrical bar magnets for several aspect ratios.

For two cylindrical magnets with radius ${\displaystyle R}$, and length ${\displaystyle L}$, with their magnetic dipole aligned, the force can be computed analytically using elliptic integrals.[6] In the limit ${\displaystyle x\gg R}$, the force can be approximated by,[7]

${\displaystyle F(x)\simeq {\frac {\pi \mu _{0}}{4}}M^{2}R^{4}\left[{\frac {1}{x^{2}}}+{\frac {1}{(x+2L)^{2}}}-{\frac {2}{(x+L)^{2}}}\right]}$

Where ${\displaystyle M}$ is the magnetization of the magnets and ${\displaystyle x}$ is the distance between them. For small values of ${\displaystyle x}$, the results are erroneous as the force becomes large for close-to-zero distance.

If the magnet is long (${\displaystyle L\gg R}$), a measurement of the magnetic flux density very close to the magnet ${\displaystyle B_{0}}$ is roughly related to ${\displaystyle M}$ by the formula

${\displaystyle B_{0}\,=\,{\frac {\mu _{0}}{2}}M}$.

The effective magnetic dipole moment can be written as

${\displaystyle m=MV}$

Where ${\displaystyle V}$ is the volume of the magnet. For a cylinder this is ${\displaystyle V=\pi R^{2}L}$, and

${\displaystyle M}$ is the magnetisation field of the dipole.

When ${\displaystyle L\ll x}$ the point dipole approximation is obtained,

${\displaystyle F(x)={\frac {3\pi \mu _{0}}{2}}M^{2}R^{4}L^{2}{\frac {1}{x^{4}}}={\frac {3\mu _{0}}{2\pi }}M^{2}V^{2}{\frac {1}{x^{4}}}={\frac {3\mu _{0}}{2\pi }}m_{1}m_{2}{\frac {1}{x^{4}}}}$

Which matches the expression of the force between two magnetic dipoles.

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 dipoles having a magnetic moments m₁ and m₂. In case of uniformly magnetized spherical magnets this model is precise even at finite size and distance, as the outside field of such magnets is exactly a dipole field.[8]

Magnetic field of an ideal dipole.

The magnetic field of a magnetic dipole in vector notation is:

${\displaystyle \mathbf {B} (\mathbf {m} ,\mathbf {r} )={\frac {\mu _{0}}{4\pi r^{3}}}\left(3(\mathbf {m} \cdot {\hat {\mathbf {r} }}){\hat {\mathbf {r} }}-\mathbf {m} \right)+{\frac {2\mu _{0}}{3}}\mathbf {m} \delta ^{3}(\mathbf {r} )}$

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

${\displaystyle {\hat {\mathbf {r} }}=\mathbf {r} /r}$ 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 function.[note 2]

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.

Frames of reference for calculating the forces between two dipoles

Force between coaxial cylinder magnets. According to the dipole approximation, the force drops proportional to ${\displaystyle 1/z^{4}}$ for large distance z, resulting in slopes of −4 in the log–log plot.

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[9]

${\displaystyle {\begin{aligned}B_{z}(\mathbf {r} )&={\frac {\mu _{0}}{4\pi }}m_{1}\left({\frac {3\cos ^{2}\theta -1}{r^{3}}}\right)\\B_{x}(\mathbf {r} )&={\frac {\mu _{0}}{4\pi }}m_{1}\left({\frac {3\cos \theta \sin \theta }{r^{3}}}\right)\end{aligned}}}$,

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:

${\displaystyle \mathbf {F} (\mathbf {r} ,\mathbf {m} _{1},\mathbf {m} _{2})={\frac {3\mu _{0}}{4\pi r^{5}}}\left[(\mathbf {m} _{1}\cdot \mathbf {r} )\mathbf {m} _{2}+(\mathbf {m} _{2}\cdot \mathbf {r} )\mathbf {m} _{1}+(\mathbf {m} _{1}\cdot \mathbf {m} _{2})\mathbf {r} -{\frac {5(\mathbf {m} _{1}\cdot \mathbf {r} )(\mathbf {m} _{2}\cdot \mathbf {r} )}{r^{2}}}\mathbf {r} \right]}$

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 force for two magnets pointing in the z-direction and aligned on the z-axis and separated by the distance z is:

${\displaystyle \mathbf {F} (z,m_{1},m_{2})=-{\frac {3\mu _{0}m_{1}m_{2}}{2\pi z^{4}}}}$, z-direction.

The final formulas are shown next. They are expressed in the global coordinate system,

${\displaystyle {\begin{aligned}F_{r}(\mathbf {r} ,\alpha ,\beta )&=-{\frac {3\mu _{0}}{4\pi }}{\frac {m_{2}m_{1}}{r^{4}}}\left[2\cos(\phi -\alpha )\cos(\phi -\beta )-\sin(\phi -\alpha )\sin(\phi -\beta )\right]\\F_{\phi }(\mathbf {r} ,\alpha ,\beta )&=-{\frac {3\mu _{0}}{4\pi }}{\frac {m_{2}m_{1}}{r^{4}}}\sin(2\phi -\alpha -\beta )\end{aligned}}}$