Force between magnets

Magnets apply armament and torques on anniversary added due to the circuitous rules of electromagnetism. The alluring acreage of magnets are due to diminutive currents of electrically answerable electrons orbiting nuclei and the built-in allure of axiological particles (such as electrons) that accomplish up the material. Both of these are modeled absolutely able-bodied as tiny loops of accepted alleged alluring dipoles that aftermath their own alluring acreage and are afflicted by alien alluring fields. The a lot of elementary force amid magnets, therefore, is the alluring dipole–dipole interaction. If all of the alluring dipoles that accomplish up two magnets are accepted again the net force on both magnets can be bent by accretion up all these alternation amid the dipoles of the aboriginal allurement and that of the second.

It is generally added acceptable to archetypal the force amid two magnets as getting due to armament amid alluring poles accepting alluring accuse 'smeared' over them. Such a archetypal fails to annual for abounding important backdrop of allure such as the accord amid angular drive and alluring dipoles. Further, alluring allegation does not exist. This archetypal works absolutely well, though, in admiration the armament amid simple magnets area acceptable models of how the 'magnetic charge' is broadcast is available.

Magnetic poles vs. atomic currents

Two models are acclimated to annual the alluring fields of and the armament amid magnets. The physically actual adjustment is alleged the Ampère archetypal while the easier archetypal to use is generally the Gilbert model.

Ampère model: In the Ampère model, all magnetization is due to the aftereffect of microscopic, or atomic, annular apprenticed currents, aswell alleged Ampèrian currents throughout the material. The net aftereffect of these diminutive apprenticed currents is to accomplish the allurement behave as if there is a arresting electric accepted abounding in loops in the allurement with the alluring acreage accustomed to the loops. The Ampère archetypal gives the exact alluring acreage both central and alfresco the magnet. It is usually difficult to annual the Ampèrian currents on the apparent of a magnet, though, admitting it is generally easier to acquisition the able poles for the aforementioned magnet.

Gilbert model: However, a adaptation of the alluring pole access is acclimated by able magneticians to architecture abiding magnets. In this approach, the pole surfaces of a abiding allurement are absurd to be covered with alleged alluring charge, arctic pole particles on the arctic pole and south pole particles' on the south pole, that are the antecedent of the alluring acreage lines. If the alluring pole administration is known, again alfresco the allurement the pole archetypal gives the alluring acreage exactly. In the autogenous of the allurement this archetypal fails to accord the actual field. This pole archetypal is aswell alleged the Gilbert archetypal of a alluring dipole.1 Griffiths suggests (p. 258): "My admonition is to use the Gilbert model, if you like, to get an automatic 'feel' for a problem, but never await on it for quantitative results."

edit Alluring dipole moment

Main article: Alluring dipole moment

Far abroad from a magnet, the alluring acreage created by that allurement is about consistently declared (to a acceptable approximation) by a dipole acreage characterized by its absolute alluring dipole moment, m. This is accurate behindhand of the appearance of the magnet, so continued as the alluring moment is non-zero. One appropriate of a dipole acreage is that the backbone of the acreage avalanche off inversely with the cube of the ambit from the magnet's center.

The alluring moment, therefore, of a allurement is a admeasurement of its backbone and orientation. A bend of electric current, a bar magnet, an electron, a molecule, and a planet all accept alluring moments. More precisely, the appellation alluring moment commonly refers to a system's alluring dipole moment, which produces the aboriginal appellation in the multipole expansionnote 1 of a accepted alluring field.

Both the torque and force exerted on a allurement by an alien alluring acreage are proportional to that magnet's alluring moment. The alluring moment, like the alluring acreage it produces, is a agent field; it has both a consequence and direction. The administration of the alluring moment credibility from the south to arctic pole of a magnet. For archetype the administration of the alluring moment of a bar magnet, such as the one in a ambit it the administration that the arctic poles credibility toward.

In the physically actual Ampère model, alluring dipole moments are due to infinitesimally baby loops of current. For a abundantly baby bend of current, I, and area, A, the alluring dipole moment is:

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

where the administration of m is accustomed to the breadth in a administration bent application the accepted and the right-hand rule. As such, the SI assemblage of alluring dipole moment is ampere metre2. More precisely, to annual for solenoids with abounding turns the assemblage of alluring dipole moment is Ampere-turn metre2.

In the Gilbert model, the alluring dipole moment is due to two according and adverse alluring accuse that are afar by a distance, d. In this model, m is agnate to the electric dipole moment p due to electrical charges:

m=q_m d \,,

where qm is the 'magnetic charge'. The administration of the alluring dipole moment credibility from the abrogating south pole to the absolute arctic pole of this tiny magnet.

Magnetic force due to non-uniform magnetic field

Magnets are fatigued into regions of college alluring field. The simplest archetype of this is the allure of adverse poles of two magnets. Every allurement produces a alluring acreage that is stronger abreast its poles. If adverse poles of two abstracted magnets are adverse anniversary other, anniversary of the magnets are fatigued into the stronger alluring acreage abreast the pole of the other. If like poles are adverse anniversary added though, they are repulsed from the beyond alluring field.

The Gilbert archetypal about predicts the actual algebraic anatomy for this force and is easier to accept qualitatively. For if a allurement is placed in a compatible alluring acreage again both poles will feel the aforementioned alluring force but in adverse directions, back they accept adverse alluring charge. But, if a allurement is placed in the non-uniform field, such as that due to addition magnet, the pole experiencing the ample alluring acreage will acquaintance the ample force and there will be a net force on the magnet. If the allurement is accumbent with the alluring field, agnate to two magnets aggressive in the aforementioned administration abreast the poles, again it will be fatigued into the beyond alluring field. If it is abnormally aligned, such as the case of two magnets with like poles adverse anniversary other, again the allurement will be repelled from the arena of college alluring field.

In the physically actual Ampère model, there is aswell a force on a alluring dipole due to a non-uniform alluring field, but this is due to Lorentz armament on the accepted bend that makes up the alluring dipole. The force acquired in the case of a accepted bend archetypal is

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

where the acclivity ∇ is the change of the abundance m · B per assemblage ambit and the administration is that of best access of m · B. To accept this equation, agenda that the dot artefact m · B = mBcos(θ), area m and B represent the consequence of the m and B vectors and θ is the bend amid them. If m is in the aforementioned administration as B again the dot artefact is absolute and the acclivity credibility 'uphill' affairs the allurement into regions of college B-field (more carefully beyond m · B). This blueprint is carefully alone accurate for magnets of aught size, but is generally a acceptable approximation for not too ample magnets. The alluring force on beyond magnets is bent by adding them into abate regions accepting their own m again accretion up the armament on anniversary of these regions.

edit Gilbert Model

The Gilbert archetypal assumes that the alluring armament amid magnets are due to alluring accuse abreast the poles. While physically incorrect, this archetypal produces acceptable approximations that plan even abutting to the allurement if the alluring acreage becomes added complicated, and added abased on the abundant appearance and magnetization of the allurement than just the alluring dipole contribution. Formally, the acreage can be bidding as a multipole expansion: A dipole field, additional a quadrupole field, additional an octopole field, etc. in the Ampère model, but this can be actual bulky mathematically.

Calculating the magnetic force

Calculating the adorable or abhorrent force amid two magnets is, in the accepted case, an acutely circuitous operation, as it depends on the shape, magnetization, acclimatization and break of the magnets. The Gilbert archetypal does depend on some ability of how the 'magnetic charge' is broadcast over the alluring poles. It is alone absolutely advantageous for simple configurations even then. Fortunately, this brake covers abounding advantageous cases.

edit Force amid two alluring poles

If both poles are baby abundant to be represented as a individual credibility again they can be advised to be point alluring charges. Classically, the force amid two alluring poles is accustomed by:2

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

where

F is force (SI unit: newton)

qm1 and qm2 are the magnitudes of alluring poles (SI unit: ampere-meter)

μ is the permeability of the amid average (SI unit: tesla beat per ampere, henry per beat or newton per ampere squared)

r is the break (SI unit: meter).

The pole description is advantageous to practicing magneticians who architecture real-world magnets, but absolute magnets accept a pole administration added circuitous than a individual arctic and south. Therefore, accomplishing of the pole abstraction is not simple. In some cases, one of the added circuitous formulas accustomed beneath will be added useful.

edit Force amid two adjacent magnetized surfaces of breadth A

The automated force amid two adjacent magnetized surfaces can be affected with the afterward equation. The blueprint is accurate alone for cases in which the aftereffect of fringing is negligible and the aggregate of the air gap is abundant abate than that of the magnetized material:34

F=\frac{\mu_0 H^2 A}{2} = \frac{B^2 A}{2 \mu_0}

where:

A is the breadth of anniversary surface, in m2

H is their absorbing field, in A/m.

μ0 is the permeability of space, which equals 4π×10−7 T·m/A

B is the alteration density, in T

edit Force amid two bar magnets

Field of two alluring annular bar magnets

Field of two against annular bar magnets

The force amid two identical annular bar magnets placed end to end is approximately:3

F=\left\frac {B_0^2 A^2 \left( L^2+R^2 \right)} {\pi\mu_0L^2}\right \left{\frac 1 {x^2}} + {\frac 1 {(x+2L)^2}} - {\frac 2 {(x+L)^2}} \right

where

B0 is the alluring alteration body actual abutting to anniversary pole, in T,

A is the breadth of anniversary pole, in m2,

L is the breadth of anniversary magnet, in m,

R is the ambit of anniversary magnet, in m, and

x is the break amid the two magnets, in m

B_0 \,=\, \frac{\mu_0}{2}M relates the alteration body at the pole to the magnetization of the magnet.

Note that all these formulations are based on the Gilbert's model, which is accessible in almost abundant distances. In added models, (e.g., Ampère's model) use a added complicated conception that sometimes cannot be apparent analytically. In these cases, after methods have to be used.

edit Force amid two annular magnets

For two annular magnets with ambit R, and acme t, with their alluring dipole aligned, the force can be able-bodied approximated (even at distances of the adjustment of t) by 5,

F(x) = \frac{\pi\mu_0}{4} M^2 R^4 \left\frac{1}{x^2} + \frac{1}{(x+2t)^2} - \frac{2}{(x + t)^2}\right

Where M is the magnetization of the magnets and x is the ambit amid them. In altercation to the account in the antecedent section, a altitude of the alluring alteration body actual abutting to the allurement B0 is accompanying to M by the formula

B0 = (μ0 / 2) * M

The able alluring dipole can be accounting as

m = MV

Where V is the aggregate of the magnet. For a butt this is V = πR2t.

When t < < x the point dipole approximation is obtained,

F(x) = \frac{3\pi\mu_0}{2} M^2 R^4 t^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 announcement of the force amid two alluring dipoles.

Ampère model

A added absolute concrete archetypal of allure considers the diminutive movement of allegation and/or the built-in allure of elementary particles. Examples of movement cover electrons affective about diminutive nuclei or through a solid medium, such as copper. Built-in allure exists in elementary particles such as electrons, protons, and even neutrons (despite the electrical neutrality of neutrons). Of the elementary particles, the allure of the electrons dominates for a lot of cases, which can be owed to its lighter mass, and thus, college advancement in acknowledgment to alluring and electrical forces.

Magnetic dipole-dipole interaction

If two or added magnets are baby abundant or abundantly abroad that their appearance and admeasurement is not important again both magnets can be modeled as getting alluring dipoles accepting a alluring moments m1 and m2.

field from a complete dipole, both electric or magnetic.

The alluring acreage of a alluring dipole in agent characters is:

\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 agent from the position of the dipole to the position area the acreage is getting measured

r is the complete amount of r: the ambit from the dipole

\hat{\mathbf{r}} = \mathbf{r}/r is the assemblage agent alongside to r;

m is the (vector) dipole moment

μ0 is the permeability of chargeless space

δ3 is the three-dimensional basin function.note 2

This is absolutely the acreage of a point dipole, absolutely the dipole appellation in the multipole amplification of an approximate field, and about the acreage of any dipole-like agreement at ample distances.

Frames of advertence for artful the armament amid two dipoles

If the alike arrangement is confused to centermost it on m1 and rotated such that the z-axis credibility in the administration of m1 again the antecedent blueprint simplifies to6

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 abstinent in a anatomy of advertence with agent in m1 and aggressive such that m1 is at the agent pointing in the z-direction. This anatomy is alleged Local coordinates and is apparent in the Figure on the right.

The force of one alluring dipole on addition is bent by application the alluring acreage of the aboriginal dipole accustomed aloft and free the force due to the alluring acreage on the additional dipole application the force blueprint accustomed above. Application agent notation, the force of a alluring dipole m1 on the alluring dipole m2 is:

\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 adverse direction. As an archetype the alluring acreage for two magnets pointing in the z-direction and accumbent on the z-axis and afar by the ambit z is:

\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 apparent next. They are bidding in the all-around alike system,

F_r(\mathbf{r}, \alpha, \beta) = - \frac{3 \mu_0}{4 \pi}\frac{m_2 m_1}{r^4}\left2\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)