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.
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.
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