1 gilbert model
1.1 calculating magnetic force
1.1.1 force between 2 magnetic poles
1.1.2 force between 2 nearby magnetized surfaces of area a
1.1.3 force between 2 bar magnets
1.1.4 force between 2 cylindrical magnets
gilbert model
the gilbert model assumes magnetic forces between magnets due magnetic charges near poles. model produces approximations work close magnet when magnetic field becomes more complicated, , more dependent on detailed shape , magnetization of magnet magnetic dipole contribution. formally, field can expressed multipole expansion: dipole field, plus quadrupole field, plus octopole field, etc. in ampère model, can cumbersome mathematically.
calculating magnetic force
calculating attractive or repulsive force between 2 magnets is, in general case, extremely complex operation, depends on shape, magnetization, orientation , separation of magnets. gilbert model depend on knowledge of how magnetic charge distributed on magnetic poles. useful simple configurations then. fortunately, restriction covers many useful cases.
force between 2 magnetic poles
if both poles small enough represented single points can considered point magnetic charges. classically, force between 2 magnetic poles given by:
f
=
μ
q
m
1
q
m
2
4
π
r
2
{\displaystyle f={{\mu q_{m1}q_{m2}} \over {4\pi r^{2}}}}
where
f force (si unit: newton)
qm1 , qm2 magnitudes of magnetic poles (si unit: ampere-meter)
μ permeability of intervening medium (si unit: tesla meter per ampere, henry per meter or newton per ampere squared)
r separation (si unit: meter).
the pole description useful practicing magneticians design real-world magnets, real magnets have pole distribution more complex single north , south. therefore, implementation of pole idea not simple. in cases, 1 of more complex formulas given below more useful.
force between 2 nearby magnetized surfaces of area a
the mechanical force between 2 nearby magnetized surfaces can calculated following equation. equation valid cases in effect of fringing negligible , volume of air gap smaller of magnetized material:
f
=
μ
0
h
2
a
2
=
b
2
a
2
μ
0
{\displaystyle f={\frac {\mu _{0}h^{2}a}{2}}={\frac {b^{2}a}{2\mu _{0}}}}
where:
a area of each surface, in m
h magnetizing field, in a/m.
μ0 permeability of space, equals 4π×10 t·m/a
b flux density, in t
force between 2 bar magnets
field of 2 attracting cylindrical bar magnets
field of 2 repelling cylindrical bar magnets
the force between 2 identical cylindrical bar magnets placed end end @ great distance
x
≫
r
{\displaystyle x\gg r}
approximately:
f
≃
[
b
0
2
a
2
(
l
2
+
r
2
)
π
μ
0
l
2
]
[
1
x
2
+
1
(
x
+
2
l
)
2
−
2
(
x
+
l
)
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 flux density close each pole, in t,
a area of each pole, in m,
l length of each magnet, in m,
r radius of each magnet, in m, and
x separation between 2 magnets, in m
b
0
=
μ
0
2
m
{\displaystyle b_{0}\,=\,{\frac {\mu _{0}}{2}}m}
relates flux density @ pole magnetization of magnet.
note these formulations based on gilbert s model, usable in relatively great distances. other models, (e.g., ampère s model) use more complicated formulation cannot solved analytically. in these cases, numerical methods must used.
force between 2 cylindrical magnets
exact force between 2 coaxial cylindrical bar magnets several aspect ratios.
for 2 cylindrical magnets radius
r
{\displaystyle r}
, , length
l
{\displaystyle l}
, magnetic dipole aligned, force can computed analytically using elliptic integrals. in limit
x
≫
r
{\displaystyle x\gg r}
, force can approximated by,
f
(
x
)
≃
π
μ
0
4
m
2
r
4
[
1
x
2
+
1
(
x
+
2
l
)
2
−
2
(
x
+
l
)
2
]
{\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
m
{\displaystyle m}
magnetization of magnets ,
x
{\displaystyle x}
distance between them. small values of
x
{\displaystyle x}
, results erroneous force becomes large close-to-zero distance.
if magnet long (
l
≫
r
{\displaystyle l\gg r}
), measurement of magnetic flux density close magnet
b
0
{\displaystyle b_{0}}
related
m
{\displaystyle m}
formula
b
0
=
μ
0
2
m
{\displaystyle b_{0}\,=\,{\frac {\mu _{0}}{2}}m}
.
the effective magnetic dipole can written as
m
=
m
v
{\displaystyle m=mv}
where
v
{\displaystyle v}
volume of magnet. cylinder
v
=
π
r
2
l
{\displaystyle v=\pi r^{2}l}
.
when
l
≪
x
{\displaystyle l\ll x}
point dipole approximation obtained,
f
(
x
)
=
3
π
μ
0
2
m
2
r
4
l
2
1
x
4
=
3
μ
0
2
π
m
2
v
2
1
x
4
=
3
μ
0
2
π
m
1
m
2
1
x
4
{\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 expression of force between 2 magnetic dipoles.
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