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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