Dynamic Dipole-Dipole Magnetic Interaction and Damped Nonlinear Oscillations

Haiduke Sarafian

The Pennsylvania State University

Motivation

Consider two permanent magnets.  Position them along their mutual common axial axis and orient their magnetic moments so that they are anti-parallel.  Drop one of the magnets vertically on top of the second stationary magnet.  Select a suitable characteristics for the magnets, namely the mass of the falling magnet and their magnetic moments, such that the balance between the weight of the falling magnet and the mutual magnetic force between the two magnets results in oscillations.  Model the problem theoretically and confirm the accuracy of the model vs. data.

Observation

Two avi files:  4150.avi and 4151.avi.

"C:\DataFiles\Mathematica_Conference_October2009\MVI_4151.AVI"

Experiment setup

Data

Axial permanent magnet-magnet interaction

1.  Magnetic field of a steady looping current.

(Overscript[B, ->])(z)=((Subscript[μ, 0])/(4π))i ((2π (R^2))/((((R^2)+(z^2)))^(3/2)))(OverHat[k])

(π (R^2)) i (OverHat[k]):=(Overscript[μ, ->])

2.  Magnetic field of a permanent magnet.

(Overscript[B, ->])(z)=((Subscript[μ, 0])/(4π)) (2/((((R^2)+(z^2)))^(3/2)))(Overscript[μ, ->])

3.  Magnet potential energy.

U=-(Subscript[(Overscript[μ, ->]), 2]).(Overscript[(Subscript[B, 1]), ->])

4.  Force on the magnet.

(Subscript[(Overscript[F, ->]), 12])=(Subscript[(Overscript[μ, ->]), 2]).((δ (Subscript[(Overscript[B, ->]), 1]))/(δ z))(OverHat[k])

(Subscript[(Overscript[F, ->]), 12 ])=-6 ((Subscript[μ, 0])/(4π))(Subscript[(Overscript[μ, ->]), 1]).(Subscript[(Overscript[μ, ->]), 2])(z/((((R^2)+(z^(2 ))))^(5/2)))(OverHat[k])

5.  "Approximated" B field.

(Overscript[B, ->])(z)∼2((Subscript[μ, 0])/(4π)) ((Overscript[μ, ->])/(z^3))

6.  "Approximate" force.

(Subscript[(Overscript[F, ->]), 12])∼-6 ((Subscript[μ, 0])/(4π))(Subscript[(Overscript[μ, ->]), 1]).(Subscript[(Overscript[μ, ->]), 2])(1/(z^4))(OverHat[k])

For identical magnets, (Subscript[μ, 1])=(Subscript[μ, 2])≡μ ?

7.  Data_method 1, Gauss meter.

(*fitdata=FindFit[cylindricalMagnetdata,α (z^(-β)),{α,β},z]
loglogplotfit=LogLogPlot[α (z^(-β))/.fitdata,{z,4,10},PlotStyle->{Thickness[0.008],GrayLevel[0.3],Dashing[{0.02}]},PlotRange->{All,{10,120}}];
Show[listplot1,loglogplotfit,s12]*)

{α->7299.62,β->2.89}

α=2μ(((Subscript[μ, 0])/(4π)))----->μ=3.65 Amp.(m^2)

8.  Data_method 2, magnetic tower.

(Subscript[μ, 22])=(Sqrt[((m g)/(6 ((Subscript[μ, 0])/(4π))(1/((Subscript[z, 2])^4))))]), ...

(Subscript[μ, 55])=(Sqrt[((m g)/(6 ((Subscript[μ, 0])/(4π))[-(1/((Subscript[z, 5])^4))+(1/((((Subscript[z, 5])-(Subscript[z, 2])))^4))-(1/((((Subscript[z, 5])-(Subscript[z, 3])))^4))+(1/((((Subscript[z, 5])-(Subscript[z, 4])))^4))]))])

μ=(3.58±0.43) Amp.(m^2)

9.  A theoretical model.

(Overscript[z, ..])(t)-6(((Subscript[μ, 0])/(4π)))((μ^2)/m)(1/((z(t))^4))+γ (OverDot[z])(t)+ g=0

10.  Kinematics

11.  Phase diagrams