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\begin{document}

A magnet with a small top placed above it that has a dipole point
opposite the magnetic field can hover in mid air for a little while.
This avoids violating Earnshaws theorem because it is a dynamic system.

\( \Omega =\frac{\mu B}{S} \) where S = spin angular momentum=\( I\omega  \). In this case \( \gamma <<\Omega <<\omega  \)= spin rate. \( \gamma = \)
typical oscillation frequency with respect to the equilibrium position.
Then \( \mu ^{i}b^{i} \) is an adiabatic invariant. \( E=-\mu _{B}B+mgz \) does give a shallow minimum.

top stable for \( \frac{1}{2}<\frac{z=height}{a=baseradius}<\sqrt{\frac{2}{5}}\sim .6325 \) and \( \frac{\triangle M}{M_{center}}\simeq 0.05 \). Then \( \frac{R_{max}}{a}\simeq .2 \).

spindown time: \( I\omega '=-T \). The torque is produced by the boundary layer around
it. \( \delta \sim (\frac{\nu }{\omega })^{\frac{1}{2}} \) then \( \sigma \sim \rho \nu \omega \frac{r}{\delta }\sim \rho \nu ^{\frac{1}{2}}\omega ^{\frac{3}{2}}r \) Then \( T\sim \rho _{a}\nu ^{\frac{1}{2}}\omega ^{\frac{3}{2}}r^{4} \). Then \( I\sim mr^{2} \). Really interested in \( \frac{\omega }{\omega '}\sim \frac{I\omega }{T}\sim mr^{2}\frac{1}{\rho _{a}\nu ^{\frac{1}{2}}\omega ^{\frac{1}{2}}r^{4}}\sim \frac{\rho h}{\rho _{a}\nu ^{\frac{1}{2}}\omega ^{\frac{1}{2}}} \) . This is \( t\sim \frac{\rho }{\rho _{a}}(\frac{h^{2}}{\nu })^{\frac{1}{2}}(\frac{1}{\omega })^{\frac{1}{2}}\sim 4*10^{3}*1*\frac{1}{10}\sim 400\sim 6 \)minutes.

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