The Levitron^TM is a toy in which a spinning magnetic top floats stably in air above a magnetized base. The potential energy of this system is V̂_B(r) = Mgz − m · B(r) if we model the top as a point magnetic dipole with moment m and mass M moving in the magnetic field of the base. A precession-averaged, adiabatic approximation is valid if the top spins much faster than it precesses and it precesses much faster than the rate of change of the magnetic field experienced by the precessing top. In this approximation,is the (constant) projection of m on the instantaneous direction of B.

(a) Suppose that B(r) = −∇ψ(r) is axially symmetric. Show that the magnetic scalar potential above the base and near the z-axis has the form

(b) Work to second order in ρ for ψ(ρ, z) and show that m_B (c) If m_B

(i) ψ₁ and ψ₂ have opposite signs; 

(ii) ψ₁ and ψ₃ have the same signs; and 

(iii) ψ² ₂ > 2ψ₃ψ₁.

VB(r) Mgz - mg B(r), where B(r) = |B(r)] and m B