The Levitating Magnetized Sphere · AI Generator #GEN-F704F5C1

Physics · electromagnetism

Problem

A small conducting sphere of radius

R = 2.0\;\text{cm}

and mass

m = 15\;\text{g}

has been magnetized to have a uniform magnetic dipole moment

\vec{\mu}

pointing vertically upward. The sphere is placed above a large horizontal conducting sheet and finds an equilibrium position at height

h = 6.0\;\text{cm}

above the sheet, where it hovers motionlessly. The conducting sheet can be considered infinite and perfect. When the magnetized sphere is at height

h

, its image in the conducting sheet creates a magnetic dipole pointing downward at position

-h

below the sheet surface. (a) Find the magnitude of the magnetic dipole moment

\mu

of the sphere. (b) If the sphere is displaced slightly upward from equilibrium and released, find the frequency of small oscillations about the equilibrium position. Use

\mu_0 = 4\pi \times 10^{-7}\;\text{H/m}

and

g = 9.8\;\text{m/s}^2

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