The Levitating Graphite Flake · AI Generator #GEN-F1483AE8

Physics · electromagnetism

Problem

A small flake of graphite with mass m=2.0×10−6  kgm = 2.0 \times 10^{-6}\;\text{kg} is diamagnetic, meaning it creates an induced magnetic dipole moment that opposes applied magnetic fields. When placed above a strong neodymium magnet, the flake can be made to levitate in stable equilibrium. The magnetic field above the magnet varies with height as B(z)=B0e−z/λB(z) = B_0 e^{-z/\lambda}, where B0=0.8  TB_0 = 0.8\;\text{T} is the field at the magnet surface, λ=5.0  mm\lambda = 5.0\;\text{mm} is the characteristic decay length, and zz is the height above the magnet surface. The graphite's magnetic susceptibility is χ=−4.5×10−4\chi = -4.5 \times 10^{-4}. The induced magnetic dipole moment of the flake is μ⃗=χVB⃗μ0\vec{\mu} = \chi V \frac{\vec{B}}{\mu_0}, where V=8.0×10−12  m3V = 8.0 \times 10^{-12}\;\text{m}^3 is the volume of the flake and μ0=4π×10−7  H/m\mu_0 = 4\pi \times 10^{-7}\;\text{H/m}. Find the height zz above the magnet surface where the graphite flake levitates in equilibrium.

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