The Levitating Paperclip · AI Generator #GEN-0921CA3F

Physics · electromagnetism

Problem

A physics demonstration involves suspending a steel paperclip in mid-air using electromagnetic forces. The setup consists of an electromagnet positioned directly above the paperclip, which induces a magnetic dipole in the paperclip that creates an attractive force. When the electromagnet is turned on, it produces a uniform magnetic field of strength B=0.15  TB = 0.15\;\text{T} that decreases with distance according to B(z)=B0e−z/dB(z) = B_0 e^{-z/d}, where d=2.0  cmd = 2.0\;\text{cm} is the characteristic decay length and zz is measured upward from the paperclip's equilibrium position. The steel paperclip has mass m=0.80  gm = 0.80\;\text{g}, length ℓ=3.0  cm\ell = 3.0\;\text{cm}, and magnetic susceptibility χ=500\chi = 500. When placed in the magnetic field, the paperclip becomes magnetized with magnetic moment M⃗=χVμ0B⃗\vec{M} = \frac{\chi V}{\mu_0} \vec{B}, where V=AℓV = A\ell is the volume of the paperclip and A=0.15  mm2A = 0.15\;\text{mm}^2 is its cross-sectional area. Find the equilibrium height zeqz_{eq} at which the paperclip hovers, measured from the position where the magnetic field strength equals B0=0.15  TB_0 = 0.15\;\text{T}.

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