The Magnetic Levitation Trap · AI Generator #GEN-2E145D03

Physics · electromagnetism

Problem

A research team designs a magnetic levitation system using a small permanent magnet suspended above a superconducting disk. The magnet has mass m=2.4  gm = 2.4\;\text{g} and magnetic dipole moment μ⃗=0.15  z^  A⋅m2\vec{\mu} = 0.15\;\hat{z}\;\text{A}\cdot\text{m}^2 pointing vertically upward. Due to the Meissner effect, the superconductor creates a perfect mirror image of the magnet below its surface, effectively producing a repulsive force. The magnet is found to hover in stable equilibrium at height h0=3.2  mmh_0 = 3.2\;\text{mm} above the superconductor surface. At this position, small vertical oscillations occur with period T=0.18  sT = 0.18\;\text{s}. The researchers then place the entire apparatus in a uniform external magnetic field B⃗ext=B0z^\vec{B}_{ext} = B_0\hat{z} pointing upward. They observe that the equilibrium height changes to a new value h1h_1. Given that the interaction energy between two magnetic dipoles separated by distance rr along their common axis is U=−μ0μ22πr3U = -\frac{\mu_0\mu^2}{2\pi r^3} (where the negative sign indicates attraction, but here we have repulsion due to the image effect), find the strength of the external field B0B_0 if the new equilibrium height is h1=4.1  mmh_1 = 4.1\;\text{mm}. Use μ0=4π×10−7  H/m\mu_0 = 4\pi \times 10^{-7}\;\text{H/m} and g=9.81  m/s2g = 9.81\;\text{m/s}^2.

Browse the problem library