The Levitating Ring · AI Generator #GEN-3345129C

Physics · electromagnetism

Problem

A superconducting ring of radius R=5.0  cmR = 5.0\;\text{cm} and mass m=12  gm = 12\;\text{g} is placed horizontally above a vertical solenoid. The solenoid has N=800N = 800 turns, length L=15  cmL = 15\;\text{cm}, and radius r=3.0  cmr = 3.0\;\text{cm}. When a sinusoidal current I(t)=I0sin⁡(ωt)I(t) = I_0 \sin(\omega t) flows through the solenoid with I0=2.5  AI_0 = 2.5\;\text{A} and ω=377  rad/s\omega = 377\;\text{rad/s} (60 Hz), the ring levitates at a stable height hh above the solenoid's top face. The magnetic field on the axis of the solenoid at distance zz above its center is approximately: B(z)=μ0NI2L[L/2+zr2+(L/2+z)2+L/2−zr2+(L/2−z)2]B(z) = \frac{\mu_0 N I}{2L} \left[ \frac{L/2 + z}{\sqrt{r^2 + (L/2 + z)^2}} + \frac{L/2 - z}{\sqrt{r^2 + (L/2 - z)^2}} \right] For the superconducting ring, the induced current exactly cancels the magnetic flux through it. At equilibrium levitation height, the time-averaged magnetic force balances the gravitational force. (a) Find the magnetic flux Φ(t)\Phi(t) through the ring when it is at height hh above the solenoid. (b) Determine the induced current Iring(t)I_{\text{ring}}(t) in the superconducting ring. (c) Calculate the equilibrium levitation height hh where the ring stably levitates. Use μ0=4π×10−7  H/m\mu_0 = 4\pi \times 10^{-7}\;\text{H/m} and g=9.8  m/s2g = 9.8\;\text{m/s}^2.

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