The Electromagnetic Levitating Ring · AI Generator #GEN-C9FEF697

Physics · electromagnetism

Problem

A horizontal circular conducting ring of radius R=5.0  cmR = 5.0\;\text{cm}, mass m=12  gm = 12\;\text{g}, and resistance ρ=0.15  Ω\rho = 0.15\;\Omega is placed above a solenoid coil. The solenoid has N=800N = 800 turns, radius r=3.0  cmr = 3.0\;\text{cm}, and height h=6.0  cmh = 6.0\;\text{cm}. The ring is positioned so that its center is directly above the center of the solenoid at a height d=4.0  cmd = 4.0\;\text{cm} above the solenoid's top face. When an alternating current I(t)=I0sin⁡(ωt)I(t) = I_0 \sin(\omega t) with amplitude I0=8.0  AI_0 = 8.0\;\text{A} and angular frequency ω=2π×60  Hz\omega = 2\pi \times 60\;\text{Hz} flows through the solenoid, the ring experiences a time-averaged upward magnetic force that exactly balances its weight, causing it to levitate. (a) Find the magnetic flux Φ(t)\Phi(t) through the ring as a function of time. (b) Calculate the time-averaged power dissipated in the ring due to the induced current. (c) Determine the time-averaged magnetic force on the ring and verify that it equals the gravitational force. Use g=9.8  m/s2g = 9.8\;\text{m/s}^2, μ0=4π×10−7  H/m\mu_0 = 4\pi \times 10^{-7}\;\text{H/m}.

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