The Magnetic Levitation Ring · AI Generator #GEN-5373B505

Physics · electromagnetism

Problem

A physicist is demonstrating magnetic levitation using a vertical solenoid with a conducting aluminum ring. The solenoid has N=1000N = 1000 turns, length L=0.50  mL = 0.50\;\text{m}, and radius R=5.0  cmR = 5.0\;\text{cm}. An aluminum ring of mass m=15  gm = 15\;\text{g}, radius r=4.0  cmr = 4.0\;\text{cm}, and resistance R=2.0×10−4  Ω\mathcal{R} = 2.0 \times 10^{-4}\;\Omega is placed coaxially above the solenoid at a distance d=2.0  cmd = 2.0\;\text{cm} from the top. When the physicist suddenly switches on a current that increases linearly in the solenoid according to I(t)=αtI(t) = \alpha t where α=500  A/s\alpha = 500\;\text{A/s}, the changing magnetic flux through the ring induces currents that create a repulsive force. (a) Find the magnetic field at the center of the ring due to the solenoid when the current is II. (b) Calculate the induced EMF in the ring at t=0.10  st = 0.10\;\text{s}. (c) Determine whether the ring will levitate (i.e., whether the magnetic force exceeds the gravitational force) at t=0.10  st = 0.10\;\text{s}. Assume the magnetic field is approximately uniform across the ring's area and equal to the field at the ring's center. Use g=9.8  m/s2g = 9.8\;\text{m/s}^2 and μ0=4π×10−7  T⋅m/A\mu_0 = 4\pi \times 10^{-7}\;\text{T·m/A}.

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