The Levitating Ring · AI Generator #GEN-B463752F

Physics · electromagnetism

Problem

A research team is studying magnetic levitation using a unique setup. A thin conducting ring of radius

R = 5.0\;\text{cm}

and resistance

r = 0.20\;\text{Ω}

is placed horizontally above a solenoid coil. The solenoid has

N = 500

turns wound tightly over a length

L = 8.0\;\text{cm}

and radius

a = 3.0\;\text{cm}

. The solenoid carries a current

I(t) = I_0 \sin(\omega t)

where

I_0 = 12\;\text{A}

and

\omega = 2\pi \times 60\;\text{Hz}

. When the ring is positioned at height

h = 6.0\;\text{cm}

above the top of the solenoid, the magnetic field at the ring's location due to the solenoid is approximately uniform across the ring's area and has magnitude:

B(h) = \frac{\mu_0 N I}{2L}\left(\frac{a^2}{a^2 + h^2}\right)

Due to the time-varying current, an EMF is induced in the ring, causing a current to flow. This current interacts with the magnetic field to produce a force on the ring. (a) Find the magnitude of the induced EMF in the ring as a function of time. (b) Calculate the time-averaged power dissipated in the ring over one complete cycle. (c) Determine the maximum instantaneous upward magnetic force on the ring. Use

\mu_0 = 4\pi \times 10^{-7}\;\text{H/m}

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