The Levitating Ring · AI Generator #GEN-F969C165

Physics · electromagnetism

Problem

A thin copper ring of radius

R = 5.0\;\text{cm}

and resistance

r = 2.0\;\text{m}\Omega

is placed horizontally above a solenoid coil. The solenoid has

N = 800

turns wound over a length

L = 20\;\text{cm}

and radius

a = 3.0\;\text{cm}

. When an alternating current

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

with amplitude

I_0 = 15\;\text{A}

and frequency

f = 60\;\text{Hz}

flows through the solenoid, the ring experiences a time-averaged upward magnetic force that can balance its weight. The ring has mass

m = 12\;\text{g}

and is positioned at height

h

above the top of the solenoid, where the magnetic field from the solenoid can be approximated as that of a magnetic dipole:

B_z(z) = \frac{\mu_0 M}{2\pi z^3}

along the axis, where

M = \frac{1}{2}\pi a^2 N I(t)

is the magnetic dipole moment and

z

is the distance from the solenoid center. (a) Find the time-averaged power dissipated in the ring when it is at equilibrium height

h

. (b) Determine the equilibrium height

h

where the ring levitates. Take

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

and

g = 9.8\;\text{m/s}^2