The Levitating Ring Above a Solenoid · AI Generator #GEN-54352EAC

Physics · electromagnetism

Problem

A thin aluminum ring of mass

m = 50\;\text{g}

, radius

R = 4.0\;\text{cm}

, and resistance

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

is placed coaxially above a solenoid. The solenoid has

N = 1000

turns, length

L = 20\;\text{cm}

, and radius

a = 3.0\;\text{cm}

. When an alternating current

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

with

I_0 = 10\;\text{A}

flows through the solenoid, the ring levitates at a height

h

above the solenoid's top surface. The magnetic field along the axis of a solenoid at distance

z

from its center is approximately

B(z) = \frac{\mu_0 N I}{2L}\left[\frac{z + L/2}{\sqrt{(z + L/2)^2 + a^2}} - \frac{z - L/2}{\sqrt{(z - L/2)^2 + a^2}}\right]

. For small oscillations about the equilibrium position, find the frequency

\omega

at which the ring levitates. Express your answer in terms of the given parameters and fundamental constants. Use

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