The Electromagnetic Levitator · AI Generator #GEN-F34FE231

Physics · electromagnetism

Problem

A research team develops an electromagnetic levitation device consisting of a horizontal conducting ring of radius

R = 5.0\;\text{cm}

and resistance

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

suspended above a vertical solenoid. The solenoid has

N = 1000

turns over a length

L = 20\;\text{cm}

and radius

a = 3.0\;\text{cm}

. When the ring is positioned at height

h = 8.0\;\text{cm}

above the top of the solenoid (measured from the ring's center), the magnetic field at the ring's location due to the solenoid is approximately uniform across the ring's area with magnitude

B = \frac{\mu_0 N I}{2L}\sqrt{\frac{a^2}{a^2 + h^2}}

, where

I

is the solenoid current. The solenoid is driven by a sinusoidal current

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

where

I_0 = 15\;\text{A}

and

f = \omega/(2\pi) = 60\;\text{Hz}

. Due to the changing magnetic flux, the ring experiences an induced EMF that drives a current, which in turn interacts with the magnetic field to produce a levitating force. (a) Calculate the amplitude of the induced current in the ring. (b) Find the time-averaged upward magnetic force on the ring. The ring has mass

m = 12\;\text{g}

- is this force sufficient to levitate the ring?

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