Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: jumping ring demo



Here is my attempt to answer Martha Takats question as to why the
nonmagnetic ring jumps off the core of the electromagnet. It is in the
spirit of the previous discussion of the problem by Rondo Jeffery.

Let I(t) be the current in the coils of the electromagnet. Choose
counterclockwise as positive current, clockwise as negative.
Choose the time t = 0 so that the dependence of this current on time t is

I(t) = I(0)sin wt,

where w is the frequency and I(0) is the amplitude.

The voltage induced in the ring is, by Faraday's Law,

V = - d(Flux)/dt = -M dI/dt,

where M is the (magnetic) flux through the ring, divided by I. M can be
regarded as the mutual inductance between electromagnet and ring.

Then, by combining these two equations, we get

V = - Mw cos wt.

This shows that V and I are out-of-phase by 90 degrees.

But the current in the ring is not in phase with V (as Jeffery has
emphasized). This is due to the self-inductance L of the ring. If we
ignore L, then we "prove" that the ring does not jump.

In fact, the current lags behind the voltage by an acute angle "theta".
This angle is the inverse tangent of L/R, where R is the resistance of the
ring. The same lag occurs in the driven RLC circuit.

Hence, the ring current can be written as

I = - I (0) cos (wt - theta),
RING RING

where I (0) is the amplitude of the ring current.
RING

Now, compare the coil current I to the ring current. If you plot them
both versus time, you see that they are of opposite sign during a time
duration of (T/2) + (2 theta/w) during a complete cycle of time duration
T. During this time period, repulsion occurs, since anti-parallel currents
attract.

Over the shorter time period of the remainder of the cycle, of duration of
(T/2) - (2 theta/w), the currents are of the same sign and attraction
occurs. But repulsion occurs over the greater portion of the cycle, and so
the ring jumps.

Allen Miller
Physics,
Syracuse University
(315) 443-5962






= On Mon, 12 Apr 1999, Martha Takats wrote:

Would someone point me to an explanation of the jumping ring demo?
This is the one where you put a nonmagnetic (aluminum) ring over the
core of an electromagnet and plug the electromagnet into an AC
outlet. The ring jumps vigorously off. My attempts at analysis
do not give me a net force in the direction of decreasing field, though
there clearly is such a force--if you try to hold the ring in place,
there is a noticeable force trying to make it jump off.
--
Martha Takats
Department of Physics and Astronomy
Ursinus College
Collegeville, PA 19426