Re: jumping ring demo

[Allen Miller <amiller@SUHEP.PHY.SYR.EDU>]

Crawford Naccallum notes in his message below that sometimes the ring will
not jump, despite the fact that the ring circuit is closed. I believe
that the reason is that the ring resistance R is high. The induced
voltage amplitude divided by R gives a good rough estimate for the
current amplitude in the ring (denoted by I    (0) in my original
posting).                                  RING

If the current amplitude in the ring is too small, then the force of
repulsion on the ring does not beat gravity.

Here is a related matter. Suppose that you put such a high resistance ring
on the bottom of the core of the electromagnet. It stays put (but heats
up.) Then bring a low resistance ring to the top of the core and lower by
hand. The high resistance ring will be observed to rise towards the the
other ring.

The reason for the attraction between the two rings is that the currents
in the two rings are (approximately) in phase with each other. The are
exactly in phase, if you ignore the very slight difference in the
self-inductance of the two rings. The two ring currents are both clockwise
or both counterclockwise, at any instant, and hence attract.

Allen Miller,
Physics
Syracuse University

 On Mon, 12
Apr 1999, crawford j maccallum wrote:

> I seem to remember doing this demo at a public showing, and using a _cut_
> ring, which didn't jump, as a control.  When we put a bit of wire across
> the cut to 'complete the circuit', however, it didn't jump either, which
> puzzled us.  Looks like Allen Miller has the explanation: it depends on
> the value of L/R.          Crawford
>
> On Mon, 12 Apr 1999, Allen Miller wrote:
>
> > 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