In Figure 11.3.2, we show a permanent magnet that is fixed at the origin with its dipole
moment pointing upward. On the
z-axis above the magnet, we have a co-axial,
conducting, non-magnetic ring with radius
a, inductance L, and resistance R. The center
of the conducting ring is constrained to move along the vertical axis. The ring is released
from rest and falls under gravity toward the stationary magnet. Eddy currents arise in the
ring because of the changing magnetic flux and induced electric field as the ring falls
toward the magnet, and the sense of these currents is to repel the ring when it is above the
magnet.
This physical situation can be formulated mathematically in terms of three coupled
ordinary differential equations for the position of the ring, its velocity, and the current in
the ring.
We consider in Figure 11.3.2 the particular situation where the resistance of the
ring (which in our model can have any value) is identically zero, and the mass of the ring
is small enough (or the field of the magnet is large enough) so that the ring levitates
above the magnet. We let the ring begin at rest a distance 2
a above the magnet. The ring
begins to fall under gravity. When the ring reaches a distance of about
a above the ring,
its acceleration slows because of the increasing current in the ring. As the current
increases, energy is stored in the magnetic field, and when the ring comes to rest, all of
the initial gravitational potential of the ring is stored in the magnetic field. That magnetic
energy is then returned to the ring as it “bounces” and returns to its original position a
distance 2
a above the magnet. Because there is no dissipation in the system for our
particular choice of
R in this example, this motion repeats indefinitely.
What are the important points to be learned from this animation? Initially, all the free
energy in this situation is stored in the gravitational potential energy of the ring. As the
ring begins to fall, that gravitational energy begins to appear as kinetic energy in the ring.
It also begins to appear as energy stored in the magnetic field. The compressed field
below the ring enables the transmission of an upward force to the moving ring as well as
a downward force to the magnet. But that compression also stores energy in the magnetic
field. It is plausible to argue based on the animation that the kinetic energy of the
downwardly moving ring is decreasing as more and more energy is stored in the
magnetostatic field, and conversely when the ring is rising.
Figure 11.3.3 shows a more realistic case in which the resistance of the ring is finite.
Now energy is not conserved, and the ring eventually falls past the magnet. When it
passes the magnet, the sense of the induced electric field and thus of the eddy currents
reverses, and the ring is now attracted to the magnet above it, which again retards its fall.
There are many other examples of the falling ring and stationary magnet, or falling
magnet and stationary ring, given in the animations at this link. All of them show that the
effect of the electric field associated with a time-changing magnetic field is to try to keep
things the same. In the limiting case of zero resistance, it can in fact achieve this goal,
e.g. in Figure 11.3.2 the magnetic flux through the ring never changes over the course of
the motion.
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