W9_parallel_resonance.eps

Week 8: Faraday’s Law and Induction 287

B(in)

v

Ieddy

copper sheet

Ieddy

Fm

Figure 106: A sheet of copper being pulled rapidly out of a field has inducededdy currents. The
forces from these currents, according to Lenz’s Law,resistthe motion, causing a magnetic “drag
force” similar to that observed in the rod on rails problem. The kineticenergy of the object is
transformed into heat by these currents (resistive Joule heating).

Eddy currents are remarkably important, as they are a source ofenergy losswhenever we attempt
to e.g. alter a magnetic field in the vicinity ofany conductor. Eddy currents produceJoule heating
of the conducting material very readily – one can actually cook foodon stoves that use rapidly
a varying magnetic field to directly heat metal pots placed in the field^78. Transformers (covered
later) rely on rapidly varying, ferromagnetically enhanced magneticfields to step up or step down
voltage, and unless care is taken to prevent eddy currents in the design of the magnetic cores, much
of the energy being transmitted through the transformer will be lost to heating the cores. Eddy
currents cancel electromagnetic radiation at the surfaces of conductors, both heating the conductors
slightly and causing the electromagnetic field toreflectfrom the surface rather than be transmitted.
It seems worthwhile to spend a moment trying to understand them.

In figure 106 above, a sheet of copper being pulled rapidly out of a strong magnetic field is
illustrated. It is moving at some speedvto the right. As it is pulled out, the magnetic flux through
theentire sheetis reduced. This creates an induced field in the conductor and its associated induced
voltage that (because it is agood conductor) can and does drive a large current in the copper. This
current is not isolated or confined in the conductor – the conducting sheet is like an entire field of
parallel resistance pathways and the current spreads out to usethem.

Note well, however, thatlike the rod on rails problem(which this greatly resembles!) the net
forceon the induced current is in a direction thatopposesv(whichever direction the sheet is moving,
in or out of the field). The current flowinthe field produces this force , while the current flowing
in the opposite direction through the part of the sheet that is out of the field does not. One expects
that the velocity of this sheet, like the velocity of the rod, will be exponentially damped, or, if the
sheet is being pulled, will reach a terminal velocity.

The current itself is like a “whirlpool” or eddy of charge swirling aroundin the material, hence
the name eddy current. There are several simple demonstrationsof eddy currents – swinging a sheet
of copper down between the poles of a powerful magnet with or without slits that break up the
conductive pathways and reduce the effect, swinging a magnet above a conducting sheet, or (my
favorite) dropping a powerful magnet down through a copper pipeand a PVC pipe at the same
time.

(^78) Wikipedia: http://www.wikipedia.org/wiki/Induction Cooker. This is actually a lovely article, and will introduce
you to the notion ofskin depth, as induction stovetops only tend to work on ferromagnetic pans (such as cast iron)
because they have a small skin depth.