(2008-05-08)
Faraday's Disk: Low Voltage, High Current (1831)
A homopolar contraption which helps clarify fundamental concepts.
Let's first imagine a device with axial symmetry whose
characteristics are easy to figure out.
(We'll end up analyzing something even simpler.)
Consider a conducting disk ("copper disk") spinning rapidly in a small
gap between two slightly smaller ring magnets
(axially magnetized). In the main,
the magnetic flux F through the copper
disk is equal to the surface area S
of the facing magnets multiplied by an average
magnetic induction B whose value is roughly
the sum of the surface fields of the two magnets.
F =
B S =
B p
( OD 2 - ID 2 ) / 4
Current may travel from the center of the disk (through a conducting axle)
to the rim, where it's collected
either by uniformly distributed brushes or by a circular
contact with a pool of mercury.
In a steady regime,
each line of current need not be straight but it's rigidly attached to the disk and will
thus cut through the above magnetic flux
once per cycle.
If the disk spins at a frequency n
expressed in hertz (1 Hz is 60 rpm) the voltage
U between the axle and the rim is thus:
U = F n
= B S n
The internal resistance of this type of generator can
be extremely low (it's basically the resistance of the axle, the disk and the
mercury contacts).
Although the voltage is modest, the current produced
can be extremely high.
Here are a few (optimistic) estimates of what would be obtained with readily available
neodymium magnets of different sizes (and prices)
using a thin disk and a small gap between the magnets.
Actual measurements could
be 20% lower, because the values
of the induction fields are overestimated.
Voltage at 10,000 rpm, vs. price of
a pair of magnets (color-coded suppliers)
Polar plates must be used to reduce the magnetic losses from the center holes.
Ring magnets which have very large center holes (like the huge
NR025 from
Applied Magnets) are
not suitable for this application, unless we
design for the inside current the same kind of circular contact
which is required for the outside current.
To be blunt,
we could even rule out entirely the units
which have holes larger than
1" (ID) in the above table.
The highlighted RY046
(from K&J Magnetics ) is nice enough.
Two stacked pairs of these, would yield 70%
more voltage (up to 570 mV at 10,000 rpm).
With 3 pairs, we would obtain 700 mV (at a cost of $96.00).
If made out of pure copper, a quarter-inch rod will have a resistance of about
0.5 mW per meter of length.
For a half-axle of length L=1.5", that's
0.02 mW (That would
be 0.01 mW if both sides of the axle are used
to carry current, but substtuting brass for copper increases the resistance by
a factor of 4.)
It would be an overkill
to have a copper disk with an axle-to-rim resistance
much below that.
Let's estimate what this entails for the thickness (e) of the disk:
Consider the disk at rest.
Let s be the resistivity of copper.
The electric field E inside the disk is radial
and the current density j
is proportional to it
(that's Ohm's law: j = sE).
At a distance r
from the axis, the current density is equal to the total current I
divided by the lateral area of the relevant cylinder. Thus:
j = I / (2pr e)
= s E
(Incidentally, by Gauss's law,
there's a static charge
eo I/s
on the axle.)
The voltage U is the integral of E dr
from axle (r0 ) to rim (r1 ) :
U = ò
E dr =
[ I / (2p s e) ] Log ( r1 / r0 )
This gives the resistance of the disk as R = U/I.
On the other hand, the resistance of the half-axle of length L is:
R = L / (p s r02 )
Using 2r0 = 0.25", 2r1 = 2"
and L = 1.5",
those two are equal when:
e = ( r02 / L )
Log ( r1 / r0 )
= 0.026 " = 0.66 mm
Such a thickness should provides good enough structural integrity
and doesn't force too large a gap between the magnets,
(which would reduce the field, the flux and the voltage).
This corresponds to
22-gauge copper
sheet (thickness 0.025" = 0.635 mm).
Also usable are
24-gauge (0.508 mm)
or 20-gauge (0.8128 mm).
18-gauge copper is just about
1 mm (0.04") which is probably too thick.
Making it Simpler :
As mentioned above, the lines of current are rigidly attached to the copper disk
(as they are related to the trajectories of charged particles which
interact with the copper lattice). However, the magnetic field lines
are not similarly bound to the magnet.
If a magnet rotates around its axis of symmetry,
the magnetic field stays the same. Thus,
nothing is induced on the copper disk if the magnets spins...
So, if we let the magnets spin at the same rate as the copper disk, we obtain exactly the same
effect as if the magnets were stationary! If we do that, we can bypass
all of the precision machining and the risky business of maintaining
a small gap between two powerful magnets:
Just sandwich the copper disk between the two magnets and spin the whole thing
as a massive flywheel!
Actually,
we don't even really need the copper disk to produce the effect.
If you're in a hurry (or can only afford one magnet)
just attach the magnet(s) to a quarter-inch screw with washers and spin
that with a drill...
Use brushes (bare wire) and a voltmeter to observe the potential difference between
the axis and the rim. (A capacitor between the leads stabilizes the voltage.)
WARNING : Spinning large neodymium magnets can be hazardous !
Neodymium magnets are brittle and their density is 7.5 g/cc.
This gives an RY046 magnet a mass of 143 g, a
moment of inertia of
0.000187 and a rotational energy of about 102.4 J
at 10,000 rpm.
A flywheel of 3 pairs would be 6 times that.
For an actual high-current generator
(with mercury contacts, casing, etc.)
a copper disk squeezed between two magnets has several advantages:
Voltage is highest on the equator of the rotor
(slightly above the surface of the magnets)
and a solid disk has lower resistance than mere magnet plating.
Incidentally, the above voltage measurement
tells you about the polarity of your magnet.
If the mechanical and magnetic north-south orientations are identical
(see sign conventions) then the
rotation tends to make the rim electrically more positive
than the axis.
In the
case of the Earth,
the North Magnetic Pole
is (currently) a south pole
(the north poles of small magnets point to it).
Thus, the rotation of the Earth creates an electromotive force
(emf) which makes the equatorial regions more
negative than the polar ones (by about 100 kV).
There were 91 episodes of "Beakman's World" made between 1992 and 1998.
Reruns have been airing again regularly since
since 2006.
Electric motors are discussed in episode 42
(the 16th in season 2).
The rotor of Beakman's motor
is entirely made from one piece of varnished wire
(even its mechanical axis just consists of the
two ends of the wire, sticking out from the coil).
The "brushes" that feed the coil for only 50% (or so)
of each cycle are merely obtained by scraping the insulating varnish off the upper part
of the wire which will touch the conducting craddle
(it's only necessary to do this on one
side, the wire on the other side can have its insulation removed completely).
Many articles and videos about this motor are now available on the Internet by several authors,
including Stan Pozmantir
and Simon Quellen Field.
Chris Palmer's original (at fly.hiwaay.net/~palmer/motor.html)
apparently vanished before June 2011
(thanks to Tim Winters
for pointing that out, on 2013-06-09)
but excerpts
and
complete
copies can still be found.
To his original article, Palmer had added the following personal comment,
which he had received from the late Mark Ritts
(1946-2009):
I play "Lester," the guy in the rat suit on "Beakman's World,"
and I'm delighted to see my personal favorite Beakman experiment
so faithfully rendered and explained on the Web. Thanks!
Mark "Lester" Ritts, Los Angeles, California
The price to pay for the great simplicity of the Beakman motor is a total lack of usefulness.
As nothing can be attached to the axis of the motor,
the success of a completed device can only be measured by how fast it runs
without load.
This was precisely the goal for the
motor contest which
Walter Lewin
ran as part of his freshman class at MIT on electricity and magnetism
(8.02) in 1984, 1988 and 2002. The 2002 award ceremony honored JungEun Lee (4700 rpm) as well as a contestant
not enrolled in the class (Daniel Wendel, 4900 rpm).
Tim Lo (1250 rpm) received a special prize for most artistic design.
The runners-up were Adam Kumpf (4100 rpm) and Ryan Damico (2500 rpm).
Only 6 of the 225 motors entered by students for extra credit ran at or above 2000 rpm.
Building this motor is a great classroom activity which can be completed in
a single class period.
Like Walter Lewin at MIT, high-school teachers are making it more fun by
letting the students compete for the fastest motor.
To measure motor speed, the traditional way with strobe lights (as used by Pr. Lewin)
is best replaced by some form of electric frequency measurement.
An adequate signal can be obtained directly from the supply voltage, because of the
battery's internal resistance (Brian Lamore).
Alternately, you can built a general-purpose probe to measure the occultations of the
light from a laser beam received by a photoelectric cell (this is another fun project
by itself).
Such a probe allows measurements
without touching the running motor of a student
(and the rotor need not be marked with white paint either).
Make the device portable and simple to use by attaching the laser pointer and the photocell
to the ends of a rigid bow (like a hacksaw frame).
In a Beakman motor, the spinning coil produces
two occulations per revolution if the laser is pointed at the fringe, but there
are twice as many if the laser is allowed to go through the coil
(the occulattion pattern is asymmetrical unless the beam goes through the center
of the rotor). Using the "fringe" methods (the only one available if the coil is not hollow)
the motor's rpm is actually 30 times the observed frequency of occultation measured in Hz.
The conversion factor to use is only 15 with the "center" method.
(2008-05-13) Magic Wheels
