Clamp-on ammeters have a ferromagnetic "clamp" that encloses a wire in
a closed magnetic circuit; the magnetic field induced in the
ferromagnetic material is proportional to the total net current flux
enclosed by the clamp, and you can measure this field precisely with a
Hall-effect sensor, thus distinguishing wires carrying a load from
wires that don't.

You could use the same approach to build a low-power electronic device
that powers itself by leeching energy from the magnetic field around a
current-carrying wire, without needing a direct electrical connection.
This could enhance safety and reliability and ease installation, since
you can "plug in" such a device without making a direct electrical
connection, and if the device shorts out, it won't cause an electrical
fire.  Not only could it harvest power without ever coming in contact
with the wire, it could harvest power without even coming near the
wire; it need only enclose the wire with a loop of ferrite without
also enclosing its return path.

Effectively this is a clamp-on transformer, with the "primary winding"
of the transformer having only one turn of wire.  If that wire is
normally carrying, say, 100 amps, then the secondary winding of the
transformer with, say, 1000 turns, could draw up to 100 milliamps; but
if the total available voltage to be dropped through that one turn is
240 volts, the secondary winding would need to handle 240kV, which is
difficult.  If we limit the secondary winding voltage to 10V, then the
voltage drop on the primary will be an insignificantly small 10mV, and
the total power being transmitted can be up to 10mV * 100 A = 1 watt;
the secondary winding then can draw up to 1 watt / 10 volts = 100 mA
still.  That's enough power to allow the use of a very simple power
supply (say, a diode, a small capacitor, and a 7805) and a relatively
inefficient electronic device.

If you don't enclose the wire in ferrite, but simply put a ferrite rod
near the wire and perpendicular to it, you'll be able to harvest
orders of magnitude less energy, but installation would be even
easier.

Inductive coupling is only one possible way to harvest energy from
power-line fields.  Capacitive coupling is also feasible.  However,
the 60Hz of typical line power presents an extremely difficult problem
for capacitive-coupling energy harvesting.  If you manage 10pF of
capacitive coupling to a power line field, which is probably about all
you can hope for, your 1/ωC capacitive reactance is 270 megohms at
60Hz.  The other side of your power-supply input is then going to be
your capacitive coupling to ground at around 100pF, I think.  (Would
it be better or worse if you had a wire to ground?  Better, I assume.)
You're going to need a power supply with an input impedance in the
hundreds of megohms in order to be able to take advantage of that.
Megohms or tens of megohms is probably feasible with CMOS; hundreds of
megohms is probably not.

However, all is not lost!  Fluorescent lights and high-intensity
discharge lights powered by AC produce much-higher-frequency
harmonics, some one to two orders of magnitude higher in frequency.
This brings the capacitive reactance down to the megohms to tens of
megohms range you need.

So you could parasitically capture a significant fraction of the line
voltage, but only in the high-frequency harmonics produced by the
discharge.

Another approach, which would be more practical in areas without
electrical lines, is to harvest atmospheric electricity, either from
actual lightning strikes or directly from the atmospheric voltage
gradient.  During a lightning storm, atmospheric voltage gradients can
reach 100kV/m, only an order of magnitude below air's ionization
strength.  In clear weather (the "fair weather condition"), it falls
three orders of magnitude to some 100V/m.

A lightning rod struck by lightning has some 30kA available, but only
for tens to hundreds of microseconds.  If you erect a ten-meter
lightning rod, you could in theory harvest up to a megavolt of the
lightning's voltage --- a few megajoules per lightning strike.  If you
put your lightning rod on a mountain top, you could perhaps get
several lightning strikes per month, for a total average power on the
order of a watt.

Harvesting the energy of a lightning strike, however, seems like a
really difficult problem.  A 1000:1 step-up transformer as discussed
above could reduce your 30kA to some 30A, at the cost of boosting your
megavolt to a gigavolt.  If you put a series of these transformers
along the lightning rod's path to ground, you could drop only a tiny
fraction of the voltage through each one, making the situation more
manageable.  If you can dump this massive amount of power through a
low-resistance path into some kind of resonating circuit, you could
then store it for milliseconds up to seconds in order to harvest it at
a more reasonable pace.

Harvesting the atmospheric voltage gradient directly seems much more
feasible, and I've heard you can do it as simply as holding up a spent
fluorescent light tube in one hand in a thunderstorm.  In the absence
of a separate source of ions, you need a corona discharge to couple
your wire to the atmospheric charge, which means that you need points
sharp enough that the electric field intensity at the point is above
air's ionization strength.  If you're working with, say, only 1000
volts, then you need micron-scale conductive sharp points to generate
ionization, and preferably enough of them to support a substantial
ionic current.  By contrast, if you have 400kV --- say, a two-meter
fluorescent tube plus a two-meter-tall person, in a thunderstorm with
100kV/m --- then any conductive point radius below around 40cm will
produce a sufficient field, if I remember my electrodynamics
correctly.  The points of the prongs on the end of the fluorescent
tube are on the order of 0.4 mm in radius, so they should work down to
about 400 volts.

The question, then, is how much current and thus power you can expect
to draw at these voltages.  It's observed that the fluorescent lamp in
the thunderstorm experiment will simply flash periodically as the
lamp's parasitic capacitance to ground charges sufficiently to ionize
its contents and discharge the capacitance.  If I SWAG this, we have a
1ms flash per 5 seconds at 40 watts with a 200V breakdown voltage,
giving 40 microamps average charging current; or maybe charging a 1pF
parasitic tube capacitance to 200V in 5 seconds, giving about 40
picoamps.  This difference of six orders of magnitude suggests that I
don't know enough about the problem even to guess.
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