Reading Steinmetz, part 3

September 20, 2012 2:43:43 AM CDT

What with getting the SMT design workflow running, I've left some of my other projects floating in the void, so let's spend some more time with Steinmetz:

Section 3:

When we left off last time, Charles had just gotten done discussing the energy stored in the magnetic field/circuit around an electrical circuit.. magnetomotive force, magnetizing force, and all that fun stuff. The central idea was that some of the energy pushing current down a wire gets transformed to magnetic energy.

Section 3 adds a new wrinkle to the discussion: motion of the wire.

Electromotive force:

If a wire moves through a magnetic field, it cuts across the magnetic lines of the field. Each of those lines, if you remember, represents one unit magnetic force, which translates to a physical force of 1 dyne (enough to accelerate one gram by one centimeter per second squared) at a distance of 1 cm.

When a wires cuts the magnetic line, the magnetic field imposes a physical force on the electrons in the wire, generating current. It's called electromotive force, or EMF.

The unit EMF is what you get when a wire cuts one magnetic line per second.

Now, in a perfect conductor, EMF would translate directly to motion of the electrons, but we don't live in a world of perfect conductors.. and Steinmetz certainly didn't. For any normal material (one which conforms to the Fermi-Dirac statistical model, if you want to get picky about it), it takes a certain amount of energy to push an electron away from an atom. If that electron gets close enough to another atom that it stops moving as electrical current and drops back to orbiting a nucleus, the new atom has to absorb the energy that pushed the electron away from the previous atom.

Some of that energy turns into motion, and 'making the atoms move faster' is what we understand macroscopically as 'heat'.

So.. EMF can't make all the electrons in a material move, the atoms push back, and you're left with a tradeoff between the amount of EMF that's trying to push the electrons and the amount of current that actually flows through the material.

The ratio of those two values (EMF to current) is the material's resistivity. If we have a material whose resistivity is 1, one unit of EMF produces one unit of current.. at least it would if we hadn't already stated specific definitions for 'unit current' (ten amps) and 'unit EMF' (the EMF produced by a wire cutting 1 magnetic line per second).

Turns out it takes a billion 'unit EMF's to produce one 'unit current'.

Now, unit current is 10 amps, and we've already agreed that if the resistivity of a material is 1, we should be able to say that our EMF is 10 somethings. Turns out the 'something' is the practical unit of EMF, the volt.

Scaling down to "1 volt produces 1 amp in a material whose resistivity is 1" says that a wire has to cut a hundred million magnetic lines per second to generate one volt's worth of EMF. That also happens to be a direct statement of Ohm's Law, by the way, so resistivity is measured in ohms.

The "one magnetic line per second" level of EMF is what we'd call "10 nanovolts" today, and in a 1 ohm material 10 nanovolts of EMF will cause 10 nanoamps of current to flow. That may not sound like much, but A) it corresponds to the motion of about 60 billion electrons, and B) the transistors in the computer you're using right now operate at roughly that level of current. Companies that make microprocessors for portable devices advertise products that work in the high-picoamp range (a few hundred million electrons per second).

Self-induction:

Now, we know that current moving through a wire creates an electric field. We know that a wire moving through an electric field will acquire EMF, and that the EMF will produce current.

Nobody said the magnetic field had to be different from the one generated by the current itself.

The number of lines in the magnetic field around a wire corresponds to the amount of current flowing through the wire, so if the current is steady, there won't be any change in the magnetic field. If the current changes, though, that change produces a change in the magnetic field. That change in the magnetic field induces EMF on the wire, producing current.

Now, before you rush off to patent an electrical generator that runs off its own self-induced current, Lentz's law says there's no free lunch. The current self-induced in a wire by a change in the current moves in the opposite direction of the change. If you try to increase the current through the wire, self-induction will push back and allow less current to flow. If you try to stop the current flowing through a wire, self-induction will keep some current flowing even after you disconnect the power.

I ran into that issue myself recently.

Generators:

At this point, Charles returns to the subject that's near and dear to his heart: generating electricity.

He describes a rather nifty DC generator..

Imagine the the standard horseshoe magnet that's been a staple of dime stores and cartoons for the last fifty-odd years. Now imagine sweeping it in a circle so the ends of the horseshoe trace two concentric rings.. the N pole being the smaller ring in the center, the S pole being the larger ring to the outside.

Obviously there will be a magnetic field between those rings.

Now imagine taking a ring of copper, putting it between the two magnetic rings, and turning it. Obviously we have a conductor moving through a magnetic field, so the thing will generate current.

Current induced by magnetism moves perpendicularly to the magnetic field (that's a byproduct of the Lentz's law thing mentioned earlier), so current in the copper ring will flow from the edge that's deepest between the magnets to the edge that's shallowest. It you put electrical taps at both edges, you'll get a DC current.

Now, Charles scales the thing up a bit from the version I just described.. he sets the magnetic flux between the poles of the magnet at 25 million lines, and spins the copper ring at 3000 rpm.

Running the math, any given point on the copper cylinder will cut 25 million magnetic lines per second, and will make 50 revolutions per second. That comes out to 1.25 billion lines per second, for an EMF of 12.5 volts.

Wrapping up:

Now that we have physically-demonstrable ways to define voltage, current, resistance, and the energy in a magnetic field, we have the basic vocabulary necessary to talk about building generators.. which will happen a lot next time.

Random brain cookies:

Larkinson's Law: All laws are basically false.