TANSTAAFL

I’m still going to do a post about transferring energy from gasoline to lawnmower blades, but a thought occurred to me while I was obsessively rereading my last post that I think might make for a good post in and of itself.

I’ve got a copy of The Way Things Work on my bookcase, and sitting right beside it is a copy of The New Way Things Work. I gobbled up the former book as a kid; it was my first introduction to the idea that electrons release photons whenever they jump down shells, to the concept of computer logic gates, how fission and fusion work, and a whole host of other scientific ideas. I know the book is primarily about machines, but for some reason it was the microscopic stuff that stuck with me more than the mechanical bits.

Anywho, there’s a page about transformers in the book, and I still remember thinking when I read it that transformers were somehow cheating.

You're the man, David Macaulay. Seriously.

You wrap wire around a magnet, wrap more wire around the other end of the magnet, and voila, you’ve increased the electricity pumping through the wire. So why is energy in such short supply? Are there not enough magnets?

I didn’t have a clear understanding of what the conservation of energy meant as a kid (and it can certainly be argued that I don’t now, either, given that I have no idea how Noether’s theorem about symmetry leads to conservation laws), but it still struck me as somehow wrong. And it struck me as wrong for a very long time, right up until about this semester, when I learned what a transformer is really doing.

I think this speaks to the fact that we (the general public) have a very fuzzy idea of what energy is. We know that climbing an electric fence that says “DANGER: 10,000 VOLTS”...

Please don't sue me, Steven Spielberg.

...is somehow more dangerous than licking a 9-volt battery. We also know that “it’s not the voltage that kills you; it’s the amperage.” So both of these are somehow electrical quantities that correspond to bigger and more powerful, but are they related to energy? Related, yes, but they aren’t synonymous with energy.

It turns out that a volt is a measure of energy per charge. You can think of it as the amount of pressure an electron is under. An amp, on the other hand, is the amount of charge moving past some point per second. If you multiply these two quantities together, the charge cancels out and you get energy per second. This quantity is known as the watt, which most people recognize as being what the power company delivers.

But more importantly, this gives us a way to measure energy over time. Because energy is conserved, we know that the amount of energy pumped into a transformer must be equal to the amount of energy pumped out (assuming some ideal transformer with no real world problems). What a transformer does, however, is increase the voltage of the current flowing through it. That is to say, it increases the amount of energy packed into each charge. In order to ensure that the same amount of energy passes through a transformer, less charge must flow out.

(It’s also true, however, that charge is a conserved quantity, so you might think just as much charge has to come out as goes in. But that’s not what conservation really means. It just means that the amount of charge does not change with time; it says nothing of where that charge must go. What’s really happening is that electrons go into the transformer, dump their energy into the transformer’s magnetic field, and then circle back to wherever they came from. The energy they dumped into the magnetic field, however, is then transferred to new electrons at the other end of the transformer. The number of electrons never changes, and the amount of energy never changes, but there are fewer electrons coming out one end than going in the other.)

Say we have a current carrying 1 watt of power at 1 volt and 1 ampere through a transformer. After 1 second, 1 watt pumps 1 joule of energy into the transformer. Then we turn the current off. We expect that 1 joule is going to come out of the transformer a second later. If the transformer increases the voltage of the current to 10 volts, then we have 10 joules per coulomb coming out of the transformer. But since we only have 1 joule to begin with, this means we can only send out 1/10 of a coulomb of charge. Our transformer, then, has turned our 1 volt, 1 ampere current into a 10 volt, 0.1 ampere current. The power is still 1 watt, however, which means that the amount of energy delivered over time is constant.

As the title of the post says, there ain’t no such thing as a free lunch. Transformers don’t cheat; they just do some clever bookkeeping. (I’ve tried to imagine a mechanical analog to a transformer but haven’t been able to come up with anything that doesn’t seem tortured and contrived. Perhaps this is one of the places where the world of electricity and magnetism doesn’t mirror the real world, or perhaps I’m simply having a failure of imagination.)

Until next time, folks.