Half an ounce of electrons

Publication date: 16 May 2009
Originally published 2008 in Atomic: Maximum Power Computing
Last modified 06-Dec-2012.

 

Everybody knows that battery technology is what's standing between us and proper electric cars, laptops that run for a month without recharging, man-portable death rays, and a variety of other things that everyone agrees would be great.

Every other technology's improved by leaps and bounds in the last hundred years, but a lithium-ion battery today only packs about ten times as much energy per kilogram as the batteries that powered early electric cars.

By "early", there, I mean back when one of electric vehicles' advantages over internal combustion was that you didn't need to turn a crank to start them.

(Petrol, by the way, has about 25 times the propulsion energy per kilogram of the best lithium-ion batteries, even after you take into account the fact that electric drive systems can be up around 90% efficient, while internal combustion is usually in the low thirties.)

Ideally, we wouldn't have to keep packing ever-more-alarming substances into close proximity with each other and somehow persuading them to have an electrochemical relationship.

Ideally, aliens would just land, and hand us some perfect batteries.

(Or, more practically, a machine that makes perfect batteries.)

When sci-fi authors need to come up with a source of outrageous amounts of power to run their funky gear, they can either just wing it altogether and say it's an "Arc Reactor" or a "Zero Point Module" or something, or they can make a grab for scientific plausibility and use a real, existent phenomenon. The most obvious such phenomenon is the one used by Star Trek - matter/antimatter annihilation, or some other way of converting mass directly to energy.

It's easy to figure out how much energy you can get by doing that, because this is where the one piece of advanced physics that everybody knows, E equals MC squared, comes in.

That equation is talking about "mass-energy equivalence" - energy in joules equals mass in kilograms times the speed of light in metres per second, squared. So a mass-conversion "battery" which weighs as much as a AAA alkaline (11.5 grams), and which converts all of its mass to energy, will yield about 1.034 times ten to the power of fifteen joules of energy.

A joule is a watt-second; if you run a 100-watt light bulb for ten seconds, it consumes 1000 joules. So if all you need from the above 11.5-gram mass-conversion device is a piddling thousand megawatts of power - the output of a good-sized nuclear power plant - then it'll be able to deliver it for about twelve days.

Or, if you prefer, it could run a 1.21-gigawatt time machine for 9.9 straight days.

From something the weight of a AAA battery.

There's a problem with this kind of fantasy battery, though: Its output isn't electricity. Mass conversion, in the real world, creates a broadband smorgasbord of hard radiation - essentially, all the stuff that hits you before the nuclear explosion's shock wave arrives. So this fantasy alien mass-conversion battery would need to have a fantasy alien radiation-to-electricity converter built in. (And yes, that awful Dan Brown book gets pretty much everything to do with antimatter wrong, too.)

(After this page went up, a reader contacted me to point out that I'd actually made the same mistake Dan Brown made, when calculating the energy yield of an antimatter power source. If you've got X kilograms of antimatter and annihilate it with normal matter, you don't just get X times c-squared energy - you get 2X times c-squared, half of  it from the annihilation of the matter and half from the antimatter. So if an 11.5-gram antimatter battery is half matter and half antimatter, you'll get 1.034 times 10^15 joules out of it. But if the battery's mass is all antimatter, and it sucks in normal matter - air, for instance - from the outside world, you'll get twice as much energy. Now I'm off to sit in the bathtub for a few hours, weeping and scrubbing myself with a wire brush and Dettol. But my soul will never truly be cleansed of the stain of sharing a mistake, as well as a name, with Dan Brown.)

A more obvious solution suggests itself. Electricity is a flow of electrons, right? So why not just make a "battery" which is nothing but a container full of electrons?

You'd expect a mere bucket of electrons to not contain nearly as much energy as an equivalent mass of matter being converted directly to energy, but surely it'd be good for something.

How much electricity could you, in fact, get from a AAA-sized alien battery containing 11.5 grams of electrons?

Now things become more complicated. Electricity, you see, certainly is a flow of electrons, but electrons are not, themselves, electricity. It's like water and rivers. Water does not itself comprise a river, unless it's in one place and there's a way for it to flow downhill to another place. Water flows in channels to somewhere lower; electrons flow in wires to somewhere with a lower electrical potential.

For the purpose of our back-of-an-envelope calculation of the amount of energy you could store in a AAA-sized 11.5 grams of electrons, though, we can side-step this issue, by employing a little lateral thinking.

To pack the electrons into such a battery, you see, you'd have to do work to push all the tiny negative charges together. They don't want to be together; like charges repel. That's why they want to flow to somewhere with a lower potential - lower potential equals less charge.

So to get a ballpark idea of the amount of energy you could get out of such a battery, you can just look at the amount of energy you'd have to expend to pack the electrons into it. Presumably you wouldn't be able to harvest all of that energy on discharge, but you'd get a fair slice of it.

The fundamental unit of electrical charge is the "coulomb", and the electron has a charge of about minus 1.6 (because electrons are negatively charged) times ten to the power of negative nineteen coulombs.

That's a very small amount.

But the electron is also very, very light. An electron at rest weighs about 9.1 times ten to the negative 31 kilograms. So there are about one thousand and ninety-eight million million million million electrons to the gram.

So 11.5 grams of electrons would give you about 1.26 times ten to the 28 of 'em. Which would, all together, have a negative charge of slightly more than two billion coulombs.

That's a lot.

For simplicity, let's model the job of packing the charges into the battery as taking two charges, each of zero size and about negative one billion coulombs, and pushing those together until they are, let's say, two centimetres apart - a reasonable guess for the average separation of points inside a AAA battery.

Here, we use Coulomb's Law, which tells us the force between two point charges. It's proportional to the product of the charges, and inversely proportional to the square of the distance between them.

If I tell you this, and also tell you that the force needed to push one-billion-Coulomb charges together is more than 300 million Newtons even if one of them is on the surface of the earth and the other is located on the moon, you may start to realise that it's possible that the AAA-full-of-electrons may be just a little bit more potent than the self-annihilating antimatter AAA.

(In the above situation, in order to push the Earth-surface charge toward the other one on the moon, you would need to be able to bench-press more than 30,600 tonnes. Unless I've slipped a digit in my calculations, anyway. Tell me if I have.)

UPDATE: Needless to say, someone did. Corrections down here!

As it turns out, actually packing that many electrons into something a few centimetres long starts you talking about joule counts of around ten to the power of thirty, if not more. Given a world power consumption of fifteen to twenty terawatt-hours per year, you're talking a AAA battery that can supply that load for tens of thousands, if not hundreds of thousands, of years. The total energy content would be equivalent to tens, if not hundreds, of billions of gigatons of TNT.

Any attempt to actually make such a device would, of course, require the sort of electromagnetic containment energies that'd be a serious challenge for entities that make amusingly-shaped black holes for a hobby. If that containment should fail, all of the energy would be released in a blast of beta rays whose secondary radiation astronomers in other parts of the galaxy would probably find very puzzling. It wouldn't puzzle anybody on earth in any way at all, though, because we would all now be part of the new Inner Asteroid Belt.

Your little battery full of electrons would also need to be insulated from the rest of the world by, uh, negaspace ultronium, to avoid all of those compressed electrons arcing out to places with a lower negative charge, like anywhere else in the entire universe. But I'm sure this sort of thing is all in a day's work for your typical little green man. I presume they do all those weird things to abductees to help them relax after their day job.

So, in closing, if a little green man offers you the choice of a matter-annihilation power source or one that just has several grams of electrons inside it, you're a mug if you don't pick the second one.

Just don't drop it.


UPDATE: It only took about two hours before someone working on a Ph.D having to do with pulsars pointed out to me that I had, in the above, fallen into the trap of using the classical, Newtonian sense of "energy" when I thought of a can full of electrons.

(My correspondent blogged her response as well as e-mailing me; I highly recommend that post to Weird Physics enthusiasts.)

Anyway, these electrons in a can may really really not like each other at all, but they're confined by magical alien superscience and not moving. And so, in the Newtonian sense, they can be said to have potential energy, but no kinetic energy at all.

The special-relativity kind of energy is what I should have been thinking of, though, because once you start talking about very large amounts of pretty much anything, the old Newtonian approximations fade away and the more precise Einsteinian physics takes over.

In special relativity, "mass" and "energy" are just two words for the same thing. In everyday situations - heck, in a lot of space-probe situations, too - a mass is seldom large enough that its energy makes much of a difference, and vice versa. When you're pushing a bunch of electrons cheek-by-jowl, though, you're no longer in an everyday situation.

I'm tempted to throw a couple of analogies in here that seem to me to make this situation - and its real-world equivalents, like how radiation behaves, or what happens when a star collapses - easier to understand. I'll resist the temptation, though, because I could fit everything I know about Einsteinian physics on one sheet of paper (or, at most, the door of one bathroom stall), so I'd be more likely to obscure the truth than to explain it. I'll try to stick to the facts, instead.

As Wikipedia currently says, the rest mass of an object is almost never the sum of the rest masses of its parts. You can't just add the rest mass of umpteen electrons, because electrons repel each other, and that repulsion adds energy to the system, even if nothing's moving.

So if you do the sums correctly and add the electron binding energy to the calculations, you'll find that your 11.5 grams of concentrated electrons will weigh... a bit more.

All right, quite a bit more.

Well, look, if I'm honest, really quite a lot more.

Something in the order of... hundreds of millions of tonnes.

Which, compressed into something you could hold in the palm of your hand, gives density in the order of that of a neutron star, and might therefore very rapidly spaghettify you, and anything else in the area, unless I've messed up my gravity calculation, too. (I did it the Newtonian way, which was probably a mistake.)

In any case, you'd probably be wanting the magic aliens to provide the battery with some sort of gravity shield, which with any luck would also cancel all of its mass. They could probably use an old one off their starship - you know, the thing that gives the ship imaginary mass, and thereby makes the speed of light the vessel's minimum speed.

Easy!

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