Built on Facts

Microwaves and Money

Posted by Matt Springer on December 2, 2008
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If you go to Capital One’s website and type in the appropriate login information, you’ll see a “Welcome Matthew Springer” message and two accounts. One’s a checking account and one’s a money market account where I’m endeavoring to accumulate some savings. (I need to get that moved to a CD or something, the interest rate is awful). Just below those two accounts is a number representing the total amount of cash I possess. It obeys what you might call the financial analogue of the continuity equation:

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This is the version you see in electrodynamics. It says that the flow of charge into or out of a particular location is equal and opposite to the rate of change of the charge at that location. Makes sense. Money is like that. The rate of flow of money into or out of my account determines how much the internal account balance changes. It is, as we say in the business, a conserved quantity.

Energy is a slippery thing to define precisely, but like many fundamental quantities it too is conserved. The idea of keeping track of where energy is flowing can answer a question posed by reader Nick in the comments of the last post:

[...]That got me to thinking… why is the vacuum of space cold? If there aren’t any atoms around to slow down the vibrations of other atoms like there are in a air-filled situation then how is cold “transferred” (even though I think that’s the wrong way to think about it)? Do you simply radiate heat faster than you can you replenish it?

Take something hot – like a fresh piece of toast – and pitch it out the space shuttle cargo bay and into the blackness of space. It contains energy in the form of random thermal motion of its constituent molecules*. Energy is conserved, and so whatever energy enters or leaves the toast will respectively increase or decrease its temperature.

Immediately energy will begin to flow out via thermal radiation. Per the Stefan-Boltzmann law, each square centimeter of toast at (say) 100 degrees will be emitting 0.11 watts of power. Energy is leaving, so there’s less energy in the toast and thus it’s cooling down. Is there any energy coming in from the outside? There’s almost no atmosphere in space and so the only way energy can enter or leave the toast is by radiation. In direct sunlight this can actually be quite a bit. The surface of the sunlit parts of the lunar surface are pretty hot (>100 C) despite being in the vacuum of space. Most of space, however, isn’t close to stars. Most of it isn’t even close to galaxies. In those places you’ll only be receiving radiation from the stray microwave photons that comprise the cosmic microwave background. Your piece of toast tossed into the vast void between the Milky Way galaxy and the Andromeda galaxy will cool down to around 2.7 K before it’s finally radiating slowly enough so that it’s absorbing energy from the microwave background at the same rate. In that sense, space is cold.

It’s a neat thing to think about, and a good way to imagine some fundamental concepts about energy.

(A while back I talked about some similar concepts in my old “Temperature Inside a Microwave” post, which you might like.)

*It also contains other forms of energy from chemical potential to rest mass and others, but those do not contribute to the temperature and will be staying constant in any case.

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Comments

  1. #1 Paul
    December 2, 2008

    So in the movie “Sunshine”, when they jump through space into the air lock, is it realistic to think that one guy could survive with heavy burns? (Assuming they don’t inhale first and their lungs dont collapse due to the pressure). Or deeper still, what would be the rate of energy loss for a person (assuming this is a function of time… or internal energy for that matter)

  2. #2 Carl Brannen
    December 2, 2008

    When you deposit money in the bank, the bank lends it out to someone. Then both you and the new borrower are convinced you’ve got the same money. So in that sense money does not satisfy a conservation law.

    You could argue that money is conserved if you count debt as negative money. That argument gets just a little iffy when you consider the actions of the Federal Reserve. They can print money without ever having to pay it back. Hyperinflatin is possible.

  3. #3 kevin
    December 2, 2008

    Paul,

    Random link: http://www.damninteresting.com/?p=741

    I’ve read that humans can survive surprisingly well in space without any space suit, that it won’t instantly kill you or anything. The link above gives about 10 seconds of conscious but uncomfortable time, followed by about 1 minute and half of unconscious time with only reversible (with immediate medical help) minor injuries, then death in a most unpleasant manner.

  4. #4 Matt Springer
    December 2, 2008

    Good point, Carl. I should say that my money is conserved because I can’t print any more of it. My suspicion is that the massive, massive amounts of cash being brought into existence at the top is going to bit us a few years down the road in the form of severe inflation.

    I guess the gold standard is an attempt to formalize conservation of money – in fact, to tie it directly to conservation of mass. I’m not sure it’s a good idea or even workable if it is, but apparently the concept has some popular support.

  5. #5 Eric Lund
    December 2, 2008

    I should say that my money is conserved because I can’t print any more of it.

    But you receive interest on that money, and that interest compounds. The interest these days isn’t very interesting, but I’m old enough to remember when 5.25% was the standard interest rate on plain vanilla savings accounts. Granted, that was less than the inflation rate at the time, but it’s still an inducement to keep your money in a bank rather than under your mattress (where it would, indeed, be conserved). Or you could invest that money in other things that offer higher potential returns, but also the risk of losing some or all of your money (i.e., stocks and bonds).

    I guess the gold standard is an attempt to formalize conservation of money

    I’ve seen people in the blogosphere advocating the gold standard, but it won’t have that effect. First, somebody could discover an additional source of gold, which is what happened to Spain when they plundered South America, and you get inflation anyway. (Alternatively, a known source of gold which is currently uneconomical to exploit could become economical to exploit due to a relative rise in the price of gold–something akin to oil from the Alberta tar sands, a money loser at today’s prices but a profitable operation at $80/bbl oil.) Second, gold has a handful of industrial uses which would cause the total available supply to decrease–not as much as for other precious metals like silver and especially platinum, but something that might throw off the calculation.