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:
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.