### Unobtainium capacitors

Continuing my train of thought from the other day, what are the theoretical limits for a capacitor? Just how much energy can you store in one?

If you try to analyze it in terms of voltages and capacitances, it becomes very tricky: you can make thinner plates closer together without increasing the weight or volume of the capacitor at all. The key to making the problem tractable is to realize that the energy is stored in the electric field. The goal in trying to store as much energy in a capacitor as possible is to get the electric field as strong as possible over as much volume as you can manage.

Since this will be an order-of-magnitude estimate anyway, let's assume that the electric field completely fills the capacitor. That leaves the maximum strength as the last parameter. Since we're looking for theoretical limits, let's assume you're limited by the tendency of the electric field to ionize the atoms in whatever you make this capacitor out of. So an electric field of about a volt per angstrom is just about all we can hope for. This gives us an energy density of about 400 Joules per cubic centimeter, or 0.1 watt-hours per cubic centimeter. This really isn't that much, and in fact according to Wikipedia there is a company claiming its capacitors do substantially better than this. Granted, it's a patent application, but where did I go wrong?

Well, there are certainly things I ignored - dielectric constant, for example - but I think the basic issue is thati in real capacitors the limit really is the strength of electric field your materials can tolerate. Since the energy density depends on the square of this number, clever engineers who can come up with extraordinarily tough materials have a fair amount of room to beat my crude estimate. But it seems clear to me that that room is far from unlimited: capacitors will continue to improve as energy storage devices, particularly in aspects I haven't touched on like leakage, but don't expect one to power your laser pistol any time soon.