Airships are pretty dang cool. Airplanes need a continuous expenditure of energy to stay in the air, but if you just fill a bag with a light gas, you can stay up in the air with no energy expenditure at all. The two lightest gases are hydrogen and helium. Though the hydrogen atom is 4 times lighter than helium, hydrogen gas is made of two hydrogen atoms bonded together, so it's only twice as light as helium gas. The difference in lifting power is even more minor: What matters for lifting power of a gas is the difference between its density and that of ordinary air. Hydrogen and helium are both much lighter than air, so the lifting power of either gas is nearly equal to the density of air.

Hydrogen and helium both have problems as lifting gases. Helium's problem is that it's very expensive. Helium comes out of the ground, usually as a byproduct of fossil fuel extraction. There's a finite amount of it in the Earth's crust, and effort must be expended to obtain more of it. On the demand side of the equation, helium is a very useful gas, with many applications in cooling, spaceflight, balloons, etc. It's quite ironic the second most common element in the universe is so hard to come by on Earth. Hydrogen is much cheaper. It can be created from methane, or by electrolysis of water. The main problem with using hydrogen in an airship is safety. Hydrogen is very flammable, and great care must be taken to avoid sparks, and to avoid allowing any hydrogen and oxygen to mix.

Some people have proposed vacuum airships. Vacuum would not be a much better lifting gas than hydrogen or helium. Its lifting power is exactly equal to the density of air, and hydrogen and helium are already pretty close to that. But maybe we'd like to use vacuum because its cost is equal to the energy cost of pumping out the air in a given volume of space, and it's not flammable. There are some problems with this idea, though.

The obvious problem is that an air sac filled with vacuum would simply collapse. The working principle of airships is that the gas inside the balloon can supply enough outwards pressure to keep the atmosphere from collapsing the balloon, while still weighing less than an equivalent amount of air would. Vacuum does not push back, and this is a fairly huge problem. The first thing one might think of is to make a rigid spherical shell instead of a balloon. The problem here is weight. To be light enough to still fly, the shell must be very thin, which in turn makes it susceptible to buckling. See this and the linked patent application for details, but this engineering is pretty hard to pull off. A simple uniform shell would not work for any known material, though there are ways to get around this.

Alternately, what if we kept the floppy structure, but made the electromagnetic field provide the counter-pressure for us? If we had a perfectly reflective material, a photon gas could be used. We don't have that, though, so an electrostatic solution is necessary instead.

The proposal is simple: Make a spherical balloon with a conductive layer and charge it up. The charges repel and this repulsion holds the balloon open, even with a vacuum inside. How much charge is needed to pull this off? We can calculate it without too much difficulty. Electric field from a uniformly charged spherical shell is

$E=\frac{1}{4\pi {ϵ}_0}\frac{q}{r^2}$

If we write in terms of surface charge density $\sigma =q/4\pi r^2$ it's

$E=\frac{\sigma }{{ϵ}_0}$

Energy density in the electric field is

$u=\frac{{ϵ}_0}{2}{E}^2=\frac{{\sigma }^2}{2{ϵ}_0}$

Energy density has the same units as pressure. Think about squeezing the balloon a bit. This would increase the amount of space occupied by the electric field (remember the shell theorem: electric field is 0 inside the balloon), and so increases the energy. So the energy density tells us the outwards pressure produced by the charge on the balloon.

If we plug in atmospheric pressure for $u$ we get the following charge density:

$\sigma =0.00134C/m^2$

This is a disturbingly large charge density. What will the electric field be at the surface?

$E=151MV/m$

Yes, also disturbingly large, nearly 100 times the breakdown voltage of air. The electrostatic airship would shoot out lightning long before it actually worked.

Okay, so what if we didn't demand a vacuum and filled the ballon with mere slightly depressurized air? Say we think we can tolerate an electric field of 1MV/m. How much pressure can that produce? Unfortunately, the quadratic scaling of energy density works against us! The supported pressure would only be about 5Pa. The air inside the balloon would need to have 99.995% of the density of the air outside. Making a hot air balloon sounds way easier!

So how do we make airships instead? Something that I think we should consider is: Just use hydrogen lifting gas! Avoiding sparks is just a matter of being careful, I'd like to think that materials science and engineering has improved since the Hindenburg days. We can also add a layer of safety by making one air sac inside another. The inside one is full of hydrogen, while the outer is full of nitrogen. A leak in either sac will not cause the mixing of hydrogen and oxygen, and sensors could be placed inside the the nitrogen region to detect if too much hydrogen or oxygen is going across the boundary.