Reader nanoAl had an interesting observation about the post on applying an electric charge to the earth. We were trying to find out how much electric charge would need to be applied to the earth and the moon to cancel out their gravitational attraction. The answer was suprisingly little. Here’s what nanoAl had to say:

Thats about 320 kg of electrons! I wonder how strong a box it’d take to contain the pressure from all that repulsive force, anyone know how to calculate it?

As a matter of fact, I do! Most people know that pressure is force per area. But fewer people know that it’s also the derivitive of energy with respect to volume (I’m going to play fast and loose with negative signs here, if you’re learning this for real don’t take this as anything formal):

And by analogy to the gravitational binding energy the energy required to pack a bunch of electrons uniformly into a sphere is:

Now that’s in terms of radius. We need it in terms of volume:

Hmm. I just realized I’ve changed Q to q, but that’s just a typesetting mistake that I’d rather not spend 15 minutes resetting and re-uploading. They’re the same thing – the total charge. Anyway, now we differentiate with respect to V:

Now let’s say our spherical bottle for all these electrons has a volume of one cubic meter. Plug in all our numbers from that old post, and I come up with

P = 4.58 x 10^{35} pascals

In English units, that’s about 66 million trillion trillion PSI. It had better be a strong bottle!