Teaching Physics 201 has me digging out some of my old favorite concept-y problems. Nothing dramatic in the mathematics, but at the 201 level you can’t even assume knowledge of derivatives. But you can try to catch their minds with interesting examples. Here’s a classic one:

You’ve got the earth and the moon. They have mass and so they attract each other with gravity. Both the earth and the moon are pretty large, and so the attraction is considerable. On the scale of earthbound undergraduate lab equipment however, gravity from anything but the earth is pretty hard to measure. Pretty much impossible, in fact. We can do a little better with electrical forces. You can charge a glass rod by rubbing it with fur, and observe the effect of the charge on little sheets of foil or similar. These electrical forces are very, very tiny. But even so we’re able to see them, which is more than we can say for gravity.

So, if you sprinkled some extra electrons on the earth and the same number of electrons on the moon, how much charge would you have to add in order to completely cancel the gravitational attraction? Let’s figure it out. Call the charge q, and we’ll put the gravitational force on the right and the electrical force on the left

Of course M_{e} of the mass of the earth, and similarly for the moon. The q^{2} really is the product of the charge on the earth and the charge on the moon, but we’ll assume they’re equal for conceptual convenience. Nicely, the distance r will cancel and solving for q we get:

Plug in the appropriate numbers to find the required charge on each body. I find that the answer is 5.71 x 10^{13} coulombs. That’s a pretty huge amount of charge in total. But for the earth it’s only about 10^{-11} coulombs per kilogram. And that itself is only sixty million or so excess electrons per kilogram. And considering each kilogram of earth contains trillions of trillions of electrons, only an absolutely tiny percentage of the total atoms in the earth would actually need to be ionized.

We often think about gravity being the strongest force in our daily experience. It isn’t. Electrical forces just have the decency to cancel themselves out to very high precision. Gravity builds up on itself without limit. Which in the final analysis is a pretty lucky thing for us – it’s nice having these solid planets which hold themselves together and solar systems which stay mostly stable over quite long time periods.

Still, next time lint clings to your shirt think about how amazing it is that electrical charge can produce forces strong enough to have visible effects from something so small. Gravity couldn’t hold lint to your shirt in a million years.