Math Made Easy: Problem of the Day 119

There are four fundamental forces which can bind matter together, or push it apart: Strong Nuclear, Weak Nuclear, Electromagnetic, and Gravity. The strong and weak nuclear forces have a very limited range, and are pretty much only felt inside the atom – they’re both what hold the atom together and keep it from collapsing in on itself. Outside the atom, the electromagnetic and gravitic forces dominate, so for our problem, they are the two forces we need to consider.

One might initially just decide that: oh, grapes are so much bigger than a single electron and proton that CLEARLY they’re going to be more strongly attracted to each other than the electron and proton will be. And you might be right. But let’s prove it, and see how right you are. The exercise might also give you a better intuition for which force is stronger: gravity, or the electromagnetic force, and why.

So let us first consider the two forces, and how we find them. Let’s start with the electromagnetic force. Coulombs Law tells us how strongly two charged objects attract each other. It looks like this:

And, if you remember from Problem of the Day 99, we find the gravitic force of attraction from the Law of Universal Gravitation as given to us by Sir Isaac Newton:

Is it not interesting that those two laws look almost identical? They are proportionally the same thing, but with different constants and charge instead of mass.

Let’s calculate the force experienced by the electron and proton pair. First, we need to remind ourselves of the Coulomb Constant and the charge of an electron and proton. It is convenient for us that electrons and protons have equal and opposite charges.

Now we plug those figures in, and find our force. It looks like a pretty small force, and the negative sign tells us it’s an attractive force. That’s important because the electromagnetic force can be attractive or repulsive.

Now, for the grapes, we remind ourselves of the Universal Gravitational Constant. The mass of a grape is about 5 grams.

Plugging those in, we find the attractive force of the two grapes. We don’t need a sign here because gravity is ONLY an attractive force. No evidence has ever been produced of a repulsive side to the gravitic force. It looks like we get a pretty small number here, too.

Comparing the two, we see that the grapes are much more strongly attracted to each other than the electron and proton are. Thirteen orders of magnitude to be precise, which is nothing to sneeze at. BUT, given how much bigger a grape is than either the electron or proton (over 28 orders of magnitude bigger), we can still use this result to illustrate how much weaker gravity is than the electromagnetic force. This is why we didn’t even factor in the gravitic attraction of the electron and proton when calculating their attractive force. We absolutely could have, but it would have been negligible, measuring somewhere on the order of magnitude of 10^-68, or 40 orders of magnitude SMALLER than the electromagnetic force.

This is why you can pick up one magnet with another from a table. The electromagnetic force just completely overwhelms the gravitic force. Even though the Earth is massive – there is WAY more of it than there is of either of those magnets – it really doesn’t take much to overcome it, comparatively. If the force of gravity were stronger, we’d be in a lot of trouble. We just think of gravity as an incredibly strong force because we’re used to living on top of a giant mass that is way, way, way bigger than us.