# Inverse square law

The fable of the butter gun. Intuitive explanation of the inverse square law. Written by Willy McAllister.
In Coulomb's Law, the distance between charges appears in the equation as $1/r^2$. That makes Coulomb's Law an example of an inverse square law. Another well-known inverse square law is Newton's Law of Gravitation. It makes intuitive sense that electric force goes down as the distance between two charged bodies increases. But why is the drop off in force precisely related to the square of the distance? Is the $2$ a coincidence? A trick of nature?
An inverse square law is characteristic of anything that spreads out in straight lines, without getting lost. Both gravity and electric force have this property. We illustrate the idea with a fable.

## The fable of the butter gun

Suppose a restaurant has the problem of buttering toast. They want to be very modern and do toast buttering with a machine. The restaurant owner invents a Butter Gun, with melted butter in the handle, which can be squirted out in straight lines of butter.
Here is a piece of toast, and the lines of butter go out and hit it all over.
Now instead of one toast, the butter lines might go on, and you can put the toast farther back, at twice the distance. Two pieces of toast wide, and two toasts high.
All together, four pieces of toast to intercept the butter. The butter will be a quarter as thick. This is the inverse square law (of buttering).
Extending the idea: At triple the distance, you can arrange $3$ toasts by $3$ toasts to fit within the spray lines, for $9$ total toasts, and you get $\dfrac{1}{9}$ the thickness of butter, for "economy" treatment.
The legend of the Butter Gun was inspired by this marvelous 3-minute video by Princeton University professor of physics, Eric Rogers, in 1959.