# Inverse square law

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.

Written by Willy McAllister.

### Contents

## Intuition

It makes intuitive sense that the electric force between two objects goes down if the distance between them gets bigger. But why is the drop off precisely related to the *square* of the distance? Is the exponent $2$ a coincidence?

No, it is not.

An inverse square law is characteristic of anything that spreads out from a point in straight lines. Both gravity and electric force act like this. 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. Butter can squirt out in straight lines from a point at the business end of the butter gun.

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 continue on. You can put the toast farther back, at *twice* the distance. Two pieces of toast wide, and two toasts high.

Altogether, four pieces of toast to intercept the butter. The butter will be a *quarter* as thick when the toast is twice as far away. 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. You get $1/9$th the thickness of butter, for “economy” treatment.

The lines of force coming out from a point charge spread out exactly the same as the lines of butter from the tip of the butter gun. At twice the distance, the electric force is a quarter as strong. This is the *inverse square law (of electric force)*.

## Inspiration

The Fable of the Butter Gun was inspired by this marvelous 3-minute video clip by Professor Eric Rogers, Princeton University Department of Physics, in 1959.

The Fable of the Butter Gun

(In addition to this YouTube version, this video can also be found here.)