If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Electrical engineering>Unit 5

Lesson 1: Electric force and electric field

# Inverse square law

The fable of the butter gun. Intuitive explanation of the 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. 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 $\frac{1}{9}$ the thickness of butter, for "economy" treatment.

## Want to join the conversation?

• Why do we use the surface area of a sphere to relate to the inverse square law? And Hyperphysics says source strength of gravity is 4piGM. Where does this come from? Here is the illustration from Hyperphysics I'm talking about http://hyperphysics.phy-astr.gsu.edu/hbase/forces/isq.html#isqg (I'm asking this question because I'm doing universal law of gravitation and I don't understand the inverse square law so please answer my above questions)
• Hello Sam,

Before we start consider a flashlight. Let's shine the light on a 1 cm^2 target on a wall. If you hold the light 5 cm from the wall the target will be very bright. As you move away from the wall the brightness of target is reduced.

Regarding the inverse square law. There is usually a theoretical point at the center. It has zero volume and produces whatever we want. This could be light, gravity, RF energy, etc.

This point source is usually surrounded by a sphere. As you recall the sphere has a surface area:

4*pi*r^2

Visualize the point source painting the inside of the sphere with light. In fact, let the point source produce one unit of light (intentionally ignoring the units).

Now we must shift gears and talk about the amount of light per surface area – remember the flashlight and target. The inside of the sphere has a surface area. We know the formula from above.

Done - that's it. If we merge the stuff originating from our point source with the equation of a sphere we have the Inverse Square law:

source_strength / ( 4*pi*r^2 )

The name inverse square law describes the r^2 in the denominator. As the diameter of the sphere grows the amount of light falling on each square meter on the inside of the sphere is reduced by r^2.

Regards,

APD
• This does not apply when we are shining a laser light on a wall and increase the distance between the wall and the laser light, as the light rays here do not scatter. Correct me if I am wrong.
• The light from a laser pointer does not spread out (at least not very much). If the laser light was butter, the toast would be buttered in one little spot.

If you point your butter laser at the moon, however, the butter beam would be spread out, and you could butter toast there.
• Hi Willy,

I don't get "Is the 2 a coincidence? A trick of nature?".
• Students often ask why the denominator is r^2. The 2 in the exponent seems 'too perfect' to some people. Like why isn't it 2.1, or 1.9? The Fable of the Butter Gun shows why the exponent is exactly 2.
• why is the gun like that
(1 vote)
• yes. This is what you should be taking from the video. Not the concept but the illustration of the gun lol.
• does this occur due to the curvature of body? is it related to center of mass?
• does inverse square law valid in matter also??
• No, we use Newton's law of Gravitation for matter.
• Why in coulomb law's formula r become r square
• Does the strength of a magnet follow the inverse square law?
• An electric charge can exist by itself. The strength of the electric field falls off as 1/r^2.

Magnetic poles always exist as pairs, a North and a South. This is called a "dipole", (two poles). The strength of a dipole falls off as 1/r^3.

The same thing happens with electric charges if you place a positive charge near a negative charge. That is an electric dipole, and its electric field strength also falls off as 1/r^3.
(1 vote)
• the distance is doubled, the toasts are doubled in width and height. how does that make up r^2 ?
(1 vote)
• When you increase the distance from 1 to 2, the area of toast increases from 1 to 4. If you go to a distance of 3 the area of toast increases to 9. What is the simplest algebraic expression you can come up with that lets you compute the area of toast if you are given the distance?

Area is proportional to distance squared.