The Inverse Square Law

To find the gravitational attraction between two objects, we use the equation:

where m₁ and m₂ are point masses, G is the gravitational constant, and r is the distance between the two masses.

To find the electric attraction or repulsion between two objects (the coulomb force), we use the equation:

where q₁ and q₂ are point charges, k_c is the Coulomb force consant, and r is the distance between the two charges.

To find the intensity of light a certain distance from a light source, we use the equation:

where P is the power of the light source (lightbulb, etc.) and r is the distance from the source. The same equation could be used for sound intensity.

All of the equations above have an r² term in the denominator, why should that be?

It turns out that any point (or point like) source which spreads its influence in all directions without limit to it's range obeys the inverse square law (Hyperphysics). The reason is a force, light, sound, etc., spreads out like a growing sphere from a point source, effectively becoming diluted in space. Take a look at the image below:

In the example above, the light spreads out in all directions. Notice the further away you get from the point source, the more area the light has to cover. In general, a constant r will produce sphere centered at the point source. That means for a given r, each point on the surface of a sphere of radius r has the same intensity. If we divide the initial power of the point source by the sphere's surface area, it tells us how much that power has been "diluted" by spreading out through space. The surface area of a sphere is:

So we divide our power by 1/4πr² and we get:

The r² is purely a geometric consequence of the initial power being diluted by space.

Taking another look at the Coulomb Force:

Noting that the coulomb constant k_c is equal to 1/4πε₀ where ε₀ is called the electric permeability of space, we can easily see the 1/4πr² term. The point source in this case is the point charge, which produces an electric field. In the gravity equation:

The 1/4π appears to be missing at first glance, but it's in fact hidden in the gravitational constant G. After 300 years, don't expect to see the 1/4π pulled out of G any time soon, but rest assured it's there. The point source is the mass which produces a gravitational field.

Fun With The Inverse Square Law:

I've always been fascinated by the idea that the Sun appears smaller when seen from other planets like Mars or Saturn and bigger from Venus and Mercury. From Uranus and beyond, the Sun looks like just another star. We can calculate the intensity of the Sun on other planets as compared to Earth by using the inverse square law:

Mercury- .39 AU - 6.57x (Sun's Intensity on Earth)

Venus - .72 AU - 1.9x

Earth - 1 AU - 1

Mars - 1.5 AU - ~1/2

Jupiter - 5.2 AU - ~1/25th

Saturn - 9.5 AU - ~1/100th

Uranus - 19.2 AU - 1/300th

Neptune - 30 AU - ~1/1000th

Pluto - 39.5 AU - ~1/2000th

Sedna (at it's farthest) - 975 AU - ~one millionth

It must be cold and dark once you get past Saturn.

Here is an image of the Sun taken from the Mars Rover on Mars, notice how much smaller it looks than on Earth:

Thanks to Wikipedia for the equations and Nasa for the image above.