When things spread out in three dimensions from their source (like light, sound, or shrapnel from a cluster bomb) intensity decreases proportionally to the inverse of the distance squared:

$LaTeX: \text{Intensity}\propto\frac{1}{r^2}=r^{-2}.$

Why is that? When things come from a point source, they spread out with spherical symmetry, so they are spread out over the surface of a sphere that gets bigger as it expands in time. Because the surface area of a sphere = $LaTeX: 4\pi r^2,$ the area is proportional to the square of the radius. So, if you double your distance to a lightbulb, the light from the bulb is spread out over 4 times the surface area, and the intensity of light at the new location is 1/4 as great as before:

If $LaTeX: R\Rightarrow2R_i,\hspace{2mm}I\Rightarrow\frac{1}{4}I_i.$

This may be better seen from the diagram below, where if light travels through a window of a sphere at distance LaTeX: R_i, the same light would need 4 windows to pass through a sphere at distance LaTeX: 2R_i: 2 windows wide, and 2 windows high.

Concentric spheres showing the change in area as the radius changes.

Figure 5.1.1: Light propagates equally in all directions from a point source in the middle. The flux of power that escapes through the smaller sphere must also pass through the larger sphere. Because the same flux is spread over a larger surface at the larger sphere, the intensity of light is weaker at the larger distance.

In the next section, we describe how Cavendish measured the relationship between mass, distance and gravitational attraction. However, even before he experimentally determined this relationship, one could imagine the gravitational force (from a planet) spreading outward into space in all directions. Thus, we might have expected the gravitational force to drop off according to the inverse square relationship.

Exercise 5.1.1

The distance between the centers of the earth and moon is about 385,000 km and the radius of the earth is about 6,400 km.

How many earth radii is the distance between the center of the earth and moon?
If the moon were to stop moving, estimate the moon’s acceleration toward the earth. It may help to know that the acceleration from gravity one earth radius away from the earth’s center is 10 m/s².
But the moon is moving! In this case, should its acceleration be the same as in "A" above? Why or why not?
From your answers above, estimate the speed of the moon in its orbit around the earth.
From your answer above, estimate the period of the moon… is it close to a month?

Exercise 5.1.2

In 1930, it was discovered that a beta decay: $LaTeX: n \Rightarrow p^+ + e^-$ didn’t conserve energy, momentum or angular momentum. Wolfgang Pauli imagined the existence of a new particle, the neutrino, with the correct angular momentum to conserve angular momentum in this process. We now estimate that 65 billion neutrinos from the sun pass through each square centimeter on earth, per second. The earth is about 150 billion meters from the sun and Venus is about 108 billion meters, or about 0.72 (or $LaTeX: \sim\frac{1}{\sqrt{2}}$ ) the earth-sun distance.

How fast does the sun produce neutrinos?
How many pass through you during a one hour class?
How about if you were on Venus?

It’s interesting to note that neutrinos interact with matter very very weakly, making them very difficult to detect. In fact, the flux of solar neutrinos is negligibly attenuated by the earth, so the flux coming out of the earth at night is about the same as that entering the earth on the other side facing the sun. This was astutely noted by John Updike in 1960:

Neutrinos they are very small.
They have no charge and have no mass
And do not interact at all.
The earth is just a silly ball
To them, through which they simply pass,
Like dustmaids down a drafty hall
Or photons through a sheet of glass.
They snub the most exquisite gas,
Ignore the most substantial wall,
Cold-shoulder steel and sounding brass,
Insult the stallion in his stall,
And, scorning barriers of class,
Infiltrate you and me! Like tall
And painless guillotines, they fall
Down through our heads into the grass.
At night, they enter at Nepal
And pierce the lover and his lass
From underneath the bed – you call
It wonderful; I call it crass.

The poem, Cosmic Gall was published before the 1998 discovery that neutrinos actually do have mass.