Greenwood press

INVERSE SQUARE FUNCTION
47
Graph of the inverse square function
y =
1
x
2
.
The distribution of light from a
light bulb according to the
inverse square law.

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neighborhood at noon on a hot summer day. Call that amount of light 
L. Mars is
about one-and-a-half times the distance from the sun as the earth, so if your 
neighborhood were on Mars, it would receive 
L
1.5
2
or 44 percent of the light it 
receives on earth. That might be enough to sustain life. The planet Neptune is
about thirty times as far from sun as the earth. If your neighborhood were on 
Neptune, it would receive 
L
30
2
, or about 0.1 percent of the light it receives on 
earth. This wouldn’t be enough to support life as we know it.
The distribution of sound follows the same rule. Just replace the light bulb in
the first illustration by an actor in an auditorium. In an auditorium, the sound
drop due to the inverse square law is usually unacceptable. It would mean that a
person standing in front of the first row of seats, who might be audible to people
in the tenth row, would be barely audible to people in the twentieth row. The
audibility of the speaker (about 70 decibels) to listeners in the first row would
drop to 50 decibels (a soft sound) ten rows behind. Acoustical engineers design
reverberation into auditoriums to focus the sound and overcome the inverse
square law. They place hard surfaces at the back of the stage and on the ceiling
and walls so that sound that would ordinarily dissipate would bounce back and
add to the intensity of that being heard by the audience. 
Gravity is an example of a force that follows the inverse square law. How
much lighter will a 160-pound astronaut feel if he or she is in a spaceship 12,000
miles above the earth? The radius of the earth is about 4,000 miles, so the astro-
naut is 16,000 miles from the center of the earth, or about four times the distance
of a person measuring weight on the surface of the earth. By the inverse square 
law, the astronaut would feel as though his or her weight were 
160
4
2
= 10 pounds,
even though the mass of the astronaut remains unchanged. 
Electric force acting on a point charge, 
q
1
, in the presence of another point 
charge, 
q
2
, is given by Coulomb’s law,
F =
kq
1
q
2
r
2
=
q
1
q
2
4πǫ
0
r
2
, where 
ǫ
0
is the con-
stant for the permittivity of free space. This law is an outcome of the inverse
square law. It is named in honor of the French scientist Charles Coulomb, who
established it in 1777 after studying the forces on magnetized needles. 
The inverse square law means that increasing the distance from a source of
nuclear radiation may be the difference between life and death. Accidental expo-
sure to radiation that produces 600 rems (a measure of radiation impact on living
tissue) is almost certain to cause death within two months. A person who is twice
as far away will absorb 600/4 = 125 rems, an amount that will result in a signifi-
cant, but temporary, reduction in blood platelets and white blood cells. If the radi-
ation distribution followed an inverse law, rather than an inverse square law, then
a person twice as far away as the one receiving the fatal dose would get 600/2 =
300 rems. This dose causes severe blood damage, nausea, hair loss, hemorrhage,
and death in many cases. Because radiation follows the inverse square law, being
twice the distance from a fatal dose may mean illness rather than death.
The inverse square law comes up in court cases. The lawyer faced the med-
ical examiner and asked suddenly, “The body wasn’t found in the bedroom. How
can you say that the fatal shots were made there?” The examiner replied, “Be-