Greenwood press

46
INVERSE FUNCTION

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INVERSE SQUARE FUNCTION
The inverse square law explains why sound drops off so quickly as you move
away from a source of noise, why porch lights do a good job of illuminating the
front of a house but not the street in front of the house, and why the forest reverts
to darkness as you move away from a campfire. Light and sound emerging from
single sources can be viewed as increasing spheres whose area increases as the
square of the distance from the source. As a result, the proportion of sound or
light reaching a specific unit of area, such as a square meter, varies inversely as
the square of the distance from the source. The standard equation of an inverse
square function is
y =
k
x
2
, where 
k is a constant of proportionality.
Light emerges from a source in all directions. The drawing below illustrates
the distribution of light from a 40-watt light bulb. We will use wattage as the
measure of pointance and illuminance to show how the inverse square works.
Imagine a sphere of radius 
r containing the light bulb. Forty watts fall on the
interior surface of the sphere. The surface area of the sphere is given by 
SA =
4πr
2
square meters. The energy falling on 1 square meter is therefore 40 watts
divided by 
4πr
2
. In the drawing, this energy is called “L.” If you go out twice as
far, the same energy is distributed over 4 square meters. At three times the dis-
tance, the energy is distributed over 9 square meters. 
A standard formula for light intensity is 
E =
I
r
2
, where 
E is called the illu-
minance, and 
I is called the pointance. Illuminance is measured in a variety of
units such as lux and footcandles. Pointance is a measure of the intensity at the
source of the light. As you can see, this is a direct statement of the inverse square
law.
The inverse square law provides information about the likelihood of other
planets having life forms similar to those on earth. Imagine the light that hits your