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inverse-square law

The inverse-square law is a principle that expresses the way radiant energy propagates through space. The rule states that the power intensity per unit area from a point source, if the rays strike the surface at a right angle, varies inversely according to the square of the distance from the source.

Imagine a 40- watt lamp at the center of a spherical enclosure. The total power striking the surface of the sphere is 40 watts, no matter what the size of the sphere. Some people find this counter-intuitive, but it becomes obvious when we consider that all the radiated power (no more and no less) from the source must strike an enclosure of any size that completely surrounds the source. Thus, the power striking the interior of a sphere 10 meters across is the same (40 watts in this hypothetical example) as the power striking the interior of a sphere 100 meters across, 100 kilometers across, or 100,000 kilometers across. The power per unit area, however, does depend on the size of the sphere.

The surface area A (in meters squared) of a sphere having radius r (in meters) is given by:

A = (4 pi) r 2

where pi is the ratio of a sphere's circumference to its diameter, and is approximately equal to 3.14159. If the radius of a sphere is multiplied by some factor n , then the surface area increases by a factor of n 2 . This decreases the power per unit area by a factor of n 2 . Another way of saying this is that the power per unit area becomes n -2 times as great.

Increasing the diameter of a sphere from 10 to 100 meters makes it 10 times as large in diameter, and gives it 100 times the surface area. This cuts the light power per unit area from a lamp at the sphere's center by a factor of 100. The power landing on, say, one square centimeter of the sphere's interior becomes 1/100, or 0.01 times, as great, if the sphere's diameter grows from 10 to 100 meters. The law applies only as long as the point source is at the center of the sphere, so the rays from the source strike the sphere's surface at right angles.

Also see meter , square meter , and International System of Units ( SI ).

This was last updated in September 2005

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