The Inverse Square Law

Or why nearby street lights are bright and distant streetlights are dim

Graphic of the inverse square law from Wikimedia Commons. Image by “Borb”

The Inverse Square Law is the mathematical description for how the brightness or intensity of radiated energy (such as light) varies with distance. In other words, it describes, mathematically, why and how lights appear brighter when you are close and dimmer when you are far away. This is simple enough. We intuitively use the concept to gauge distances (at least with familiar light sources such as car headlights).

Because this precise mathematical description exists, comparing the known brightness of a light (such as a light bulb) to its measured brightness at a known distance allows you to calculate the distance.

Say you are sitting next to a roaring campfire. Most of the heat you feel is due to infrared radiation, a form of electromagnetic radiation like light. If you are uncomfortable, you move a few feet away and the temperature, which is a measure of the amount of infrared radiation you are receiving, drops considerably. The amount that the radiation drops can be precisely calculated with the inverse square law.

Now let’s say that you are sitting 2 feet from a lamp. With a light meter (photometer) you measure the light intensity at a level 8. (You don’t need to know what the units are here, just that it is at a level 8). Now you move twice as far away, to 4 feet, and re-measure the intensity. If this were a simple “inverse” relationship (without the “square” part), then you would expect the light meter to measure 4, because since you are twice as far away, “one over two” is 1/2 or one-half. However, light followed the “inverse square” law, so instead of “one over two” you have “one over two squared.” Two squared is two times two, or 4. So you have 1/4, or one fourth the original intensity. Thus, the new intensity is 2.

With this precise mathematical knowledge, we can work backwards to figure out distances. Suppose that we know that at 10 feet a bright lamp gives and intensity on the light meter of, say, 16. Then we move to some distance where the light meter reads 4. Since we know the inverse square law, we know that 4 is 1/4 of 16, so that means that we must be twice as far away. Thus we are 20 feet from the lamp. We can use the same method to determine the distances to some stars.

The inverse square law works with radiated electromagnetic energy whether it be visible light, infrared, radio waves or even high powered x-rays and gamma rays. It even applies to gravity.

(I should note that this applies to unfocused or omni-directional light. That is, light that is radiated equally in all directions, say from a star or a round light bulb. Light that is focused or “collimated,” such that light rays are sent out in parallel beams, do not follow the inverse square law. This would include projectors, flashlights and lasers. However, since there really is no such thing as a perfectly collimated light source, even lasers spread out a bit. This can be described mathematically, just not with the common inverse square relationship.)

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