Resource: Inverse Square Law

Print Background Essay

Have you ever noticed that light from a flashlight seems much brighter when it shines on something nearby than when focused on something far away? Maybe you’ve also noticed that when you use spray paint the thickness of the coat is related to how close you hold the can to the wall. These effects are due to a physical property known as the Inverse Square Law, which states that the strength of a given physical quantity is inversely proportional to the distance from the source squared or I → 1/d² (I=Intensity). Inversely proportional means that as distance increases, intensity decreases.

The most well-known story about the Inverse Square Law comes from one of the most famous stories in all of science—Sir Isaac Newton and the apple. According to the story, Newton was trying to understand the planetary motion of the heavens when an apple fell from a tree, hit him on the head, and let loose a stream of ideas that would change physics forever. Newton wondered how the gravity that caused the apple to fall was related to the gravity that acted on much more distant objects like the Moon. He made some calculations and found that the Moon was falling toward Earth with an acceleration that was 3,600 times smaller than the apple’s acceleration. He knew the Moon was about 60 times as far from Earth's center as the falling apple was. Newton realized the inverse square of the distance was correlated to the change in the effects of gravity, 1/60² = 1/3600. Newton also saw that because he was squaring the inverse distance, the relationship was non-linear. For example, a satellite half as far away as the Moon experiences gravity at 1/30² = 1/900, not 1/1800. Note that the distance is one-half the distance to the Moon, yet the intensity is four times as strong as the intensity at the Moon—not twice as strong as you might expect.

The Inverse Square Law applies to anything that radiates in all directions. Gravity obviously follows this law because our spherical planet tugs on objects all across Earth's surface. The light intensity from a bulb, the radiation escaping a brick of Uranium, the force on an electron in an electric field, and the screeching sounds of a passing ambulance are all under the command of the Inverse Square Law. From a far distance these things seem to be weak—a faint light on your face or the subtle sounds of the sirens. But as you get closer to the emanating source, the effect grows larger, and the closer you get, the more rapidly the effect increases.

Print Questions for Discussion

• We've seen from the video that the Inverse Square Law applies to light. To what other types of energy can the Inverse Square Law be applied?

Answer

Gravity, sound, electric force, and radiation because they can radiate in all directions.

• Describe in your own words what is meant by, “The intensity falls in proportion to the inverse of the distance squared.”

Answer

The intensity of a source of energy is dependent on the distance from the source squared, and since it is diminishing with distance it must be dependent on the inverse of this number so that when the distance increases, the intensity decreases.

• Does a laser beam diminish in intensity according to the Inverse Square Law? Why or why not?

Answer

No. A laser beam of light does not spread vertically or horizontally, so there is no diminishing of its intensity with distance.

• Would you see a larger difference in light intensity if you

  a. moved from 1 meter away to 2 meters away or

  b. moved from 10 meters away to 11 meters away?

Answer

The answer is "a." The intensity would go from I/1² = I to I/2² = I/4, or a change of 400%. In "b" the intensity falls from I/10² = I/100 to I/11² = I/121, or a change of 21%.

• If you can hear a single speaker comfortably from 1 meter away, how many speakers would you need to listen comfortably to the same music from 100 meters away?

Answer

Approximately 10,000 speakers. If the initial intensity was I/1² = I, and the new intensity is I/100² = I/10,000, you’d need (10,000 speakers)*(I/10,000) = I to have the same intensity as you originally had with 1 speaker at 1 meter.

• According to the Inverse Square Law, the effects of gravity (and thus an object’s weight) are affected by the distance from the center of the Earth. Using this information, would a company make more money if they sold apples by the pound in a city of higher or lower elevation?

Answer

Lower elevation. At a lower elevation the distance from the center of the earth is smaller so the intensity of gravity would be greater, and thus fewer apples would be needed for a pound.

Print Teaching Tips

Have you ever noticed that a campfire illuminates the faces of all who sit around it quite well, but when you go off to your tent for more marshmallows the light dwindles down to almost nothing? You've probably observed the same with sound—your headphones seem to be loud enough when up against your ears, yet you can only barely hear them when you rest them against your neck. This is all due to an effect known as the Inverse Square Law, and it applies to many natural phenomenon including light, sound, radiation, and gravitation.

The Inverse Square Law can be explained in several ways, but it is probably best to take a step-by-step approach with something simple like a light bulb. You can try this with your class.

Cover a light source with some opaque material that lets no light escape. Then, put a small pinhole in the material to let out a one-dimensional beam of light. Now, whether you put your hand an inch from the light source or a few yards away, the light intensity in your palm from the small beam escaping from the pinhole is about the same.

Add a second dimension by cutting a slit in the material so that a fan of light escapes. The light now has another dimension to spread out into. If you put your hand a few inches from the source and slowly back away, you'll see that the intensity changes quite a bit. While you will see the entire slit on your hand when it is very close to the source, at a large distance you will see that your hand only covers a small part of the arc of light and therefore only receives a small part of the energy. The light's intensity is inversely proportional to the distance, I = 1/d or I = I₀/d.

Finally, if you cut a square from the material to allow a third dimension, you'll see even more spread. Now the light, and thus its energy, can spread in two directions. Again, if you place your hand near the cutout you will see the full bright square in your palm, but as you move away, the intensity drops quickly and the square no longer fits in your hand. Since it is spreading in two directions and s throughout a square area, the intensity falls in inverse proportion to the distance squared, I = 1/d² or I = I₀/d².