What Is the Inverse Square Law?

by Carson
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An illustration of the inverse square law

The inverse square law is the common phenomenon where the intensity of a propagating wave decreases according to the distance squared. Where does it apply? Why is it so valuable? How is this used in different physical equations? Let’s explore this in this article.

How Does the Inverse Square Law Work?

As mentioned before, the inverse square law means that the intensity of that wave dissipates proportional to the square of the distance. This actually makes sense if you consider a sphere around the object whose radius is r. Then, mark a patch of the sphere’s surface where it radiates from the center at a specific range of angles. After that, use the same range of angles, and increase the sphere’s radius to 2r. Does the area of the patch increase by a factor of four? If it’s hard to follow, don’t worry. We have an illustration for you below.

This is the essence of the inverse square law. The area where the wave projects to becomes larger, but the total energy put into the system remains unchanged. Therefore, the intensity at any given point must decrease in proportion to the increase in the surface area (i.e., the distance squared).

Where Does the Inverse Square Law Apply?

The inverse square law concerns the radiation of energy waves. This means there is a source that emits waves to the surrounding environment evenly in every direction. For example, if you drop an object into a pond, you can see ripples that are being spread on the water. The whole picture looks circular because the waves are uniform — they propagate at the same speed and intensity everywhere. Thus, this fulfills the criteria and obeys the inverse square law.

Other examples, perhaps more well-known, are light, sound, and gravity. We’ll discuss more about them in the following sections.

How Do We Use the Inverse Square Law?

Now that we’ve discussed the concepts, it’s time to use them in different physical formulas. Take the formula for gravitational force as an example:

\[F = {GmM \over r^2}\]

where F is the force, G is the gravitational constant, and m and M are the two masses. But we’ll be talking about the r^2 in the denominator, which denotes the distance between the two masses. This shows that the gravitational force dissipates according to the distance squared, and that’s how the inverse square law works for gravity.

As another example, if the intensity of light from a point source is the variable I from a unit distance away, the new intensity from any distance (d) can be denoted with this equation:

\[intensity = {I \over d^2}\]

Again, the denominator, d^2, is used according to the inverse square law. To conclude this section, if we want to use the inverse square law in an equation, we need to add a denominator of distance squared.

Conclusion

In this article, we’ve briefly explored what the inverse square law is. It works for any waves that dissipate from a point source and spread out proportional to the distance squared. This adds distance squared to the denominator. If we’ve missed anything important, or if you have any questions, please leave your suggestions in the comments below.

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