The Inverse Square Law

Definition: As light moves away from a source it spreads out and its intensity is inversely proportional to the distance from the source.

Background

Light can be thought of as a collection of particles called photons. When a flashlight is turned on, it begins emitting photons. As these photons leave the flashlight, they grow more and more spread out. Since the same number of photons are emitted each second, if we imagine taking cross sections of the spreading beam of photons, we would always measure the same number of photons, no matter the distance.

As the area of the beam spreads out by a factor of \(d^2\) where \(d\) is the distance from the flashlight, we can say that the intensity must decrease by the same factor. If we write it out, we find the following:

\[\mathrm{Intensity} = \dfrac{\mathrm{\#\ of\ photons}}{\mathrm{Area}}\]

Since the # of photons is constant and Area is in the denominator, we can reason that

\[\mathrm{Intensity} \sim \dfrac{1}{d^2}\]

where \(d\) is the distance from the source. This makes intuitive sense, since all it is saying is that something appears dimmer as it moves farther away from you (\(d\) gets bigger the intensity shrinks) and that something appears brighter when it moves closer to you (\(d\) gets smaller and the intensity grows). In fact, it allows us to use numbers to describe how bright stars are relative to each other!

3D Visualization

The 3D visualization included below allows you to easily view the spreading out of light eminating from a source. Click the two arrows pointing left to hide the equation panel. Then click the orange star to emit a wave of light. Click it again to emit another!

Since the total energy of the light is conserved, the intensity must drop at each new distance from the center!

Questions

  • Is this relationship easier to show mathematically with another shape? What about a sphere?
  • What does this mean for stars?
    • If two stars emit the same amount of light but one is twice as far away, which one looks brighter?
    • By how much?

Takeaway

The intensity of a light emitting source drops off as \(1\) over the square of the distance to the source. This is very important in astronomy as once we know how bright something is and how far away it is, we can calculate how much light it is truly producing. This is the final step in unlocking the mysteries of stars!

Activity

Examine the 3D plot above. Assume the star emits 3600 photons at each click. This means the first blue square contains all 3600 photons. Calculate the intensity (number of photons in one box) of each successive distance from the star.