guide: inverse square law

The Inverse Square Law is defined as the intensity of a physical quantity (light, sound, etc.) being inversely proportional to the square of the distance from the source.

Simplified, this means that our lights don’t reduce linearly with distance, but exponentially.

But you knew the math behind this ever since you were a kid!

We can derive the Inverse Square Law from the formula for the surface area of a sphere.

To make this a little easier, let’s redefine “light” as “a certain number of photons” because that’s basically what watts is if you dive into the units.

So a sphere with radius of distance “r” from our point source would have a certain number of photons covering its surface area, let’s say the surface area is 4 for sake of ease.

If we double the radius of our sphere to “2r”, we don’t double the surface area, we quadruple it due to that squared portion. And we now have a sphere with surface area of 16.

The number of photons being ejected from our light source didn’t change, but we did just increase our surface area by a favor of 4.

So the same number of photons that covered a sphere with a surface area of 4 must also cover a sphere with a surface area of 16.

Which means, that any corresponding portion of the second sphere is receiving 1/4 the number of photons as the initial sphere!

This works whether we’re talking about the whole sphere as an idealized point source radiating into space, or we’re talking about a specific beam angle of that sphere, which would be more akin to our lights, because units cancel in ratios!

A 35° beam angle is really a ratio of 35/360, and since both sides of our ratio carry that same fraction, they’d cancel out, and we’d still be left with a ratio of 1 to 4!

So no matter how you want to calculate it, DOUBLING THE DISTANCE REDUCES INTENSITY BY 1/4.

Hope this helped you see the inverse square law in a new light!