// HACKER NEWS — CYBERSECURITY
Inverse-Square Law
If you’re dabbling in electronics, you might have heard of what’s known as the inverse-square law. This law says that the intensity of many types of physical phenomena decreases with the square of the distance to the origin. For example, if you get twice as far from a lightbulb, a handheld light meter will register a four-fold drop in luminance. The same goes for sound, radio transmissions, and so forth.
But… why? Wouldn’t it be more natural if the intensity dropped in direct proportion to how far you are from the source? Or, if it’s about scattering in a three-dimensional space, shouldn’t it be the cube of distance? Let’s try to figure it out.
If we want an answer that doesn’t merely kick the can down the road, we need to start with the formula for the surface area of a sphere. From school, you might recall that the equation is:
That said, I bet your teachers have never explained where the formula comes from. And as with many other “obvious” concepts in elementary mathematics, the answer is not as obvious as it seems — even though it’s been known since the times of Ancient Greeks.
To get going, imagine a thin tube (a cap-less cylinder) constructed by gluing the edges of a rectangular sheet of paper:
To make the cylinder, we didn’t need to squish or stretch the rectangle, so the process preserved the surface area. The outer surface of this cap-less cylinder is just height (h) multiplied by what used to be width (w) and is now circumference.
Next, take a sphere with a radius r and place it inside such a fitted cylinder of the same circumference and the same height:
The sphere has a radius of r, so its circumference — and thus, the circumference of the cylinder — is 2𝜋r. It’s just a matter of how 𝜋 is commonly defined.
The height of the cylinder is equal to the sphere’s diameter, or twice the radius. We know that the cylinder’s outer surface area is just circumference times height, so we can write that Stube = 4𝜋r2. Lo and behold — that’s the same formula as what we’ve been given in school for the inscribed sphere. But… how come?
To get to the bottom of this, imagine tiling tiling flat squares on the outer surface of the cylinder, each square measuring a × a: