This works for all waves but it’s convenient to think about EM waves. Imagine a point source of EM wave energy, such as a candle, a light bulb or even the Sun, radiating energy in all directions. We know that a sphere has surface area = 4πr². If we move twice as far away, the energy is then smeared out over an area FOUR times the size, reducing the intensity by a factor of FOUR. Thus, intensity is inversely proportional to the square of the radius.
Here’s a problem. Imagine a 40W light bulb, which we can approximate to a point source. How far away would we have to be for the intensity (in watts per square metre) to drop to a) 5mW/m² b) 0.05mW/m²?
a) 40W/0.005W = area of the sphere = 8000m² = 4πr², hence r=25.23m
b) 40W/0.00005W = area of sphere = 800,000m², thus r = 252.3m
If I could measure the intensity at a distance of 100km, what would it be?
r = 100,000m, I = 40/4π.(100,000)² = 0.32nW/m². Is this measurable?