Inverse square law

Radiation from a small source spreads over the surface of an expanding sphere. The area of that sphere goes as the square of the radius, so the intensity at any point falls as 1/d^2. Written as a ratio between two positions, I1 x d1^2 = I2 x d2^2. This single relationship drives both halves of a radiographer's day.

For exposure planning the film needs a fixed quantity of radiation, so if the intensity drops the time must rise to compensate: t2 = t1 x (d2/d1)^2. Doubling the source-to-film distance to improve geometric unsharpness therefore quadruples the exposure. This is the trade that decides whether a shot is a two-minute job or a twenty-minute one.

For radiation safety the same law works the other way: D2 = D1 x (d1/d2)^2. Distance is the cheapest and most reliable form of protection there is, because it costs nothing and cannot be left in the van. Moving from 1 m to 10 m cuts the dose rate by a factor of 100.

The law assumes a point source in air with no scatter and no attenuation. It is accurate for gamma projectors at working distances, slightly optimistic in a confined bay where scatter adds back, and it breaks down close to a physically large source or behind heavy shielding where build-up dominates.

I1 x d1^2 = I2 x d2^2
Exposure time:  t2 = t1 x (d2/d1)^2
Dose rate:      D2 = D1 x (d1/d2)^2
Distance for a target rate:  d2 = d1 x sqrt(D1/D2)