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.

Worked example

Known distance d1	600 mm
New distance d2	900 mm
Known exposure time at d1	2 min
Known dose rate at d1	400 uSv/h
Exposure time at d2	4.5 min
Dose rate at d2	177.78 uSv/h
Exposure time factor	2.25

Moving from 600 mm to 900 mm: (900/600)^2 = 1.5^2 = 2.25, so t2 = 2 min x 2.25 = 4.50 min. The dose rate falls by the reciprocal: r2 = 400 x (600/900)^2 = 400 x 0.4444 = 177.78 uSv/h.

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