Inverse square law calculator
Enter an intensity measured at one distance and a new distance to see how strong it is there. Or enter the intensity you want and find how far away it is reached. It works with any unit (lux, W/m², µSv/h, counts per second) as long as both distances share one unit.
Runs in your browser: nothing you enter is sent anywhere.
How to use it
- Choose what to find: the intensity at a new distance or the distance for a target intensity.
- Enter the known intensity I₁ and the distance d₁ where it was measured.
- Enter the new distance d₂, or the target intensity I₂.
- New intensity: I₂ = I₁ × (d₁ ÷ d₂)².
- Distance for a target: d₂ = d₁ × √(I₁ ÷ I₂).
- Level change: ΔL = 10 × log₁₀(I₂ ÷ I₁) dB.
Worked example
A lamp gives 1,000 lux at 0.5 m. At 2 m it gives 1000 × (0.5 ÷ 2)² = 1000 × 0.0625 = 62.5 lux, 6.25 % of the original, a change of −12.0412 dB.
A radiation reading of 100 µSv/h at 1 m falls to 4 µSv/h at d₂ = 1 × √(100 ÷ 4) = 5 m.
Tips
- Measure from the source. Distances are from the centre of the source, not from the front of a housing.
- Photography flash. Moving a flash from 1 m to 2 m costs two stops of light, because a quarter of the light is two halvings.
- Radiation safety. Distance is one of the cheapest forms of protection: stepping from 1 m to 3 m cuts the dose rate from a point source to a ninth.
Frequently asked questions
What is the inverse square law?
The energy from a point source spreads over a sphere whose area grows as the square of the distance, so the intensity falls as 1 ÷ d². Twice as far gives a quarter of the intensity, three times as far a ninth.
Why is doubling the distance −6 dB?
The level change is 10 × log₁₀(I₂/I₁). A quarter of the intensity gives 10 × log₁₀(0.25) = −6.02 dB. That is the familiar rule that sound pressure level drops about 6 dB per doubling of distance from a small source in open space.
When does the law not hold?
Close to a large source (a long tube light, a big speaker array, a wide panel), with lenses or reflectors that focus the beam, indoors where walls reflect sound, and where the medium absorbs energy, such as fog or shielding. It works best far from a small source in open space.
Which units can I use?
Any. The ratio does not care about units, so the answer comes out in the unit of I₁. The two distances must use the same unit, metres with metres or feet with feet.