Skip to content
Omnikalc

Inverse square law: how intensity falls with distance

Calculate how the intensity of light, radiation or sound from a point source changes with distance, and the distance needed to reach a target intensity.

Updated on
Your inputs

Value out of range

Value out of range

Value out of range

Value out of range

Results

Intensity at the new distance

250

Share of the reference intensity

25.0%

Twice the distance gives a quarter of the intensity

Distance for the target intensity

9.49

Change in level
-6.02 dB
About −6 dB each time the distance doubles
Distance ratio (new ÷ reference)
2.00 ×

Why things fade with distance

A light bulb, a campfire or a small radioactive source sends energy out in every direction. At a distance d, that energy is spread over the surface of a sphere of area 4πd². Double the distance, and the same energy covers four times the area. The intensity, the energy per unit area, therefore falls with the square of the distance.

The formula

I₂ = I₁ × (d₁ ÷ d₂)²

  • I₁ is the intensity measured at distance d₁.
  • I₂ is the intensity at the new distance d₂.

To find the distance that gives a target intensity: d = d₁ × √(I₁ ÷ I_target).

In decibels, the change is 10 × log₁₀(I₂ ÷ I₁), which is about −6 dB every time the distance doubles.

Example

A lamp gives 1,000 lux at 3 feet:

  • At 6 feet: 250 lux, a quarter of the light
  • Change: −6.02 dB
  • To get 100 lux, move to about 9.5 feet

The intensity can be in any unit: lux or foot-candles for light, microsieverts per hour for a dose rate, watts per square meter for radiant heat. The calculator keeps the unit you enter.

Everyday uses

  • Photography: moving a flash or a light twice as far from the subject needs two more stops of exposure. Studio photographers use this to control how fast the light falls off across a scene.
  • Grow lights: plants receive four times more light at half the distance, so the height of the lamp matters as much as its power.
  • Radiation protection: distance is one of the three basic protections, with time and shielding. Radiographers and nuclear medicine staff step back from sources for this reason.
  • Sound: outdoors, the sound level from a small source drops about 6 dB per doubling of distance.

When the law breaks down

  • Near large sources: close to a long fluorescent tube or a wide panel, intensity falls more slowly. A rule of thumb is to stay at least five times the size of the source away before using the law.
  • Beams: lasers, spotlights and car headlights concentrate their output, so the intensity falls much more slowly.
  • Reflections: inside a room, walls and ceilings bounce light and sound back, so levels drop less than predicted.
  • Absorption: over long distances, air, fog or water absorb part of the energy, so levels drop faster.

Frequently asked questions

What is the inverse square law?

For a small source radiating equally in all directions, the energy spreads over a sphere whose area grows with the square of the radius. The intensity therefore falls as 1 ÷ distance²: twice as far gives a quarter of the intensity, three times as far a ninth.

Where does it apply?

To light from a bulb, radiation from a small radioactive source, sound from a point source in open air and gravity or electric fields. It applies once you are far enough away that the source looks like a point, usually at least five times its largest dimension.

When does it not apply?

Close to large sources such as a panel light or a long fluorescent tube, in a room where walls reflect sound or light, with a focused beam such as a laser or spotlight, and when the air absorbs the energy over long distances.

How is it used for radiation safety?

Distance is one of the three basic protections, with time and shielding. Stepping back from 1 to 3 meters from a small source cuts the dose rate to about a ninth. Always follow the radiation protection rules of your workplace.

Was this calculator helpful?

Thanks for your feedback!

Results are estimates provided for information purposes. They do not replace professional advice.

Related calculators