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Inverse Square Law Calculator

Find intensity at a new distance.

How it is calculated

For a point source spreading in all directions, such as light or radio waves in open space, the energy is spread over a sphere whose area grows with the square of the distance. So the intensity falls as I2 = I1 × (d1 ÷ d2)². I1 is the known intensity at distance d1, and I2 is the intensity at the new distance d2. Both distances must be above zero, and I1 cannot be negative. The distances can use any unit as long as both are the same, and the answer has the same unit as I1. Absorption and reflection are ignored.

Worked example

A lamp gives 100 (any unit, for example lux) at 2 m. What is it at 5 m? Ratio of distances: 2 ÷ 5 = 0.4. Square it: 0.4² = 0.16. Intensity: 100 × 0.16 = 16.0000. Moving 2.5 times farther away leaves only 16% of the intensity.

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Common mistakes

  • Using the simple ratio d1 ÷ d2 without squaring, which wrongly gives 40 instead of 16.
  • Putting d1 and d2 the wrong way round. The measured intensity goes with d1; the new distance is d2.
  • Applying it to a line source (a tube light seen from close by) or a narrow beam, where intensity does not fall as the square.
  • Mixing units, such as d1 in centimetres and d2 in metres.
  • Measuring distance from the casing of a lamp instead of from the light source itself at close range.

About this calculator

This calculator applies the inverse-square law, I₂ = I₁·(d₁/d₂)², which describes how the intensity of light, sound, radiation, or gravity spreads out and weakens as the same total energy is distributed over the surface of an ever-larger sphere as distance increases — doubling the distance quarters the intensity, tripling it cuts intensity to one-ninth.

Photographers and lighting technicians positioning a flash or studio light, audio engineers setting speaker placement, radiation safety officers calculating dose at a distance from a source, and physics students verifying the law experimentally all use this. Enter a known intensity at one distance and a new distance to find the intensity there.

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