Find how intensity changes with distance from a point source using the inverse square law: I2 = I1 x (r1/r2)^2. Enter an initial intensity and two distances to get the new intensity, the ratio, and the percent change.
Your Results
Calculated
New intensity (I2)
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Intensity at the new distance r2
Intensity ratio (I2 ÷ I1)
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Fraction of original intensity remaining
Distance ratio (r2 ÷ r1)
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How many times farther or closer
Change in intensity
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Percent increase or decrease from I1
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Enter an initial intensity and two distances, then press Calculate.
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Understanding the Inverse Square Law
The inverse square law describes how the intensity of a quantity spreading out from a small ("point") source falls off as you move away from it. It applies to light, sound, ionizing radiation, gravitational force, and the electric field of a point charge. The relationship between the intensity I1 at distance r1 and the intensity I2 at a different distance r2 is I2 = I1 × (r1/r2)².
Where the formula comes from
A point source radiates its output equally in all directions, so at any distance r the total energy is spread over the surface of an imaginary sphere of radius r. A sphere's surface area is 4πr², which grows with the square of the radius. Since intensity is power (or energy) per unit area, the same total output divided by a surface area that scales as r² means intensity scales as 1/r². Doubling the distance quarters the intensity; tripling the distance cuts it to one ninth.
Interpreting the results
New intensity (I2) is the calculated intensity at your new distance. Intensity ratio shows what fraction of the original intensity remains — 25% means only a quarter of the original intensity is left. Distance ratio shows how many times farther (or closer) the new distance is. Change in intensity expresses the same ratio as a percent gain or loss, which is often the easiest number to communicate.
Common applications
Photographers and lighting designers use it to predict how much a light source dims as a subject moves away. Audio engineers use it to estimate how sound pressure level falls off with distance from a speaker (roughly 6 dB per doubling of distance). Radiation safety relies on it to show that stepping back from a source sharply cuts exposure. Physics students apply the same 1/r² relationship to Newton's law of gravitation and to Coulomb's law for the force between electric charges.
Frequently Asked Questions
What is the inverse square law?
The inverse square law states that the intensity of a quantity spreading out from a point source — light, sound, radiation, gravity — is inversely proportional to the square of the distance from the source: I2 = I1 × (r1/r2)². Double the distance and intensity drops to one quarter; triple it and intensity drops to one ninth.
Why does intensity fall off as distance squared, not distance itself?
A point source radiates energy outward through the surface of an expanding sphere. That sphere's surface area grows with the square of its radius (4πr²), so the same total energy is spread over an area that grows as r². Intensity, which is power per unit area, shrinks as 1/r² as a result.
What physical quantities follow the inverse square law?
Light intensity from a point source, sound intensity, radiation exposure (gamma and X-ray), gravitational force between two masses, and the electric field from a point charge (Coulomb's law) all follow this relationship, provided the source is small compared to the distance and nothing absorbs or scatters the signal along the way.
Does the inverse square law apply at any distance or to any source?
It is exact only for an idealized point source radiating uniformly in every direction through empty space, with no absorption, reflection, or obstruction. Very close to an extended source, or inside an absorbing medium such as fog or water, the real falloff differs from the ideal 1/r² curve.