Inverse-Square Law Calculator
Work out how light, sound and radiation weaken with distance. Solve I₂ = I₁(d₁/d₂)² for the intensity or the distance, see the decibel drop for sound, and add half-value-layer shielding for radiation.
How to Use
- Pick a mode: the Intensity at a new distance, the Distance for a target intensity, Sound level in decibels, Shielding with half-value layers, or Distance + shielding together.
- Enter the intensity you know and its unit: lux or foot-candles for light, W/m² for power, µSv/h for dose rate, or counts per minute.
- Enter the distance where it was measured and the new distance, each with its own unit.
- For shielding, pick a material (typical half-value layers are offered) or type your own half-value layer, then the thickness, or the target, to get the other.
- Read the readouts and the drawing, and check Show Work for each factor. Radiation results are educational estimates only.
Worked Example
A lamp moved back. A lamp gives 500 lux at 1 m. At 2 m, I₂ = 500 × (1 ÷ 2)² = 500 × 0.25 = 125 lux (11.61 foot-candles), a drop of 6.02 dB.
Distance and shielding. A gamma source reads 100 µSv/h at 1 m. Stepping back to 2 m multiplies it by (1 ÷ 2)² = 0.25; adding 1.3 cm of lead, two half-value layers for caesium-137 at a typical 0.65 cm each, multiplies it by 0.5² = 0.25 again. The result is 100 × 0.25 × 0.25 = 6.25 µSv/h, sixteen times lower.
The common mistake: halving instead of quartering. Doubling the distance from the lamp does not halve the light to 250 lux; it spreads the same light over four times the area, giving 125 lux. The distance ratio must be squared. Decibels are the reverse trap: a sound level falls by 20 × log₁₀ of the distance ratio (6.02 dB per doubling), not 10 ×, because the decibel already turns the square into a factor of 2.
Show Work
Formulas
One Law for Light, Gravity and Radiation
Johannes Kepler stated in 1604, in his book on optics Ad Vitellionem Paralipomena, that the strength of light falls off with the square of the distance from its source, reasoning that the same light is spread over ever larger spheres. Ismaël Bullialdus suggested in 1645 that the Sun’s pull on the planets might fall off the same way, and Isaac Newton made that the law of gravity in 1687; Charles-Augustin de Coulomb measured the same law for electric charges in 1785.
The units follow the same geometry. One lux is one lumen spread over a square metre, and one foot-candle is one lumen over a square foot, so 1 foot-candle = 1 ÷ 0.09290304 = 10.764 lux exactly. In radiation protection the inverse-square law joins time and shielding as the three basic ways to cut a dose, and the half-value layer, the thickness that halves a beam, became the standard way to compare shielding materials early in the X-ray era.
The law is exact only for a point source in empty space. Close to a large lamp or source, in a reflective room, or behind a shield that scatters radiation, the real values differ, which is why the radiation results here are for learning, not for safety decisions.
About This Tool
This calculator applies the inverse-square law to whatever you measure: illuminance in lux or foot-candles, power flux in W/m² or mW/cm², a radiation dose rate in µSv/h or a count rate. It finds the intensity at a new distance or the distance for a target intensity, gives sound levels in decibels with the 20 log rule, and handles shielding with half-value layers, alone or together with distance. Each answer shows the factors involved, so you can see how much comes from distance and how much from the shield.
The shielding values offered are typical narrow-beam textbook figures; real half-value layers depend on the beam, geometry and scattered radiation. This is an educational tool, not radiation-safety advice. Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Decibel Calculator, Black-Body Radiation Calculator, and Coulomb’s Law & Electric Field Calculator.
Frequently Asked Questions
What is the inverse-square law?
Anything that spreads out evenly from a point, such as light, sound or radiation, is shared over a sphere whose area grows with the square of the distance, so I₂ = I₁ × (d₁ ÷ d₂)². A lamp giving 500 lux at 1 m gives 125 lux at 2 m and 55.56 lux at 3 m.
Why does sound drop 6 dB each time the distance doubles?
Sound intensity follows the inverse-square law, and in decibels a factor of 4 in intensity is 10 × log₁₀ 4 = 20 × log₁₀ 2 = 6.02 dB. So L₂ = L₁ − 20 × log₁₀(d₂ ÷ d₁): a speaker giving 100 dB at 1 m gives 80 dB at 10 m in the open. Indoors, reflections make it fall more slowly.
How far away should a light be for a set brightness?
Rearrange to d₂ = d₁ × √(I₁ ÷ I₂). A lamp that gives 800 lux at 0.5 m gives 300 lux at 0.5 × √(800 ÷ 300) = 0.8165 m.
What is a half-value layer?
The thickness of a material that halves a beam of radiation: I = I₀ × (½)^(x ÷ HVL). For the 662 keV gamma rays of caesium-137 a typical value for lead is about 0.65 cm, so 1.3 cm of lead lets through 25% and a tenth-value layer is 2.159 cm. Real shielding also scatters radiation, so actual values are higher.
Why is distance such good radiation protection?
Because it costs nothing and works as the square. A source giving 100 µSv/h at 1 m gives 11.11 µSv/h at 3 m. Moving from 1 m to 2 m and adding 1.3 cm of lead cuts it 16 times, to 6.25 µSv/h. Time, distance and shielding are the three basic rules; this page is a learning aid, not safety advice.
How do I use the Inverse-Square Law Calculator?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Is it free? Does it work without internet?
Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.
Where does my data go?
Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.
Common Use Cases
Photography and lighting
Moving a light from 1 m to 2 m cuts it to a quarter (500 to 125 lux), two stops, so it needs four times the power to look the same.
Desk and task lighting
A lamp giving 800 lux at 0.5 m reaches a 300 lux target at 0.8165 m; at 2 m it would be 50 lux.
Sound
Standing 10 m from a speaker that makes 100 dB at 1 m gives 80 dB; to get down to 85 dB you need 5.623 m.
Radiation physics labs
A source reading 100 µSv/h at 1 m reads 11.11 µSv/h at 3 m, and 25 µSv/h behind two half-value layers.
Sunlight on other planets
The 1,361 W/m² of sunlight at Earth (1 au) falls to 586.2 W/m² at Mars’s average 1.524 au.
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