Inverse square law calculator
How intensity falls with distance from a point source.
Fill in the fields and the result will appear here automatically.
At constant source power, point-source intensity falls as 1/r². Enter linear intensity I₁ and two positive distances in matching units. The result retains the unit of I₁ and reports the intensity ratio. The model follows one direction with an unchanged radiation pattern, without absorption or reflections; it may fail close to an extended source.
How it works
Formula and logic
The geometric factor is G=(d₁/d₂)², and I₂=I₁G. Positive distances define G even when I₁=0, which gives I₂=0. The ratio I₂/I₁ equals G only for I₁>0; that ratio is not defined when I₁=0.
Example
1000 lux at one metre becomes 111.11 lux at three metres.
Fields and units
- Intensity at the original distance — units of I₁
- Original distance — length unit
- New distance — length unit
How to use
- — Use a linear intensity such as W/m², or illuminance in lux under otherwise matching observation conditions. Do not enter a dB level: it is logarithmic and cannot be multiplied by the squared distance ratio.
- — Both distances must share a unit; intensity-unit conversion is not performed.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- OpenStax §17.3: inverse-square linear intensity and logarithmic dB
- Limitation
- The model follows one direction with an unchanged radiation pattern, without absorption or reflections; it may fail close to an extended source. Do not enter a dB level: it is logarithmic and cannot be multiplied by the squared distance ratio.
FAQ
Why a square specifically?
The source shines in every direction, and all its energy crosses a sphere around it. A sphere's area is 4πr², growing as the square of the radius — so each unit of area is left with that much less.
Does a directional source follow this law?
Directionality alone does not remove 1/r²: a beam with a fixed solid angle also has a cross-sectional area proportional to r². For a spotlight or laser, actual beam geometry and distance matter; this model has no focusing or near-source calculation.
How are sound intensity and decibels related?
For free propagation with other conditions unchanged, doubling distance leaves one quarter of the intensity. The level difference is 10 log₁₀(1/4)≈−6.0206 dB. An initial dB level is not a linear I₁.
What if the new distance is smaller?
This model gives a greater intensity. Both distances must be positive and within the model’s applicable region; moving very close to an extended source cannot be evaluated from this ratio alone.