Library / Foundations: Physics / FPH-45
Foundations: Physics
Inverse-square law for intensity
Intensity from a point source spreading in all directions falls with the square of distance.
- Ref
- FPH-45
- Rev
- v1
- Field
- Foundations: Physics
- Result
- I [W/m²]
P is the total power from a point source that radiates equally in all directions. At range r the power is spread over a sphere of area 4πr², so the intensity I is power per unit area. The same law applies to light, sound, radio and radiation from a small source.
Intensity from a point source spreading in all directions falls with the square of distance. Doubling the range cuts it to a quarter.
Calculator
- Intensity at twice the distance0.4974 W/m²
- Area of the sphere at this range, 4πr²50.27 m²
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Doubling the range from 2 m to 4 m cuts the intensity from 1.99 to 0.50 W/m², a fall of 75 %.
- A radar signal travels out and back, so the echo falls as 1/r⁴, not 1/r². That is why radar range grows only as the fourth root of power.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (W/m²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| P Source power | 0.001 to 1,000,000 W | 1.989e-5 | 19,890 | 19,890 | +10% | +10 % |
| r Distance | 0.1 to 10,000 m | 7.958e-8 | 795.8 | 795.8 | +10% | −17.4 % |
Sweep of Distance
| Distance (m) | Intensity (W/m²) |
|---|---|
| 0.1 | 795.8 |
| 0.4217 | 44.75 |
| 1.778 | 2.516 |
| 7.499 | 0.1415 |
| 31.62 | 0.007958 |
| 133.4 | 4.475e-4 |
| 562.3 | 2.516e-5 |
| 2,371 | 1.415e-6 |
| 10,000 | 7.958e-8 |
Sweep one variable
Variables and how each one contributes
Intensity
What the equation gives you. Everything else feeds it.
Source power
The total power radiated by the source. Directly proportional: doubling the power doubles the intensity at every range.
Distance
Distance from the source. Inverse square: doubling the range cuts the intensity to a quarter. A 10 % rise in r lowers it by about 17 %.
Worked example with the values above
- Area of the sphere: 4πr² = 4π × (2)² = 50.27 m².
- Spread the power over it: 100 W / 50.27 m².
- I = 1.989 W/m².
Assumptions and limits
- A point source radiating equally in all directions (isotropic).
- No absorption, scattering or reflection between the source and the point of interest.
Common mistakes
- Using the diameter as r.
- Applying it to a directional source such as a searchlight or an antenna dish, where the on-axis intensity is higher by the antenna gain.
Statistics and uncertainty
Intensity goes as P/r², so a 1 % error in range gives a 2 % error in intensity. Range is the input to measure carefully. For sound or light the result is a mean value; reflections and absorption make real measurements differ.
Where it is used
- Estimating signal strength, noise or light level at a range.
- Safe distances from a radiation source or a loud noise.
Maths you need
- Area of a sphere: 4πr²
- Inverse-square law: a quantity that falls as 1/r²