Table of Contents
What Is the Inverse Square Law?
The inverse square law states that the intensity of a physical quantity (such as light, sound, radiation, or gravitational force) from a point source is inversely proportional to the square of the distance from the source. When you double the distance from a point source, the intensity drops to one-quarter. Triple the distance, and it drops to one-ninth. This relationship arises because the energy spreads out over the surface area of a sphere, which grows as the square of the radius.
This law is one of the most universal principles in physics. It applies to gravity (Newton's law of universal gravitation), electrostatics (Coulomb's law), electromagnetic radiation (light intensity), sound (in free-field conditions), and nuclear radiation. Understanding it is essential for lighting design, acoustics, radiation safety, astronomy, and telecommunications.
The Formula
Where I1 and I2 are intensities at distances d1 and d2 from the source. The units are consistent: if I is in W/m² and d in meters, the product I x d² is constant for a given source.
Physical Applications
| Field | Quantity | Equation |
|---|---|---|
| Gravity | Force | F = GMm/r² |
| Electrostatics | Force | F = kq1q2/r² |
| Light | Illuminance | E = I/d² (lux) |
| Sound | Intensity | I = P/(4πr²) |
| Radiation | Dose rate | ∝ 1/r² |
Quick Reference Table
| Distance Multiplier | Intensity Fraction | dB Change |
|---|---|---|
| 1x (reference) | 100% | 0 dB |
| 2x | 25% | -6.02 dB |
| 3x | 11.1% | -9.54 dB |
| 5x | 4% | -13.98 dB |
| 10x | 1% | -20 dB |
Frequently Asked Questions
Does the inverse square law apply to laser beams?
No, not for the collimated portion of a laser beam. Lasers produce highly collimated (parallel) beams that maintain nearly constant intensity over long distances. The inverse square law applies to point sources that radiate in all directions. However, even laser beams eventually diverge due to diffraction, and at very long distances, their intensity does decrease roughly as 1/r².
Why does sound sometimes not follow the inverse square law?
The inverse square law assumes a point source in free-field conditions (no reflections, no barriers). In enclosed spaces, reflections from walls, ceilings, and floors create a reverberant field where sound intensity remains roughly constant. Outdoors, ground absorption, atmospheric effects, wind, and temperature gradients can cause deviations from the ideal inverse square law behavior.
How is the inverse square law used in radiation safety?
In radiation protection, the inverse square law is used to calculate safe distances from radioactive sources. Doubling your distance from a source reduces your radiation exposure to one-quarter. This principle, along with time limitation and shielding, forms the three pillars of radiation protection (distance, time, shielding). It is the simplest and most effective way to reduce radiation dose.