Inverse-square law

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Inverse-square law: a short, easy guide

- What it means: For a point source in three-dimensional space, many quantities that spread out from the source get weaker with the square of the distance. In simple terms, if you double the distance, the quantity becomes four times smaller. The general relationship is I ∝ 1/r^2 or F ∝ 1/r^2, where r is the distance from the source.

- Why it happens: The source’s output fans out over the surface of a growing sphere. The surface area of a sphere is 4πr^2, so the same amount of energy or force is spread over a larger area as you move away.

- Key examples:
- Gravity: The force between two masses follows F = G m1 m2 / r^2.
- Electricity: The electric force between two charges follows F = k q1 q2 / r^2.
- Light and other radiation: The intensity at distance r is I = P / (4π r^2). If you double the distance, the light intensity falls to one quarter.
- Sound: The sound intensity in air also falls roughly as 1/r^2 (the sound pressure itself drops as 1/r).

- Real-world notes:
- The inverse-square law works best for point sources, or when the source is much smaller than the distance to you.
- In practice, absorption, scattering, or a non-point (extended) source can make the drop with distance deviate from exactly 1/r^2.

- Quick takeaway: Doubling distance reduces intensity by a factor of four; halving distance increases it by four. The law applies widely—from gravity and electric forces to light and sound—whenever energy, force, or another conserved quantity spreads out evenly from a tiny source in open space.