How this instrument works
Intensity is power spread over area: how many watts land on each square metre of a surface at some distance from a source. For a point source radiating equally in every direction, that surface is the skin of an expanding sphere, and its area is 4πr² — so I = P ⁄ (4πr²). The formula is not a fact about light or sound specifically; it's a fact about geometry. Anything that spreads undiminished from a point in three-dimensional space, unabsorbed and unreflected, obeys it: starlight, a loudspeaker's output, gamma rays from a source in a lab, even Newtonian gravity's pull.
The square comes directly from the sphere's area formula, not from any property of the radiation itself. Double the radius and the sphere's surface grows by a factor of four, because area scales with the square of a linear dimension. The same fixed power now has four times as much surface to cover, so each square metre gets a quarter of what it got before. Triple the distance and the area grows ninefold, so intensity drops to a ninth. There is no separate 'inverse-square constant' to look up — the 2 in r² is simply the number of dimensions the sphere's surface spans.
The formula assumes a source small enough, relative to r, to be treated as a single point, and a medium that neither absorbs nor scatters the energy in transit. Both assumptions break down in practice. Stand closer to a source than its own physical size and you're in the near field, where the point approximation fails and readings can be far higher than the formula predicts. Send sound through humid air over hundreds of metres, or light through haze, and atmospheric absorption removes extra energy on top of the geometric spreading, so the real drop-off outpaces 1/r². A directional source — a spotlight, a parabolic microphone, a phased antenna — concentrates its power into a narrower cone than a full sphere, so it too departs from this idealised isotropic case.
- Enter Source power (or intensity source) — the total power the point emits, in watts, or the equivalent output for sound, radio, or radiation.
- Enter Distance from source — how far the point of interest sits from that source, in metres or centimetres.
- Read Intensity at that distance — the power landing on each square metre of the sphere at that radius.
- Change only the distance value and watch intensity move by the square of the ratio, not the ratio itself — halving r quadruples I.
Worked example — a 100 W source measured 2 m away
Set Source power (or intensity source) to 100 W and Distance from source to 2 m. The sphere swept out at that radius has surface area 4π(2)² = 4π(4) = 50.2655 m². Dividing the full 100 W across that area gives I = 100 ⁄ 50.2655 = 1.98943678865 W/m², the figure Intensity at that distance reads back — five significant figures, and every digit reproducible from the same two inputs.
Now move the same source twice as far away, to 4 m. The sphere's surface area doesn't double; it quadruples, to 4π(4)² = 4π(16) = 201.06 m², so the 100 W spreads thinner and intensity falls to 0.497359197162 W/m² — a quarter of the reading at 2 m, not a half. That fourfold drop for a twofold increase in distance is the entire content of the word 'square' in the law's name, and it's the number a photographer, an audio engineer, or a radiation-safety technician actually needs when deciding how far is far enough.
Questions
Why is it an inverse-square law instead of a simple inverse one?
Because intensity divides by the sphere's surface area, and that area scales with the square of the radius, not the radius itself. A plain inverse relationship (I ∝ 1/r) would halve intensity when distance doubles; this law quarters it, since doubling r quadruples 4πr². Mistaking one for the other is the single most common error people make when estimating brightness, loudness, or dose at a new distance.
Does this apply to light, sound, gravity, and radiation in the same way?
Yes, wherever the source is small compared with r and nothing between source and receiver absorbs or scatters the energy — the geometry doesn't care what's spreading. Newtonian gravity's field strength, a lamp's illuminance, an isotropic loudspeaker's intensity, and a gamma source's dose rate all fall as 1/r² for exactly this reason. Sound in humid air and light in haze pick up extra losses on top of the geometric term, so their real-world falloff can be steeper than the pure formula.
What happens as distance approaches zero?
The formula predicts intensity climbing toward infinity, which is unphysical — real sources have finite size, and the point-source assumption stops holding once you're closer than roughly the source's own dimensions. Inside that near-field region the field behaves differently and this instrument's readout should not be trusted; the inverse-square law only describes the far field of a source small compared with r.
Can I use the same formula for gravitational field strength, not just power sources?
Structurally yes. Newton's law of gravitation reduces to the same 1/r² geometric falloff; only the numerator's meaning changes, from radiated power to GM, the product of the gravitational constant and the attracting mass. Enter GM in place of Source power and the result reads as gravitational acceleration in place of intensity — the sphere-of-influence geometry underneath is identical.
Why does a photographer moving a light from 1 m to 2 m need four times the power, not two?
Because the light's intensity on the subject falls to a quarter, not a half, once the sphere it illuminates has four times the surface area at twice the radius. To keep exposure constant at the new distance, the source must output four times the power, or the lens must open two full stops — a rule lighting technicians use daily and one this instrument reproduces exactly, with the numbers to prove it.
Why does zero source power always give zero intensity, regardless of distance?
Because intensity is a fixed fraction of the power being divided — I = P ⁄ (4πr²) — and dividing zero by any nonzero sphere area still returns zero. Distance changes how the power is distributed, not whether there is any power to distribute; with nothing radiating from the source, every point in space, near or far, reads zero.