Inverse Square Law Calculator — Intensity & Distance with I₁·r₁² = I₂·r₂²

What is the inverse square law? Where do we find the inverse square law?

The inverse square law describes how physical quantities radiating from point sources dissipate as a function of the distance. The core concept is that if the emission is radial, we can map it onto a spherical surface. The radius of the sphere corresponds to the distance from the point source.

We calculate the area of a sphere as it grows according to the formula:

A = 4πr²

where we find a dependency on the square of the distance rather than on the distance itself. Because the same amount of energy is spread over an area that grows with r², the intensity received at any point falls off as 1/r².

A geometrical justification for the inverse square law formula

Imagine turning on a lightbulb inside a deflated balloon (don't do it if the bulb is prone to get hot!). The bulb emits light with a given luminous flux measured in lumens. This flux distributes on the skin-tight (bulb-tight?) balloon with a specific illuminance (luminous flux per unit area) that equals the flux divided by the area of the balloon itself.

Now inflate the balloon so that it roughly approximates a sphere with a given radius r. How did the quantities we introduced above change?

• The luminous flux remained the same, since we are not "missing out" on any bit of radiation emitted by the bulb.
• The illuminance is reduced by a factor controlled by the change in the surface area of the balloon.

This means that, while we are catching all the energy emitted by the bulb, every unit area on the balloon receives less energy than before. In jargon, we call this characteristic behavior of radiating phenomena geometric dilution, as the power of the source per unit area reduces with the distance.

The formula for the inverse square law equation

For a source whose total emission stays constant, the intensity measured at two different distances satisfies:

I₁ · r₁² = I₂ · r₂²

where:

• I₁ — the intensity at the first distance r₁;
• I₂ — the intensity at the second distance r₂; and
• r₁, r₂ — the distances from the point source.

Rearranging the equation lets you solve for whichever quantity you need:

• Intensity at a new distance: I₂ = I₁ · (r₁ / r₂)²
• Distance for a target intensity: r₂ = r₁ · √(I₁ / I₂)

Where do we find the inverse square law equation?

The inverse square law appears throughout Nature whenever something radiates uniformly from a point:

• Light and illumination — the brightness of a lamp falls as 1/r² (lumens, lux, candelas, foot-candles).
• Radiation — the dose from a point radioactive source decreases with the square of the distance; this is the basis of the "time, distance, shielding" safety rule.
• Sound — the intensity of sound from a point source obeys the inverse square law in a free field.
• Gravity — Newton's law of universal gravitation: the force between two masses scales with 1/r².
• Electrostatics — Coulomb's law: the force between two charges scales with 1/r².

How to use our inverse square law calculator: explanation and practical example

1. Select your unit system: Metric (m, km, cm, mm; W/m², lx) or American/Imperial (ft, mi, in, yd; W/ft², foot-candle).
2. Choose what to calculate:
   • Intensity at a Distance — enter the source intensity I₁, the reference distance r₁, and a new distance r₂ to get the intensity I₂ there.
   • Distance for an Intensity — enter I₁, r₁, and a target intensity I₂ to find the distance r₂ at which that intensity occurs.
3. Click Calculate. The result is shown alongside the formula used and the values you entered.
4. Use the Share button to get a direct link to your calculation.

Worked example: a lamp produces an illuminance of 100 W/m² at 1 m. What is the illuminance at 2 m?

I₂ = I₁ · (r₁ / r₂)² = 100 · (1 / 2)² = 100 · 0.25 = 25 W/m²

Doubling the distance quarters the intensity — a hallmark of the inverse square law.

Related calculators

Our lumen calculator and lighting tools can help you grasp luminous-flux quantities, while the Coulomb's law calculator and Gauss's law calculator show the inverse square law at work in electromagnetism.

FAQs

What does "inverse square" mean?

It means a quantity is inversely proportional to the square of the distance: if you double the distance, the quantity drops to one quarter; triple it, and it drops to one ninth.

Why does the intensity fall off with the square of the distance?

The radiated energy spreads over the surface of an expanding sphere whose area is A = 4πr². Since the total energy is constant, the energy per unit area must decrease as 1/r².

Do the units of intensity matter in the calculation?

Because the law compares the same quantity at two distances, the intensity unit cancels out in the ratio. Your result I₂ is returned in the same unit you entered for I₁.

Does the inverse square law always hold?

It holds for an idealized point source radiating uniformly into free space, with no absorption, reflection, or focusing. Real-world beams, lenses, and enclosed spaces can modify the behavior.