Inverse Square Law
Calculate how intensity decreases with distance using the inverse square law formula I₂ = I₁ × (r₁/r₂)². Free online physics calculator with charts.
About This Calculator
The Inverse Square Law Calculator helps you compute how a physical quantity (intensity, force, or radiation) changes as you move away from a point source. Whether you are a physics student, engineer, or researcher, this tool makes it easy to understand the fundamental inverse-square relationship that governs gravitational, electrostatic, light, sound, and radiation phenomena.
The inverse square law formula is: I₂ = I₁ × (r₁/r₂)², where I₁ is the intensity at distance r₁, and I₂ is the intensity at distance r₂. This means the product of intensity and the square of the distance is constant: I₁ × r₁² = I₂ × r₂² = k. The geometric reasoning is that energy from a point source spreads uniformly over a spherical surface area of 4πr², so the energy per unit area decreases as 1/r².
This law applies to many physical phenomena: Newton's law of universal gravitation (F ∝ 1/r²), Coulomb's law for electrostatic force (F ∝ 1/r²), light illuminance from a point source, sound intensity in free field, X-ray and gamma-ray radiation intensity, and magnetic field strength from a monopole. As a practical example, the solar irradiance at Earth's surface (1 AU from the Sun) is about 1361 W/m². At Mars (1.52 AU), it drops to about 1361 × (1/1.52)² ≈ 589 W/m².
Regional Notes
Global: The inverse square law is universal and applies identically across all regions. The calculator works with any consistent unit system — use metric (meters, W/m²), imperial (feet, lumens/ft²), or astronomical units (AU, solar flux).
Frequently Asked Questions
What is the inverse square law?
The inverse square law states that a physical quantity (such as intensity, force, or radiation) from a point source is inversely proportional to the square of the distance from the source. This means if you double the distance, the intensity drops to one-quarter of its original value.
What is the formula for the inverse square law?
The inverse square law formula is I₁/I₂ = (r₂/r₁)², where I₁ is the initial intensity at distance r₁, and I₂ is the final intensity at distance r₂. Alternatively, I₂ = I₁ × (r₁/r₂)².
What are examples of the inverse square law in physics?
The inverse square law applies to gravitational force (Newton's law of gravitation), electrostatic force (Coulomb's law), light intensity from a point source, sound intensity, radiation intensity (X-rays, gamma rays), and magnetic field strength from a point source.
Why does the inverse square law happen?
The inverse square law arises geometrically because energy from a point source spreads out uniformly over the surface of an expanding sphere. Since the surface area of a sphere is 4πr², the energy per unit area decreases proportionally to 1/r².
What happens to intensity when distance is doubled?
When the distance from a point source is doubled, the intensity drops to one-quarter (25%) of the original value. This is because intensity is inversely proportional to the square of the distance (1/2² = 1/4).
How do I use this inverse square law calculator?
Enter the initial intensity (I₁), initial distance (r₁), and final distance (r₂) in any consistent units to calculate the final intensity (I₂). The calculator uses the formula I₂ = I₁ × (r₁/r₂)² and provides a breakdown table and charts showing how intensity changes with distance.
Does the inverse square law apply to all types of radiation?
The inverse square law applies to any radiation that propagates radially from a point source in free space, including light, radio waves, X-rays, gamma rays, and sound. However, it does not apply to guided waves (like in fiber optics) or to sources with directional emission (like lasers).
What units can I use with this calculator?
You can use any consistent units for distance (meters, feet, astronomical units, etc.) and any unit for intensity (W/m², lumens, lux, N, etc.). The calculator works with dimensional analysis as long as you use the same unit system consistently.