Inverse Square Law Calculator
Free calculate intensity at different distances from a point source using i₁d₁² = i₂d₂². Get instant, accurate results with our easy-to-use calculator.
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What is the Inverse Square Law?
The inverse square law describes how the intensity of a physical quantity (light, sound, radiation, gravity) decreases as the distance from the source increases. Intensity is inversely proportional to the square of the distance.
This law applies to point sources that radiate uniformly in all directions. As you move away, the same amount of energy spreads over a larger area (proportional to distance²), so intensity decreases as 1/d².
The inverse square law is fundamental in physics, explaining why stars appear dimmer with distance, why sound fades, why radiation exposure decreases with distance, and why gravitational and electrical forces follow this relationship.
Inverse Square Law Formula
Where:
- • I₁, I₂ = Intensities at distances d₁ and d₂
- • d₁, d₂ = Distances from source
Rearranged:
I₂ = I₁ × (d₁/d₂)²
d₂ = d₁ × √(I₁/I₂)
How to Calculate
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1
Identify known values
Determine which values you know: I₁, d₁, I₂, or d₂.
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2
Convert distances to same units
Ensure all distances use the same units (meters, centimeters, etc.).
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3
Apply the formula
Use I₁d₁² = I₂d₂², rearranging as needed to solve for the unknown.
Practical Examples
Example 1: Light Bulb
Light intensity is 128 lux at 5 m from a lamp. What is intensity at 20 m?
Solution:
I₂ = I₁ × (d₁/d₂)² = 128 × (5/20)²
I₂ = 128 × (0.25)² = 128 × 0.0625
I₂ = 8 lux (1/16 of original!)
Example 2: Sound Source
Sound intensity is 0.1 W/m² at 10 m. What distance gives 0.01 W/m²?
Solution:
d₂ = d₁ × √(I₁/I₂) = 10 × √(0.1/0.01)
d₂ = 10 × √10 = 31.6 m
Applications
Lighting Design
Calculating illuminance at different distances, designing lighting layouts, and ensuring adequate light levels.
Radiation Safety
Determining safe distances from radiation sources, calculating exposure limits, and radiation protection.
Audio Engineering
Understanding sound propagation, speaker placement, and calculating sound intensity at different distances.
Photography
Understanding flash falloff, exposure calculations, and how distance affects light intensity in photos.
Frequently Asked Questions
Why is it called "inverse square"?
Intensity is inversely proportional to distance squared (I ∝ 1/d²). Doubling distance reduces intensity to 1/4, tripling distance reduces it to 1/9.
Does this apply to all light sources?
Only to point sources radiating uniformly. Extended sources, collimated beams (lasers), and directional sources don't follow this law exactly.
What happens if you double the distance?
Intensity becomes 1/4 (25%) of the original. This is because the same energy spreads over 4× the area (area ∝ d²).
Does this work for gravity and electric fields?
Yes! Both gravitational force (F = GMm/r²) and electric force (F = kq₁q₂/r²) follow inverse square laws, though they're forces, not intensities.
What about in a medium (not vacuum)?
The inverse square law still applies, but absorption and scattering in the medium cause additional intensity loss beyond the 1/d² relationship.