Inverse Square Law Calculator

The general inverse power relationship is: I₂ = I₁ × (d₁/d₂)ⁿ

• I₁ is the known intensity at distance d₁.
• I₂ is the intensity at distance d₂.
• n is the exponent (use 2 for inverse square behavior).

The ratio form is I₂/I₁ = (d₁/d₂)ⁿ. For intensity-like quantities, change in dB is 10·log₁₀(I₂/I₁).

1. Select what you want to solve (I1, I2, d1, or d2).
2. Enter any three known values in matching units.
3. Keep distances in the same unit for d1 and d2.
4. Use exponent 2 for classic inverse square spreading.
5. Click Calculate to see results above the form.
6. Use the download buttons to export CSV or PDF.
7. Optional: enable the comparison table for a distance sweep.

These examples assume consistent distance and intensity units.

1) Meaning of the inverse square law

The law connects intensity to distance for a spreading point source. As distance grows, the same power covers a larger spherical area. Because area scales with d², intensity scales like 1/d² when n = 2. It is used for light, radiation, and many flux measurements.

2) A simple doubling example

Doubling distance reduces intensity to one quarter with n = 2. If I1 = 100 at d1 = 1, then at d2 = 2 you get I2 = 25. This quarter‑rule is a fast check for calculation mistakes. Halving distance raises intensity by four, all else equal.

3) Where the relationship shows up

Light levels (lux) drop as you step away from a lamp. Dose rate from a small radiation source drops similarly. In open air, sound power per area can also follow this pattern. Many instruments report power or flux per unit area, matching the model.

4) Why the exponent n can change

Real systems are not always ideal point sources. A long line source often trends closer to n = 1. Directional emitters, shielding, absorption, and reflections change decay. Use a fitted n when measurements show a different slope.

5) Solving for intensity at a new distance

Choose I2 when you know I1, d1, and d2. The calculator applies I2 = I1 × (d1/d2)^n and returns the ratio I2/I1. Percent change and dB output help compare setups quickly. Keep the same intensity unit for both I1 and I2.

6) Solving for distance to reach a target

Choose d2 when you know I1, I2, and d1. Rearranged, d2 = d1 × (I1/I2)^(1/n). Example: to reduce 100 down to 10 with n = 2, you need 3.162× farther. This is handy for setting safe standoff distances and exposure limits.

7) Reading ratio, percent, and dB values

Ratio expresses scaling directly: 0.25 means “one quarter as strong.” Percent change shows relative increase or decrease from the reference. dB uses 10·log10(I2/I1) for power‑like quantities. A 0.25 ratio equals about −6.02 dB.

8) Common pitfalls and safe inputs

Keep d1 and d2 in the same distance unit. Intensities should be non‑negative, and distances must be positive. If reflections dominate, results become an estimate, not a guarantee. Use the comparison table to see trends over a full range. Validate with a few real readings when accuracy is critical.

1) What does the inverse square law mean?

It states intensity falls with distance raised to a power. For a point source in open space, intensity is proportional to 1/d², meaning doubling distance quarters the intensity.

2) What inputs do I need?

Select what to solve, then provide any three of I1, I2, d1, and d2. Keep distance units consistent, and keep intensity units consistent for I1 and I2.

3) Why can I change the exponent n?

Some setups are not ideal point sources. Geometry, directionality, absorption, and reflections can change how fast intensity decays. Adjust n to match measurements or expected behavior.

4) What does the dB value represent?

It is 10·log10(I2/I1) for power‑like intensity quantities. Negative dB means intensity decreased from the reference. A ratio of 0.25 corresponds to about −6.02 dB.

5) Can I use lux, W/m², or arbitrary units?

Yes. The calculator only uses ratios, so any consistent intensity unit works. Choose a label you prefer, and keep I1 and I2 in that same unit.

6) When is the result only an estimate?

Indoors or near large surfaces, reflections can dominate. In air or water, absorption can matter over long distances. Use the calculator as a model, then validate with real readings.

7) What is the comparison table used for?

It generates intensities across a distance range using your I1 and d1 reference. This helps you see trends, pick thresholds, and export multiple scenarios to CSV or PDF.