Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

Does light really follow an inverse-square law?

AA SLAI SL Accessible Also in: Statistics

Research question

Does the illuminance measured by my phone fall as 1/d² as I move away from a lamp, and at what distances does the model fail?

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Why it makes a good exploration

The inverse-square law is fundamental to light, sound and gravity, and your phone contains a light sensor. Testing it — and finding where it breaks down (near the bulb, with reflections) — is a clean, personal experiment.

The mathematics you'll need

Power models and inverse variation
Log-log linearisation
Residual analysis
Transformations for background light

Modelling the data, step by step

This idea usually needs power or reciprocal models. See it worked step by step, with a criterion tip at every step: Power y = a·xⁿ (log–log) · Reciprocal y = a/x + b and a/(x − h) + k · Choosing and comparing models.

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for you.

Where the data comes from

In a dark room, measure illuminance with phyphox's light sensor at 20+ distances from a small bulb, three readings each; record background light.

phyphox (RWTH Aachen) — Free app that turns your phone's accelerometer, microphone, barometer and gyroscope into data loggers with CSV export.

A possible outline

Explain why intensity should fall with 1/d² from geometry.
Collect data, subtracting background light.
Fit y = axⁿ via logs and compare n with −2.
Examine residuals close to and far from the bulb.
Reflect on the sensor, reflections and the bulb not being a point source.

Pitfalls that cost marks

Measuring from the wrong point (the sensor position on the phone matters).
Ignoring background light.
Only testing over a narrow range of distances.

Showing personal engagement

Compare an LED and an incandescent bulb.
Use your model to design desk-lamp placement for a target lux level.
Explain the discrepancy near the bulb with a finite-source model.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Power models and inverse variation; Log-log linearisation
AA HL	Not a natural fit	The mathematics is mainly from the AI course; at AA HL the exploration would need an AA-level approach (calculus, proof or probability theory) to reach the top of Criterion E.
AI SL	Good fit	Power models and inverse variation; Log-log linearisation
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

Level: Accessible. A good first extended piece of maths, with room to go deeper. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: measuring from the wrong point (the sensor position on the phone matters); and ignoring background light. Say how each one affects your answer.

Use of mathematics (E)

SL: power models and inverse variation; log-log linearisation; residual analysis; transformations for background light, each explained in your own words and carried out correctly.

How modelling explorations reach the top bands in general: Modelling with functions IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Model the bulb as a line or disc source and show how the inverse-square law emerges at large distances.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL modelling exploration adds.

See a complete IA, marked

Should I lease or buy my first car? (AI SL) is a complete modelling exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL and AI SL. We rate it Accessible: a good first extended piece of maths, with room to go deeper.

What mathematics does this IA use?

Power models and inverse variation, log-log linearisation, residual analysis and transformations for background light.

Where can I get data for this IA?

In a dark room, measure illuminance with phyphox's light sensor at 20+ distances from a small bulb, three readings each; record background light. Checked sources and tools: phyphox (RWTH Aachen).

Turn this idea into your IA

The planner opens with “Does light really follow an inverse-square law?” filled in: the question, the maths and the data sources.

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