Illuminance and distance from the source

This activity allows students to experimentally verify the inverse square law that governs the propagation of light. It develops the ability to make systematic measurements and identify mathematical relationships.

Why does a campfire seem so much dimmer when you walk away from it? The answer lies in one of the most universal principles in physics: the inverse square law. When light radiates outward from a point source, it spreads over the surface of an ever-expanding sphere. Since the surface area of a sphere grows as the square of its radius, the light intensity at any point decreases as 1/r². This principle governs not just light but any phenomenon that radiates from a point source into three-dimensional space: sound, gravity, electric fields, and radiation all obey the same law. The inverse square law was first established for light by Johann Heinrich Lambert in 1760 and later confirmed by many experiments. Remarkably, modern smartphones contain light sensors (lux meters) precise enough to verify this fundamental law with a simple lamp and a tape measure. This experiment guides students through a systematic measurement of illuminance at various distances, revealing the mathematical beauty of the inverse square relationship.

Learning objectives:

The student uses the brightness sensor of an Android smartphone and the FizziQ application to measure illuminance at different distances from a point light source. By taking measurements at regular intervals and organizing the data in a table, the student can draw an illumination curve as a function of distance and then determine the mathematical relationship that models this propagation.

Level:

High school

FizziQ

Author:

Duration (minutes) :

35

What students will do :

- Measure illuminance at multiple distances from a point light source using the smartphone light sensor
- Verify the inverse square law: E ∝ 1/r² for light from a point source
- Plot illuminance versus distance and illuminance versus 1/r² to demonstrate the relationship
- Determine the luminous intensity of the source from the graph
- Understand the geometric origin of the inverse square law and its universality

Scientific concepts:

- Inverse square law
- Bright illuminance
- Light propagation
- Mathematical modeling
- Data interpolation

Sensors:

- Light sensor (lux meter / ambient light sensor)

What is required:

- Android smartphone with FizziQ app and light sensor
- A lamp without a shade as a point light source
- A room that can be darkened
- A tape measure
- FizziQ experience notebook

Experimental procedure:

Set up the experiment in a room that can be completely darkened (close blinds, turn off all lights except the test source).
Choose a small, bright lamp without a shade to approximate a point source. A bare LED bulb or a small desk lamp works well.
Place the lamp on a stable surface. Set up a tape measure extending outward from the lamp along a straight line on a table or the floor.
Open FizziQ on an Android smartphone and select the Light Sensor (Lux) measurement. (Note: on iPhones, this sensor may not be directly accessible.)
Position the smartphone sensor at 10 cm from the lamp, facing the light source directly. Record the illuminance value after it stabilizes (wait 5 seconds).
Move the smartphone to 15 cm, then 20 cm, 30 cm, 40 cm, 50 cm, 75 cm, and 100 cm, recording the illuminance at each distance.
Record at least three readings at each distance and calculate the average to reduce noise.
Enter all data pairs (distance r, illuminance E) in the FizziQ notebook.
Plot E versus r. The curve should show a steep decrease that flattens at larger distances.
Now plot E versus 1/r². This graph should be a straight line through the origin if the inverse square law holds.
Determine the slope of the E versus 1/r² line. This slope equals I/(4π), where I is the luminous intensity of the source in candelas.
Calculate the luminous intensity I and compare with the manufacturer's specification for the lamp if available.

Expected results:

The illuminance should decrease rapidly with distance, showing a characteristic 1/r² profile. At 10 cm from a typical desk lamp (10-15 W LED), the illuminance may be 5000-20000 lux; at 100 cm, it drops to 50-200 lux. The plot of E versus 1/r² should be approximately linear, with a coefficient of determination R² above 0.95 for clean data. Deviations from perfect linearity are expected at very short distances (where the source is not a point) and at large distances (where ambient light becomes significant relative to the source). Reflections from walls and ceiling also contribute stray light that adds a roughly constant offset to the readings, slightly degrading the linearity at larger distances.

Scientific questions:

- Why does the illuminance decrease as 1/r² rather than as 1/r?
- What geometric property of spheres explains the inverse square law?
- In what situations would the inverse square law not apply (non-point sources, focused beams)?
- How does the inverse square law apply to sound intensity? To gravitational force?
- If you double the power of the light source, how does the illuminance at a fixed distance change?
- Why is it important to perform this experiment in a darkened room?

Scientific explanations:

The inverse square law is a fundamental physical principle that describes how the intensity of a phenomenon decreases with distance from its source. For light, the illuminance E (measured in lux) is inversely proportional to the square of the distance r from the source: E = I/(4πr²), where I is the light intensity of the source.

This relationship is explained by the dispersion of light energy on increasingly large spherical surfaces as we move away from the source. On a graph, this relationship is manifested by a hyperbolic curve of type y = k/x².

The light meter built into some Android smartphones uses a photocell that converts light into an electrical signal. Its precision, although lower than that of a professional light meter, is sufficient to verify the inverse square law.

Some experimental difficulties can affect measurements: the source is never perfectly point-like, reflections off walls can add stray light, and the exact position of the sensor in the smartphone can be difficult to determine. The FizziQ interpolation tool allows you to confirm the relationship in 1/r² and estimate the proportionality coefficient.

This same law applies to other phenomena such as sound and gravitational or electric fields, illustrating the universality of certain physical principles.

Extension activities:

Frequently asked questions:

Q: My Android phone does not seem to have a lux sensor. Can I still do the experiment?
R: Most Android phones from 2012 onward include an ambient light sensor. In FizziQ, it appears as the "Brightness" or "Illuminance" sensor. If it is not available, check the phone specifications. iPhones may not expose this sensor to third-party apps.

Q: The illuminance values fluctuate even when the phone is stationary. Why?
R: Some light sensors have automatic gain adjustment that can cause fluctuations. Also, AC-powered lamps may flicker at 50/60 Hz, causing rapid oscillations. Use a DC-powered LED or battery lamp for more stable readings. Average over several seconds.

Q: The E versus 1/r² graph does not pass through the origin. What does this mean?
R: A non-zero y-intercept typically indicates residual ambient light that you have not fully eliminated. This constant background illuminance adds to every measurement. Subtract this offset from all readings to improve the fit.

Q: At very close distances (5 cm), the measurement deviates from the 1/r² law. Is this expected?
R: Yes. The inverse square law assumes a point source, which is only valid when the distance is much larger than the physical size of the source. At distances comparable to the lamp dimensions, the source cannot be treated as a point and the law breaks down.