Abstract
How does the intensity of a light source change as you move away from it? This project describes a method to verify the inverse square law: how light, sound, electrical signals, and gravity each decrease with distance from their source. It does not matter if you are talking about a lightbulb or the sun; this law still applies!Summary
Written by Richard Blish, PhD 
Edited by Andrew Olson, PhD, and Ben Finio, PhD, Science Buddies
Thanks to David Aziz, PhD, for helpful suggestions on photocell measurement methods.
Recommended Project Supplies
Objective
Determine how the intensity of light changes with increasing distance from the light source.
Introduction
As you move away from a light source, the light gets dimmer. No doubt you have noticed this with reading lamps, streetlights, and so on. Figure 1 shows what is happening. The blue area, marked "S," represents a point source of light. Imagine the light from the star spreading out into empty space in all directions. Now imagine the light that falls on a square at some arbitrary distance from the star (r). Move away, doubling the distance from the star (2r). The light from the original square has now "spread out" over an area of 4 (= 22) squares. Thus, at twice the original distance, the intensity (power per square meter) of the light passing through a single square will be 1/4 of the original intensity. Going out still farther, tripling the original distance (3r), and the light from the original square now covers an area of 9 (= 32) squares. Thus, at three times the original distance, the intensity of the light passing through a single square will be 1/9 of the original intensity. This is what is meant by the inverse square law. As you move away from a point light source, the intensity of the light is proportional to 1/r2, the inverse square of the distance. Because the same geometry applies to many other physical phenomena (sound, gravity, electrostatic interactions), the inverse square law has significance for many problems in physics.
The figure shows directional light originating from a point source that covers a larger area the further away it is from the source. As the light travels it has a specific brightness and size at any given point. The inverse square law shows that when light travels twice the distance its area grows four times as large and the brightness decreases by four times. The rate a light grows in area and decreases in brightness is related to the distance it travels from another point squared.
Figure 1. An illustration of the inverse square law. Image credit Wikimedia Commons user Borb.
But do not just take our word for it! Why not see for yourself if light really behaves this way? This project shows you how you can use a light-sensitive resistor, called a photoresistor, which has an electrical resistance (measured in ohms (Ω)) that changes with exposure to light, and a digital multimeter to see if light intensity really does decrease according to the inverse square law. You will measure the resistance of the photoresistor at different distances from a light source. Using information from the photoresistor's datasheet, you can convert the resistance measurement to lux, the SI unit of illuminance (a measure of intensity that accounts for how different wavelengths are perceived by the human eye). You can then create a graph to see how illuminance changes with distance from the light source, and verify if it follows the inverse square law.
Terms and Concepts
To do this project, you should do research that enables you to understand the following terms and concepts:
- Intensity
- Inverse square law
- Resistor
- Photoresistor
- Resistance
- Ohms (Ω)
- Multimeter
- Lux
- Illuminance
Questions
- How is light measured in the metric system? What is the difference between intensity and illuminance?
- How does the photoresistor's resistance change with increasing illuminance?
- How do you expect the resistance to change as you move the photoresistor away from a light source (decreasing illuminance)?
- In what other situations in physics does the inverse square law apply?
Bibliography
- Exploratorium. (n.d.). Inverse Square Law. Exploratorium Science Snacks. Retrieved May 5, 2014.
- The Physics Classroom. (n.d.). Inverse Square Law. Retrieved May 5, 2014.
- Nave, C.R. (n.d.). Power per Unit Area of Surface. HyperPhysics, Department of Physics and Astronomy, Georgia State University. Retrieved May 23, 2016.
- Jameco Electronics (n.d.). Photoconductive Cell CdS P/N: CDS001-8003. Retrieved May 23, 2016.
- Science Buddies. (n.d.). How to use a Multimeter. Science Buddies. Retrieved May 23, 2016.
Materials and Equipment 
Recommended Project Supplies
The following specialty items are available from our partner Home Science Tools:
- Electronic Sensors Kit (1). You will need the following items from the kit.
- Photoresistor
- Digital multimeter
- Alligator clip leads (2)
You will also need the following items (not included in the kit):
- Lamp and lightbulb
- Tape measure
- Tape
- Cardboard box, or other item to make a platform so you can rest the photoresistor level with the lightbulb
- Dark room in which to do the experiment; the room should have as few external light sources as possible.
- Lab notebook
Disclaimer: Science Buddies participates in affiliate programs with Home Science Tools, Amazon.com, Carolina Biological, and Jameco Electronics. Proceeds from the affiliate programs help support Science Buddies, a 501(c)(3) public charity, and keep our resources free for everyone. Our top priority is student learning. If you have any comments (positive or negative) related to purchases you've made for science projects from recommendations on our site, please let us know. Write to us at scibuddy@sciencebuddies.org.
Experimental Procedure
- Set up your multimeter to measure the resistance of the photoresistor, as shown in Figure 2.
- Plug the black multimeter probe into the port labeled COM.
- Plug the red multimeter probe into the port labeled VΩmA.
- Connect the multimeter probes to the leads of the photoresistor using alligator clips.
- Set the multimeter dial to measure resistance in the 200 Ω range.
- Turn the multimeter's power switch to ON.
- If this is your first time using a multimeter, see the Science Buddies reference How to Use a Multimeter, particularly the section How do I measure resistance? to learn more.
Figure 2. How to connect your multimeter to the photoresistor.
- Set up your experiment, as shown in Figure 3.
- If possible, set the experiment up in a room with no windows, or do the experiment at night. If this is not possible, find a room with as little external light as possible. The photoresistor is very sensitive; even a little bit of light leaking under a door can affect your readings.
- Turn off all other lights in the room except the single lamp you will use for the experiment.
- If necessary, remove the lamp shade from your lamp.
- Set up your tape measure to measure distance from the lamp.
- Tape your photoresistor to a cardboard box so it is level with the lamp. The face of the photoresistor (the side with the squiggly lines) should be facing directly toward the lamp.
Figure 3. Experimental setup.
- Measure the resistance of the photoresistor as you increase the distance from the lightbulb (for example, every 10 cm).
- Take at least three readings at each distance and calculate an average.
- Try to take measurements over a range of at least several meters. If you have space, you can go even farther, but remember that stray ambient light will affect your readings.
- Note: The resistance will increase as you move away from the light source. If the screen of your multimeter reads "1 .", then the resistance has exceeded the range of the multimeter dial setting. Rotate the dial up one resistance setting (for example, from "200" to "2000") to increase the range. Make sure you pay attention to units. A prefix of "k" means "kilo-ohms" (kΩ).
- Convert the resistance values to illuminance in lux using Equation 1 (hold your mouse cursor over the equation to magnify it). Note: This equation is an approximation. See the Variations section to learn more about the source of Equation 1.
Equation 1: where- E is the illuminance in lux
- R is the resistance in kilo-ohms (so make sure to convert to kΩ before using Equation 1 if you took your readings in Ω)
- Make a graph of illuminance versus distance. Does the relationship follow the inverse square law? Advanced students should generate a best-fit curve and determine the R-squared value.
Ask an Expert
Variations
- How does the resistance of the photoresistor change with distance? Make a plot of resistance versus distance. What is the mathematical relationship between resistance and distance? If you just glance at the graph, it might appear quadratic (resistance increases with distance squared), but is that actually the case?
- Equation 1 is derived from the graph on the photoresistor's datasheet.
- Can you derive Equation 1 yourself using the graph? This will require knowledge of log-log plots.
- The graph only shows a calibration range for the sensor from 1–100 lux. The readings in your experiment may have exceeded this range. Can you be sure that the calibration curve is still linear (on a log-log plot) at higher or lower illuminance values? Use a lux meter to create your own calibration curve for the photoresistor over the range of illuminance values from your experiment. Does it match the graph from the datasheet? You can also do more research into the typical relationship between resistance and illuminance for photoresistors and examine the shape of the curve. What does this say about the validity of your results using Equation 1?
- Do your results change if you do the experiment with different types of lightbulbs; for example, incandescent, compact fluorescent (CFL), or light-emitting diode (LED)? What about lightbulbs of different color temperature (for example "warm white" or "daylight"—look on the lightbulb's packaging to find the color temperature)? Does the wavelength of light emitted appear to have an effect on the inverse square law?
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