Where Is 'Full Sun' No Brighter than Twilight?
Abstract
How far would you have to travel so that the light of the full sun would provide "daylight" no brighter than twilight on Earth? 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. When you have finished your experiment, you can use your results to calculate an answer.Objective
The goal of this project is to use an electronic photosensor to verify that the inverse square law applies to light. In other words, verify that intensity of light decreases in proportion to the square of the distance from the light source.
Credits
Written by Richard Blish, Ph.D.
Edited by Andrew Olson, Ph.D., Science Buddies
Thanks to David Aziz, Ph.D. for helpful suggestions on photocell measurement methods.
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Introduction
As you move away from a light source, the light gets dimmer. No doubt you've noticed this with reading lamps, streetlights, and so on. Figure 1 shows what is happening with a picture. 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 (= 2^{2}) squares. Thus, at twice the original distance, the intensity of the light passing through a single square will be 1/4 of the original intensity. Going out still further, tripling the original distance (3r), and the light from the original square now covers an area of 9 (= 3^{2}) 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/r^{2}, 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.
Figure 1. An illustration of the inverse square law. Image credit Wikimedia Commons user Borb.
But don't 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 an inexpensive photosensor and digital multimeter to see if light intensity really does decrease according to the inverse square law.
You will use cadmium sulfide (CdS) photocells to measure light intensity (see Figure 2). CdS photocells act as lightsensitive resistors. The squiqqly stripe across the face of the photocell is the lightsensitive part. As the intensity of the light falling on the stripe increases, the resistance of the photocell decreases.
Figure 2. A selection of CdS photocells.
One way to do the experiment would be to calibrate precisely how much the resistance changes with a given change in light intensity. Such a calibration would be difficult to perform accurately unless you had access to expensive test equipment (which you could use to do the experiment, instead of the CdS photocell, so there wouldn't be much point!). But there is another way to use the photocell that avoids the calibration problem.
The idea is to mask off (cover) a portion of the photocell so that only a fraction of the active area is exposed to light. With the mask in place, you make two measurements:
 the resistance of the photocell, and
 the distance between the light source and the photocell.
Then, leaving the light source unchanged, you remove the mask from the photocell and change the distance between the light source and the photocell until you match the previous reading. Now you are exposing the entire photocell to light. Since you adjusted the distance so that the two readings match, you know that the light intensity falling on the exposed surface of the photocell was equal for each measurement. What was changed between the two measurements was the portion of the sensor that was exposed. All that is left to do is to measure the fraction of the photocell that was left uncovered by the mask, and the ratio of the distances for the two readings.
You can see that the experimental approach is analogous to the star diagram with which we started, above. Think of the squares as representing the exposed area of the photocell. At 3r, a reading was taken with the sensor unmasked. Now if we move closer to the light source, what fraction of the sensor do we need to expose to the light in order to get the same reading? If the Inverse Square Law holds true, the answer is shown at 2r: a matching reading would be obtained with 4/9 of the sensor area exposed. The ratio of the distances is 2/3, and the ratio of the sensor areas is equal to the distance ratio squared. Again at r (if the Inverse Square Law holds true), another matching reading would be obtained with 1/9 of the sensor area exposed. Now the distance ratio is 1/3, and again, the ratio of the sensor areas is equal to the distance ratio squared.
The experimental procedure shows you how to mask off a variable fraction of the sensor area, and also how to measure the area of the sensor that is left exposed. You'll be able to see for yourself if the Inverse Square Law holds true for light!
Terms and Concepts
To do this project, you should do research that enables you to understand the following terms and concepts:
 Inverse square law
 Power law regression fit
 Correlation coefficient
Bibliography
 Exploratorium. (n.d.). Inverse Square Law. Exploratorium Science Snacks. Retrieved May 5, 2014, from http://www.exploratorium.edu/snacks/inverse_square_law/.
 The Physics Classroom. (n.d.). Inverse Square Law. Retrieved May 5, 2014, from http://www.physicsclassroom.com/class/estatics/Lesson3/InverseSquareLaw.
 Here is an Excel tutorial if you need help using a spreadsheet program:
Excel Easy. (n.d.). Excel Easy: #1 Excel tutorial on the net. Retrieved May 5, 2014, from http://www.exceleasy.com.
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Materials and Equipment
To do this experiment you will need the following materials and equipment:
 Digital voltmeter, available at Amazon.com
 CdS photosensors
 2 singleedge razor blades
 Card stock
 Tape
 Tape measure
 Notebook hole punch
 Light source: Maglite^{™}type flashlight works well, (regular flashlight can also be used)
 Computer with spreadsheet program (Microsoft Excel or similar) to plot data and extract regression fit parameters (power law exponent).
 Optional: if you have a feeler gauge (collection of metal shims marked with known thicknesses), this can be a handy way to set the gap width. You do not need to buy one especially for this project.
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I Did This Project! Please log in and let us know how things went.Experimental Procedure
 Fold two pieces of cardstock in half and punch a hole at the seam of each.
 Tape the photosensor in place behind the hole on one card. The photosensitive area (squiggly line) should be visible through the hole.
 Tape a pair of single edge razor blades over the hole in the other card, with a small gap (about 0.5 mm to start) between them. The razor blades should be parallel to one another, so that the gap has the same width along its entire length. The card with the razor blades will be the mask for the photosensor. (This is where the feeler gauge comes in handy if you have one.)
 Fasten the cardstock containing the razor blade slit in front of the light source. Measure the distance from the light source filament to the slit (e.g., 30 mm).
 The slit should be parallel to the hot filament in the clear, incandescent lamp.

To measure the slit width accurately, project an image of the slit on a screen (a piece of paper or cardboard will work fine).
 Position the screen roughly 0.3 meter from the slit and measure the distance from slit to filament.
 Measure the magnified slit width from the divergent rays from the lamp on the paper. (The appropriate distance to measure on the paper is perpendicular to the long direction of the slit.)

The slit width w is the product of the measured distance on the screen (the magnifed width, w_{m}) and the ratio of the distance from slit to filament d_{sf} over the distance from slit to screen d_{ss}. In equation form:
w = w_{m} ×d_{sf} / d_{ss}.
 Figures 3 and 4, illustrate the procedure for measuring the slit width.
Figure 3. Diagram at left shows how projected light magnifies the width of the slit on the screen. The photo at right shows an example of a slit made with razor blades taped to cardstock. Also shown is a set of feeler gauges, which can be used (optional) to set the approximate slit width.
Figure 4. The photo on the left shows the slit attached to the front of the lantern flashlight. The slit image is projected onto a screen about 0.3 meter away. The photo on the right is the same view in the dark, showing the projected image of the slit (a little fuzzy in the photo). Measure the projected slit width and follow the instructions in the text to calculate the actual slit width.

Measure the average total width for the meandering stripe on the front of the CdS sensor. This is the active area of the photosensor (see Figure 5). To calculate the fraction of the sensor that is exposed, you will divide the slit width by the average total width of the sensor.
Figure 5. The meandering stripe on the front of the CdS sensor is the active area. Measure the average width (see white arrows) of the sensor. The long axis of the slit should be aligned with the long axis of the photocell (turquoise line over orange photocell at lower left). The slit should be centered, and its width should not exceed the average width of the sensor. To calculate the fraction of the sensor area that is exposed, divide the slit width by the sensor's average width.
 Position the photocell right behind the slit, with its long axis aligned with the long axis of the slit. In Figure 5, the long axis of the large orange CdS photocell is marked with a turquoise line. This (imaginary) line should be parallel with the long axis of the slit. The slit should be centered on the sensor, and the slit width should not exceed the average sensor width.

Use the digital multimeter to measure the resistance of the masked photocell. If you need help using a multimeter, check out the Science Buddies Multimeter Tutorial. Make sure that the slit is aligned with the length of the filament (as in the photo in Figure 5). Adjust the distance between the masked sensor and the filament until you reach a chosen resistance value (e.g., 50 kohms). Note the distance. See Figure 6 for a diagram of how to make the measurement.
Figure 6. The diagram is a schematic side view showing the setup for measuring the resistance of the masked photosensor. Note that the lamp filament is oriented "endon," parallel to the length of the slit.
 Now unmask the sensor and note the new (longer) distance such that the same resistance is observed. Again, make sure that the slit is aligned with the length of the filament.
 Now change the slit width and repeat steps 6–10. Do this for at least 5 different slit sizes. Slit widths in the range of about 0.5 to 3 mm should work well. (The slit width should not exceed the average sensor width; see Figure 5.)

Make a data table similar to Table 1 (roughly simulated data).
Table 1. Data in blue represent your measurements. Data in red and green are calculated from blue and plotted in Figure 7 as an example. (simulated data).

Use your spreadsheet program to graph the data (linear x and yaxes are fine). Figure 7 shows a sample graph (using roughly simulated data).
Figure 7. Graph of the distance ratio vs. fraction of the masked sensor exposed (simulated data from Table 1). The equation shows the powerlaw regression fit and correlation coefficient (r^{2}).

You can use a power law regression fit to see how closely your data matches the inverse square law.
 To add a power law regression in Excel, click on any datapoint in the series,
 click "Add Trendline",
 click to choose Power Law (rather than linear, polynomial, exponential, etc.),
 click the Options tab,
 click the checkboxes to display the equation for the regression line and its Correlation Coefficient.
 That's it!
 For the simulated data shown, you can see that the power law exponent is 1.9883, close to the value of 2 which you would expect from the Inverse Square Law. The correlation coefficient (r^{2}) is a measure of systematic error, not the inevitable random measurement error. If there were no systematic error, the r^{2} value would be unity (one), whereas if the r^{2} value were near zero, one would be forced to conclude that the x and yvariables are not related in the chosen fashion (power law in this case).
 Now let's get back to the question we pose in the title of this project! How far would you have to travel so that the light of the full sun would provide "daylight" no brighter than twilight on Earth? To find the answer, you'll need to know the relative intensity of full sun and twilight. In units of lux: full sun is roughly 100,000 and twilight is about 10. With what you've learned in this project, how many times the earthsun distance would you have to travel for full sun to look like twilight on earth?
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I Did This Project! Please log in and let us know how things went.Variations
 Advanced. You can use Fisher's Z transformation (available in Excel) to quantify whether the r^{2} value is sufficient to declare a good fit or not. You can even know the percentage degree of confidence. We encourage students to read about R. A. Fisher, a brilliant statistician who invented many valuable techniques when he worked on agricultural crossbreeding in the 1920s.
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