As we move away from a light source, the illumination from that source decreases. If the light comes from a single point (which it can't) or from a source which is nearly a point, and is allowed to spread out equally in all directions, then we can imagine the light spreading out over concentric spheres. The light will be more spread out further from the center, since further from the center it is spread over larger spheres.
We can easily measure the illumination from a light source at various distances, using an illumination meter and a meter stick. This device is very simple to use.
Be sure the meter is on the highest scale (the '500' scale). Set up the meter by plugging in the photosensor, with the red plug on positive (+) and black plug on negative (-).
When using the meter be sure the scale is high enough that the light you are measuring doesn't 'peg' the meter, causing it to go all the way past the end of the scale.
Be sure the room is dark enough that you aren't detecting a significant amount of light from anything but the source. As a less desirable alternative, measure the illumination of the ambient light in the room and subtract it from your readings.
Using a meter stick and the illumination meter, obtain approximately six illumination vs. distance data points for the clear bulb.
Analyzing dataYour distances should be fairly evenly spaced, and the minimum illumination measured should be less than 1/10 of the maximum.
Be sure the distance is measured accurately from the center of the source (the filament of the bulb) to the photosensor.
Read the meter as accurately has possible.
Author and Plot
Author your data in the form of a y vs. x data set, with y being the illumination and x the distance.
Plot your data and speculate on whether the data is polynomial, exponential, quadratic, or power-function in nature.
Attempt quadratic, polynomial, exponential and power function fits
Attempt to fit your data with a quadratic function of the form ax^2 + bx + c. How well does it fit?
Attempt a fit with a cubic polynomial a x^3 + b x^2 + c x + d and assess how well your model fits.
Repeat with a quartic polynomial a x^4 + b x^3 + c x^2 + d x + e.
Attempt to linearize your data with a ln(y) vs. x transformation and obtain the best fit to your data. Use the inverse transformation to 'unlinearize' your data. What model do you obtain? How well does it fit your data?
Attempt to linearize your data with a ln(y) vs. ln(x) transformation and obtain the best fit to your data. Use the inverse transformation to 'unlinearize' your data. What model do you obtain? How well does it fit your data?
Evaluate your models and compare with data characteristics
You have obtained exponential, power-function and polynomial models. Of the models which seems to work best?
What characteristics of light illumination vs. distance from point source would lead you to conjecture an exponential model?
What characteristics would lead you to conjecture a power-function model?
What characteristics would lead you to conjecture a quadratic model?
Using the ModelUse your model to answer the following questions. Use DERIVE to solve the necessary equations:
1. What illumination would be predicted by each model for distances of 9 cm, 17 cm, 37 cm and 73 cm?
2. At what distance would your each model result in the following illumination values: 500, 400, 300, 200, 100, 50, 25, and 10 foot candles?
3. For the quadratic, exponential and power-function models, work out by hand the illumination at the 17 cm distance and the distance at which illumination would be 100 foot candles.
4. Use DERIVE to make plots to give approximate answers to the first two questions for your best model.
A proportionality modelImagine a sphere of radius 1 meter surrounding the light source. The total intensity of the source is spread out over this sphere.
Now imagine that the surface of the sphere is covered with tiny square tiles. Then imagine the sphere expanding to double its radius, as the tiles expand to accomodate the inflating sphere. What happens to the area of the tiles? And what therefore happens to the illumination of a tile?
Each tile doubles in length and in width. It therefore quadruples to 4 times area. Since the light is spread over 4 times the area, the illumination will become 1/4 as great.
This image leads us to the conclusion that the area of the sphere is proportional to the square of its radius, with the illumination proportional to 1 / the square of its radius. That is, we have an inverse square function y = k / x^2, or y = k x^ (-2).
Use one of your data points to evaluate the proportionality constant k. Compare the resulting function with the results of your curve fitting exercise.
Use two more data points and average the proportionality constants obtained for the three points. Compare the function you obtain for this average k value with the results of your curve fitting exercise.
A line sourceA 'light stick', a relatively long flourescent bulb, provides something like a long straight-line source.
Repeat the above experiment, measuring illumination from the closest possible position out to about 10 cm from the center. Be sure all your distances are measured from the center of the bulb.
What is your best model, and how might you explain the reason for this model in terms of proportionality?
