# Investigation Light Intensity Decay Over Distance

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```					Why Is Alaska So Cool?                                       Angle Intensity Investigator

Activity 7: Angle Intensity Investigator
Investigations in Light Intensity Change Through Angular
Displacement

Guiding Question
How does the intensity of light change as the angle of incidence to the light source
increases?

Prediction of Results
Predict what you think will happen and what type of equation and graph might best fit
the data representing the intensity of a light as the angle of incidence increases.

Objective
After completing this lesson, a student should be able to analyze light intensity striking
a surface at varying angles of incidence.

Materials
CBL, TI-83 Plus calculator, light sensor, lamp with light bulb (60 to 100 watts), tape
measure, Scotch tape (or similar), protractor, sheet of paper, ANGLE program, daily log

Vocabulary
the normal   angle of incidence   periodic      sinusoidal regression

Introduction
You have completed an investigation of light intensity changes with varying distance.
You will continue investigating intensity changes, but now you will note the change of
intensity as light strikes a surface through increasing angles of incidence. As Figure 1
indicates, the angle of incidence is the angle at which light strikes a surface. The normal
is an imaginary line perpendicular to the plane on which light strikes. The angle of
incidence is the angle between the normal and an incident (that is, an incoming) light
ray.

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Angle of incidence

Normal
Incident Light
Ray

Figure 1

As the angle of incident light increases, is it more reasonable to think that its intensity
more likely to get a sunburn in the late evening when the sun is low on the horizon
(greater incidence angle) or at noon time when the sun is more overhead (smaller
incidence angle), as sketched below? Is it usually hotter at noon or in the late evening?

Evening Sunlight                                         Noon Sunlight

If you suspect that intensity decreases with increasing angle of incidence, you are on the
right track. Assuming for the moment, then, that the light intensity (I) decreases as the
angle of incidence (a) increases, we can say the relationship between intensity and
incidence angle are inversely proportional over the angles involved.

But you may note that as the sun rises through the morning and then sets toward the
evening, its angle of incidence decreases, reaches a minimum at noon, and then begins
to increase again into the evening. That is, the incidence angle starts at sunrise at a
maximum of 90, decreases to a minimum until noon (near the equator it decreases to
0, but not in Alaska), and then increases back to a maximum to 90 at sunset. You can
see that this cycle repeats itself time and again: the incidence angle begins at a
maximum, then decreases to a minimum, increases to a maximum, and so forth. Figure
2a below shows just such a cycle. Intensity, however, changes as the inverse of the
incidence angle, so is represented in the Figure 2b cycle.

Morning                                         Morning
Noon                                         Noon

Evening                                Evening

Figure 2a – Changing                         Figure 2b – Changing
Incidence Angle                               Light Intensity
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This cycle of repeating events is called periodic and you might recognize the curves as
sine waves. Here is an equation that describes intensity as a sine wave:

I = sin(a) (intensity is proportional to the sine of the incidence angle, a).

It is important to remember in this comparison that the angle with respect to the sun
will always be between 90 and 0. This means that if we determine one quarter of the
period (or cycle), then we can know through regression analysis what the rest of the
curve should look like. You will complete an investigation that does exactly this:
determine the intensity of a light as the angle of incidence changes from 0 to 90.

Prelab Prep
1. Please gather the materials indicated at the beginning of this lesson. You will also
use the ANGLE program. This program may be present in your TI-83 calculator.
If it is not, you may download it from this web site.
2. Take a sheet of 8 ½ in. x 11 in. paper and a protractor. Position the paper so that
it is in “landscape mode,” that is, so that it is turned at a right angle from the way
you might usually write a letter. From the top left hand corner, draw lines on it at
the following angles: 0, 10, 20, 30, 40, 50, 60, 70, 80, and 90. Note that 0
and 90 are actually the top and left edges of the sheet, respectively. Label each
line with its associated angle. You will have 10 lines in all (counting the top and
left edges as lines). The sketch below is not to scale and does not include all the
angles, but gives an idea of the way the paper should look when you finish.

10

90                 50

Equipment Setup Procedure
Note: In the associated figures below, for clarity none of the sketch elements is to scale.

1. Place the flashlight on a table or other suitable level surface, and in a position similar
to that of your previous intensity versus distance investigation.
2. Stretch out the tape measure so that the measuring tape forms a (straight) line
beginning at the bulb to about 1.5 m away from the bulb. Use Scotch tape (or
similar) to secure the ends of the tape measure, and also the 0.5 m mark.
3. Place the sheet of paper carefully along the tape measure with a corner at the 0.5 m
mark, as in Figure 2. Tape the paper to the table so that it does not slip.

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Angle Intensity Investigator                                   Why Is Alaska So Cool?

Tape measure

0.5 m

Figure 2

4. Place the light sensor along the edge of the paper and tape measure with the front of
the light sensor barrel exactly at the 0.5 m mark. The light sensor should be in a
direct line with the light, as in Figure 3.
Light Sensor

Figure 3

5. For the entire procedure, as you pivot the light sensor along various angles, always
keep the front of the barrel at the 0.5 m mark so that light may strike the photocell
inside the sensor, as Figure 4 indicates.

Figure 4

6. Turn on the light.
7. Connect the CBL, TI-83 Plus calculator, and light sensor as is customary to do.

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8. Turn on the CBL and the calculator.

The CBL is now ready to receive commands from the calculator. Some of the next

Experiment Procedure
1. Make sure the CBL and the calculator are turned on.
2. On the calculator, press PRGM to see the programs available to you. You will see a
screen similar to the next one (you may have other programs also):

3. Choose the ANGLE program by pressing the up/down (         ) cursor keys, and
pressing the calculator ENTER key when the cursor is on ANGLE. You will see the
screen below:

4. Press ENTER ENTER , the following screen shows up:

5. Choose 1: Collect Data by pressing ENTER ENTER , and follow the set up
instructions through the next several screens as you have done before.
6. When the screen says STATUS: OK, continue by pressing ENTER .

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7. The next screen is as follows. Do what it says, if you have not done so already, and
press ENTER .

8. Follow the next screen’s instructions, and press ENTER .

9. And now, you are almost ready. Press ENTER , the next screen shows as follows:

10. You are now ready to begin data collection. Ensure once again that your sensor is at
the proper angle with respect to the light. You will take a total of ten samplings at
angles increasing in 10 increments beginning with 0 (i.e., 00) going through 90.
When you are ready, press ENTER .

11. At this point you did not notice it, but when you pressed the Enter button, the CBL
instantly sampled the light intensity three times, averaged the samplings, and stored
its value in the calculator. The next screen pops up:

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Notice that the calculator screen shows TRIAL 2, and you are instructed to move the
sensor to the 10 angle. Follow the instructions. As you continue to press ENTER
on the calculator, the calculator will prompt you at each subsequent trial to move the
light sensor and to ready you for each of the subsequent samplings. Continue to
follow the instructions on the calculator screen now.

12. After Trial 10, you are presented with a screen similar, but not identical, to the
following:

Figure 5 – Intensity vs. Angular Displacement
You have completed collecting your data, and are now ready to analyze the data. You
may turn off the CBL and disconnect it from the calculator. Your data are stored in the
calculator so that you may turn the calculator off with the data safely inside. If you do
this investigation again, you will find that new data replaces the older data.

Data Record
Data Table: Observations
You noticed that after the data were collected, a plot of light intensity (mW/cm2) versus
angle degrees appeared on the calculator screen. The plot looked similar to the one
shown in Figure 5.

The calculator has stored the distance and light intensity values in its data tables. To
view these values, turn on the calculator. Press the STAT button, and the next screen
shows up:

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Choose EDIT and 1:Edit. To do this, simply press ENTER . The screen that shows up is
similar but not identical to the following.

The column L1 shows the incident angles in degrees from the light bulb. Column L2
shows the light intensity values in milliwatts per square centimeters (mW/cm2). (Other
columns, such as L3 are of no interest in this study.) Though your L1 will be the same as
the screen shot above, it will surely not be the same as L2. The paired values are in
rows, so that in the above investigation (not yours), the light intensity at 0 was 0.89103
mW/cm2, at 10 was 0.8825 mW/cm2, and so on.

You may scroll through the values using the up/down cursor keys.

Mathematical Analysis
You will conduct now a regression analysis. Recall that a regression analysis is an
attempt to develop a mathematical equation that best describes the data points
collected. Note: To review how to do regression analysis with the calculator, refer to
Activity 1: TI-83 + DATA = MODEL, Introduction to Computational Science and
Mathematical Modeling Using the TI-83 Plus Graphing Calculator.

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As you have seen in other investigations, several types of mathematical equations may
be used as models. Remember our original equation, I=sin(a). You need to find a
regression that is based on an equation of that type; it is called a sinusoidal regression.

To do this, press STAT       then scroll down using the down arrow (         ) repeatedly
until SinReg is highlighted. Press ENTER .

The calculator returns to the home screen with the regression called SinReg ready to be
invoked. Since you want to use the values in the data tables L1 and L2 for the regression
analysis, you need to enter those tables.

To do this, press 2nd [LI]  2nd [L2] ENTER .

After some moments, your calculator will render an equation of a sine wave:
y=a*sin(bx+c)+d. It will be similar, but with different values, to the screen below.

Note the coefficients a and b, and the constants c and d. These are constants for your
particular wave form, and address the amplitude, the vertical and horizontal
displacements of the wave. Does this equation agree with the mathematical model
relating intensity and angle that was described above in the introduction section to this
experiment? Yes. The equation is of the same type, though the constants are different
from our earlier equation, I=sin(a). (There is no r2 value computed by the calculator for
a sine regression.) Record the sinusoidal regression equation and coefficients in your
daily log.

Note: If you have difficulty running the sine regression or get an error message on your
calculator something like SINGULAR MAT, then your calculator is having difficulty
finding a solution for your data. You can run the program again with a sample data set
to perform the regression equation. To do this, press PRGM 1:ANGLE, then on the
****OPTIONS**** screen, choose 2:USE SAMPLE instead of 1:COLLECT DATA. You
will then have a data set that should allow you to see a meaningful regression equation
and graph.

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Angle Intensity Investigator                                   Why Is Alaska So Cool?

Now, you will graph the equation itself, superimposed on the data points you collected.

Press Y= and move the cursor to the first available function. Press VARS 5
ENTER to copy the regression equation to the cursor position. Your screen should now
look similar to the next screen shot, but with your values.

Press GRAPH to see the scatter plot and regression curve together. How well do they
match? They should be fairly close as the following screen demonstrates. (Note: if the
graph does not look like a curve similar to the one below, your calculator may need to
be set to radian mode instead of degree mode. To do this, press MODE , then move
the cursor to highlight RADIAN and press ENTER , then GRAPH .)

If you zoom out one step ( Z00M 3 ENTER ), you may see a further extension of the
sine wave that includes your data points. This would imply that as the angle continues
to increase beyond 90 that the light’s intensity would now begin to increase again.

In fact, in the real world the graph would have limits at 90 before noon and 90 after
noon; that is, we could say the reasonable interval is between -90 and 90 inclusive.
Consequently, set the window view so that you can see only the portion of the graph
that has real world meaning.

To do this, press WINDOW and enter the following values in the next screen shot (use
the cursor keys and press ENTER after typing each entry):

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When your Window screen looks like the one above, press GRAPH . Your model
should look similar to the one below. Here is the section of the model that has real
world meaning for us:

You can see the section of the model now from -90 through 0 to 90.

Enter and Analyze Data in Excel

Enter the data into two columns in an Excel worksheet. This will serve two purposes: it
will be a way to record and save the data from each of your experiments, because the
data in the calculator will be replaced with new data for each experiment you perform,
and it will give you an opportunity to graph the data in Excel and print the chart.

The first column should include the angle of incidence in degrees. The second column
should include the intensity of the light, measured in mW/cm2. Follow the procedure
outlined in Activity 2: Excelling with Data, Mathematical Modeling with Excel to create a
scatter chart of the data, but this time choose the middle chart sub-type, “Scatter with
data points connected by smoothed lines” as shown below.

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Angle Intensity Investigator                                     Why Is Alaska So Cool?

You will not be able to add a trendline to the chart that is a sinusoidal regression,
because Excel does not offer this as an option.

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Why Is Alaska So Cool?                                        Angle Intensity Investigator

Concluding Questions
1. What is your sinusoidal regression equation and correlation coefficient? Record
these values in your daily log as well. Use labels or descriptions appropriately.
Include the equation generated by the TI-83 as well as that generated by Excel.

2. Did you accurately predict the type of equation and graph that would best
represent the intensity of light as the distance from the light source increases? If
not, what type of equation did you predict?
3. Using any of the analytical methods discussed elsewhere (evaluating for
function, creating a table of values, tracing the graph), determine the intensity at
35. Which method did you use?
4. Consider carefully: We stated earlier that in the real world, it is unreasonable to
associate increases of angle of incidence beyond 90 with further changes of
intensity. Why is this unreasonable? (Hint: If you put the light sensor at 100,
what would be the angle of incidence?)
5. You examined intensity varying inversely with the angle of incidence, based on a
sinusoidal mathematical model. A point of interest is that sunlight strikes the
earth at varying angles of incidence immediately related to varying latitudes. For
example, at noon on the equator, the sun is directly overhead so that the solar
angle of incidence is 0; but at noon in more northerly regions the angle of
incidence increases the farther north one travels. Indeed, north of the arctic circle
in winter, such as at Barrow, the incidence angle is beyond 90 and there is no
direct sunlight at all.
a. What does your investigation of light intensity relative to angle of
temperatures in Mexico in the summertime?
b. Why does it suggest this to you?

Print a copy of the Excel graph to submit.

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