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EJERS, European Journal of Engineering Research and Science
Vol. 2, No. 11, November 2017
DOI: http://dx.doi.org/10.24018/ejers.2017.2.11.517 23
Abstract—Many concepts in the physics curricula can be
explained by the inverse square law. Point-like sources of
gravitational forces, electric fields, light, sound and radiation
obey the inverse square law. This geometrical law gives the
ability of unifying educational approach of various cognitive
subjects in all the educational levels. During the last years we
have been using engaging hands-on activities to help our
students in order to understand the cohesion in Nature and to
export conclusions from experimental data. The development
of critical thinking is also stimulated by student ‘s
experimental activities. Teaching students to think critically is
perhaps the most important and difficult thing we do as science
teachers. In this paper three activities are described, which
were executed by students. These activities are concerning the
electromagnetic radiation and the main goal is to confirm the
inverse square law. We used three activities entitled as:
“Inverse Square Law-Light”, “Photometer construction” and
“Radioactive source”. The significant motive for this work
constituted the following question: “Is it possible to find lab
activities which bring out unification and a non-piecemeal
description of physical phenomena, helping students to think
critically?”.
Index Terms—Inverse Square Law; Light; Photometer;
Radiation; Science Teaching.
I. INTRODUCTION
The radiation intensity from a point-like source with
unlimited range, which effects in all directions, in a specific
distance r is equal to the quotient of the power to the surface
of an imaginary sphere with radiant r.
In the following figure, I is the intensity in r distance, that
corresponds to a surface A. At a 2r distance the same
amount of energy pass through the surface 4A. So the
intensity becomes I/4 etc.
Fig.1. A specified physical quantity or intensity is inversely proportional to
the square of the distance from the source of that physical quantity.
Published on November 27, 2017.
Nikolaos Voudoukis is with Department of Electrical and Electronic
Engineering Educators, School of Pedagogical and Technological
Education (ASPETE), Athens, Greece (e-mail: nvoudoukis@aspete.gr).
Sarantos Oikonomidis is High School Principal at Ralleio Geniko
Lykeio Thileon Pirea (e-mail: sarecon@gmail.com).
Therefore, the power is proportional to the inverse square
of the distance. Being strictly geometric in its origin, the
inverse square law applies to diverse phenomena. Newton's
law of gravity, Coulomb's law for the forces between
electric charges, light, sound and radiation obey the inverse
square law. This geometrical law gives the ability of
unifying educational approach of various cognitive subjects
in all the educational levels. This paper describes simple
experiments that verify the inverse square law.
Students know intuitively that intensity decreases with
distance. A light source appears dimmer and sound gets
fainter as the distance from the source increases. The
difficulty is in understanding why the intensity decreases as
1/r2 rather than as 1/r or 1/r3, or even as 1/ √r, where r is the
distance from the source.
In a recent paper [1] it is shown how to obtain the
inverse-square law of the distance to the light intensity
emitted from a small source in a simple, fast and with good
precision way. In another recent paper P. Papacosta and N.
Linscheid describe a simple experiment that verifies the
inverse square law using a laser pointer, a pair of diffraction
gratings, and a ruler [2].
The development of critical thinking (CT) is widely
claimed as a primary goal of science education [3]. A
method for development of critical thinking skills is the
Socratic questioning method. Its implementation provides
opportunity to help students in appropriate manner to
understand concepts and phenomena. The development of
critical thinking is also stimulated by student ‘s
experimental activities. For the educational approach of the
different actions that take place in this paper, we suggest the
educational model that includes the following steps: 1.
Trigger of interest 2. Hypothesis expression 3. Experiments
– Measurements, 4. Formulation of conclusions and
proposals - recording 5. Generalisation - feedback – control.
It is an important part of learning that a person sees and
engages a concept several times before mastery is attained
[4]. This is very useful in clarifying concepts, as well as
when predicting the course of the experiment and its
subsequent explanation. An example is the inverse square
law.
II. EXPERIMENTS
A. 1st Experiment: Inverse square law – Light
1) Materials
A cardboard with grid, a cardboard with a hole,
supporting clips, ruler, candle.
Students set the device shown in the following picture so
that the cardboard with the hole to be at the middle of the
distance between the candle and the cardboard with the grid.
Inverse Square Law for Light and Radiation: A Unifying
Educational Approach
Nikolaos Voudoukis, and Sarantos Oikonomidis
EJERS, European Journal of Engineering Research and Science
Vol. 2, No. 11, November 2017
DOI: http://dx.doi.org/10.24018/ejers.2017.2.11.517 24
They observe and they count the lighted squares on the
cardboard with the grid.
Fig. 2. The apparatus used in the 1st experiment.
Fig. 3. The 1st experiment.
We can make, for example, the following questions to the
students for hypothesis expression from them.
What do you think will happen if we redouble the
distance between the first cardboard with the hole and the
second one with the grid?
When the distance between the candle and the hole is
equal to the distance between the hole and the cardboard
with the grid, how many squares are lightened?
2) Procedure
1) Keep the distance between the bulb and the card with
the 1 cm square hole constant at 10 cm. Put the bulb at
different distances from the graph paper and count how
many squares on the graph paper are lit at each distance.
Record the number of squares illuminated in the data
table. (Comment: Be sure to measure the distance from
the bulb, not the card.)
2) Measure the size of the squares in the graph paper to
determine the area of each square. If you use the graph
paper provided with this activity they should be 1/2 cm
on a side, and thus each has an area of 1/4 cm2.
Calculate the area illuminated at each distance
measured, and record it in your data table.
3) The amount of light received per area is called
brightness. The amount of light given off by the bulb
and passing through the hole in the card always remains
constant. So, what we want to calculate is the brightness
relative to some standard brightness (say the brightness
of the bulb on the graph paper at 10 cm). We call
brightness B, Area A, and the amount of light (also
called power or luminosity) L, and we can write the
following:
B = L/A for any distance and B0 = L/A0 for the standard
distance (10 cm)
So relative brightness is B/B0 = A0/A (L cancels out
because it is the same for both)
But, at a distance of 10 cm the area illuminated was 1 cm2
So, A0 = 1 and we have B/B0 = 1/A
Calculate the relative brightness for each distance, and
record it in your data table.
TABLE I: DATA TABLE OF THE 1ST EXPERIMENT
Distance from
bulb (cm)
Number of
squares
illuminated
Area
illuminated
(cm2)
Relative
brightness
(cm-2)
10
4
1.00
1
13
6.7
1.68
0.6
15
9.2
2.30
0.43
17
11.5
2.88
0.35
20
16.5
4.13
0.24
23
22.2
5.55
0.18
25
26
6.50
0.15
27
28.5
7.13
0.14
30
36.5
9.13
0.11
Using the data from the above table students can make the
graph of relative brightness vs distance (data as points and
plotting the theoretically line). As a conclusion we have that
the relative brightness should obeys the low B/B0 = k/ r2.
(Comment: The constant of proportionality is k = 1/100,
because for r = 10 cm, A = 1 cm2)
B. 2nd Experiment: Photometer construction
1) Materials
Two paraffin blocks, ruler, two similar lightings, four
lamps and aluminium foil.
Building a photometer. Verification of the inverse square
law for the light. The aim is to create a photometer and to
verify the relation between the power of light and distance.
2) Procedure
1) Put the aluminium foil between the two pieces of
paraffin.
2) Put the two lamp holders in one-meter distance between
them.
3) Both lamp holders have lamps of 100W. Close all the
other lightings and put the photometer between the two
lamp holders so that the two pieces of paraffin have the
same luminosity.
4) Fill the data table.
5) Replace one lamb of 100W with another of 75 W and
repeat the second and the third steps.
6) Repeat the second and the third steps with other
combinations of lamps and we fill the table.
7) Check if the data (measurements) follows the inverse
square law.
EJERS, European Journal of Engineering Research and Science
Vol. 2, No. 11, November 2017
DOI: http://dx.doi.org/10.24018/ejers.2017.2.11.517 25
Fig. 4. Description of the 2nd experiment.
Fig.5. The photometer with aluminum foil between two pieces of paraffin.
What the two paraffin pieces will look like if they receive different amounts
of light.
Fig.6. The photometer lightened with the two lightings. How the two blocks
of paraffin will appear if they receive equal amounts of light.
Fig.7. The 2nd experiment.
TABLE II: DATA TABLE OF THE 2ND EXPERIMENT
P1(W)
P2(W)
P1/P2
d1(cm)
d2(cm)
d1/d2
100
100
1/1
50
50
1/1
75
100
3/4
46
54
46/54
40
100
2/5
39
61
39/61
40
75
8/15
43
57
43/57
P1: power of Lamp1
d1: distance between lamb1 and photometer
d2: distance between lamb2 and photometer
The same luminosity means the same intensity I of light
incident on each one of the paraffin blocks. If the intensity
I= k / r² (k a constant depends on source accordingly from
its power).
For the second case (P1=75W, P2=100W), P1/P2=3/4.
From the experimental data it emerge that intensities are
equal at distances d1=46cm and d2=54cm. When the
intensity is the same on both paraffin blocks (as shown in
Fig. 5) then these two intensities can be put into an equation.
So we have:
Ι=k/(d1)² Ι=k΄/(d2)² k΄=3/4 k
(d1/d2)2 = (d1/d2)² = (46/54)²=0.73 ~ ¾
The same is for the other two cases.
(d1/d2)² = (39/61)²=0.41~2/5
(d1/d2)² = (43/57)²=0.57~8/15
Thus the law is verified.
(Comment: There is an error of about 7%. The theoretical
reading of the ratio of the two intensities (using light bulbs
of 40W and 75W) should be 0.53 and not 0.57 as measured.
One reason for this is that the students did not take into
account the fact that the overall luminous efficiency (% of
light energy/heat) of incandescent light bulbs changes with
the wattage of the bulb. For example, a 40W tungsten
incandescent light bulb has a luminous efficiency of only
1.9% (only 1.9% of its 40W power is converted into visible
light). For a 60W light bulb the luminous efficiency is 2.1%
and for a 100W light bulb is 2.6%.)
C. 3rd Experiment: Radioactive source.
1) Materials
Radio-active Cobalt-60 5μCi, Geiger-Müller, ruler.
The inverse square law in a radioactive source of gamma
rays, using a Geiger- Müller is studied. The aim is to
ascertain the validity of the law also in electromagnetic
radiation that emits from radioactive sources.
2) Procedure
1) Record the measurements from the Geiger-Müller
for two minutes.
2) Repeat the measurement four times and calculate
the mean rate per minute.
3) Rotate the tube of the meter 900 (it is to eliminate
any effect of Alpha and Beta particles that may distort the
reading of the Gamma rays) and repeat 2 and 3 steps.
4) Compare the results from the different directions of
the meter. This is the stand radioactivity.
5) Put the Geiger 8 cm away from the source.
6) Measure for every minute.
7) Repeat step 3 for 16 cm, 24 cm and 32 cm.
8) Check if the data (measurements) follows the
inverse square law.
Using the Geiger Muller we took stand radioactivity
measurements for two minutes in two vertical directions.
There has been taken five different measurements in each
direction.
Continuously we used a radioactive source Cobalt-60
5μCi and took five measurements for 2 minutes period in
two different distances 20 cm and 40 cm. The data
confirmed satisfactory the inverse square law.
TABLE III: DATA TABLE OF THE 3RD EXPERIMENT
Intensity I
(lux)
Distance r
(m)
160
0,42
140
0,50
EJERS, European Journal of Engineering Research and Science
Vol. 2, No. 11, November 2017
DOI: http://dx.doi.org/10.24018/ejers.2017.2.11.517 26
130
0,52
120
0,56
100
0,62
87
Ο,67
80
0,70
60
0,84
40
1,12
30
1,40
Measurements are with the radioactive source of Cobalt-
60 5μCi without the background radiation.
Fig. 8. Graphic plot of intensity (I) vs distance (r).
As a conclusion we have that the intensity of gamma rays
radiation decreases as we go away from the source of
radiation and obeys the low I=k/r2.
Safety and technical notes: Note that 5μCi is equivalent to
185 kBq. Cobalt-60 is the best pure gamma source.
However, students can use sealed radium source. This gives
out alpha, beta and gamma radiation. Students can use it for
this experiment by putting a thick aluminium shield in front
of it. This will cut out the alpha and beta radiations. An
alternative is to try using a Geiger-Muller tube sideways.
The gamma radiation will pass through the sides of the tube
but alpha and beta will not.
III. GENERALIZATIONS
For generalizations we can apply the following subjects
for further study:
1) The inverse square law for gravitational and electrical
forces and it’s relation to the gravitons and photons
respectively.
2) The magnitude of a star. When the Absolute Magnitude
of a star is known (as in the case of standard candles)
then the distance to such a star can be calculated by the
use of the Inverse Square Law. (As Edwin Hubble did
in 1924 and 1929. He used the Luminosity – Periodicity
law of Cepheid stars discovered by Henrietta Leavitt.).
The Inverse Square Law is a powerful tool for
astronomers that help to calculate distances to stars and
galaxies near and very far away (using Supernovae of
the Ia type).
3) The absolute magnitude of a star.
4) The inverse square law for sound. The sound intensity
from a point source of sound will obey the inverse
square law if there are no reflections or reverberation.
IV. ASSESSMENT
By the end of the activity students should be able to [5]:
• Explain what the inverse square low is.
• Identify the mathematical expression of an inverse
square low.
• Describe an experiment for checking the inverse square
low for the light.
• Do a quick mathematical check for given data (e.g. by
doubling and tripling the distance and seeing if the data
follows an inverse square law by dropping to a quarter
and a ninth).
• Predict a measurement (comparatively) for a given
distance from the source.
• Predict the gravitational and electrostatic forces
between objects.
The intervention was performed on high school students
(17years old) in Athens, Greece during the school year
2016-2017. The number of students participating in this
study was forty seven (47) students - two (2) classes, one of
twenty four (24) students and the other of twenty three (23)
students - divided in sixteen (16) teams of three (3) students
each (there was one team of two students). For the
assessment of the proposal they took pre, post and final
tests. We find that the quality of the students' reasoning
about the inverse square low is improved by this approach.
A comment of a student summarizes the main attitude of
all students “These activities were particularly interesting
and helped us to better understand the concepts learned. All
showed interest. I think it is good all students to learn in
this way.”
V. CONCLUSION
The activities used to teach students the inverse square
low support a unifying approach for this low. The unifying
approach enhances learning, helping students to think
critically. The development of critical thinking is stimulated
by student ‘s experimental activities which lack strict
instructions.
The experiments are carried out by students and can be
used for supporting the teaching of the inverse square low in
an inquiry- based approach, as well as helping students to
approach the nature of science by guiding them to realize
the relationship of experiment and theory in scientific
investigations and also the way scientists work.
Our didactical approach seems, from the assessment, to
be quite encouraging and we suppose that it is appropriate
not only for high school students. We think that it will be
beneficial and for non-major science university
undergraduate students too.
REFERENCES
[1] L. Pereira Vieira, V. de Oliveira Moraes Lara (2014). Dayanne
Fernandes Amaral, “Demonstration of the Inverse Square Law with
the aid of a Tablet/smartphone” Physics Education, 2014.
[2] P. Papacosta, N. Linscheid, “The Confirmation of the Inverse Square
Law Using Diffraction Gratings” Phys. Teach. 52, 243. 2014
[3] Bailin, S. “Critical thinking and science education” Science &
Education, 11, 361–375, 2002.
[4] A. B. Arons, Teaching introductory physics, NY: Wiley, 1997.
[5] C. Gipps. Beyond Testing. Towards a theory of educational
assessment. London. Washington, D.C.: The Falmer Press, 1994
EJERS, European Journal of Engineering Research and Science
Vol. 2, No. 11, November 2017
DOI: http://dx.doi.org/10.24018/ejers.2017.2.11.517 27
Nikolaos Voudoukis received a BSc degree in
Physics from Athens National University, Greece, in
1991, a BSc in Electrical and Computer Engineering
from the National Technical University of Athens,
Greece, in 2012, his MSc degree in Electronics and
Telecommunications from Athens National
University, in 1993, and his PhD degree from Athens
National University, in 2013. He has worked as
telecommunication engineer in Greece. Dr.
Voudoukis now is Assistant Director at a high school
and a part-time Lecturer at the School of Pedagogical & Technological.
Education, Athens, Greece.
Sarantos Oikonomidis received a BSc degree in
Physics from University of Patras, Greece in 1983,
his MSc degree in Physics Education from Athens
National University, in 1993, and his PhD degree
from Athens National University, in 2010. Dr
Oikonomidis is High School Principal at Ralleio
Geniko Lykeio Thileon Pirea.
