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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.