E LEKTROTEHNIŠKI VESTNIK 90(5): 272-276, 2023

O RIGINAL SCIENTIFIC PAPER

The problem of Inverse Square Law and Inverse Square Root

Matej Babič

Faculty of Information Studies, Novo mesto, Slovenia

E-mail: babicster@gmail.com

Abstract. The inverse-square law is the holy grail of understanding the laws of light, and man is a being of light.
Some things are used inversely as the square of something (usually divided) and some things are not. For
example, if a point light source sends light uniformly in all directions, the amount of light energy per second
falling on a small area is inversely proportional to the square of the distance from the source. Double the distance
for the same area and the energy per second will be four times less. Triple the distance and it will be nine times
less. (It should really be part of the area of a sphere centered on the point source so that the distance is
unambiguous. However, a small planar area perpendicular to the black from the point source approximates this.)
Thus, light follows an inverse square law (at least in Euclidean space). In this article, I will present a very old
problem of our civilization: the inverse square law and some solutions to this problem. The normalization of a
vector is a calculation that frequently takes place in programs like graphics. This calls for two pricey operations—
calculating a square root and performing a floating-point division. The approach that computes an inverse square
root very quickly utilizing simpler operations is described in the following. In this article, I will present how the
problem of inverse square law and inverse square root are connected.

Keywords: inverse square law, speed of light.

Problem inverznega kvadrata razdalje in problem

inverznega korena
1 INTRODUCTION
Zakon inverznih kvadratov je sveti gral razumevanja zakonov As energy spreads out from a point, the same energy
svetlobe in človek je bitje svetlobe. Nekatere stvari se passes through a larger and larger spherical surface. The
uporabljajo obratno kot kvadrat nečesa (običajno razdeljen), area of this surface is given by S=4πr2. Thus, the energy
nekatere pa ne. Na primer, če točkovni vir svetlobe pošilja per unit area decreases as 1/r2.
svetlobo enakomerno v vse smeri, je količina svetlobne The one who was the first to extend the paradigm
energije na sekundo, ki pade na majhno površino, obratno management perfectly is the founder of answer theory,
sorazmerna s kvadratom razdalje od vira. Podvojite razdaljo za the Jesuit priest Prof. Dr. Ruđer Bošković. Bošković
isto površino in energija na sekundo bo štirikrat manjša.
assumes that the earth breathes unobserved [1]. As a
Potrojite razdaljo in energija na sekundo bo devetkrat manjša.
(Resnično bi morala biti del območja krogle s središčem na result, our body size constantly changes between day and
točkovnem viru, tako da je razdalja nedvoumna, toda majhno night. During the day, the radiation is greater than at
ravninsko območje, pravokotno na črno od točkovnega vira, se night and we are smaller without being capable of
temu približa.) Torej svetloba sledi inverznemu kvadratnemu discovering the realization. What applies to the
zakonu (vsaj v evklidskem prostoru). V tem članku bom acceleration of light also applies to motion, which is also
predstavil zelo star problem naše civilizacije: zakon inverznega measured in m/s. This indicates a lower speed on the day
kvadrata in nekaj rešitev tega problema. Normalizacija vektorja side where the meter is correspondingly smaller and a
je izračun, ki se pogosto izvaja v programih, kot je grafika. To higher hurry on the night side. As ensues, the Earth will
zahteva dve dragi operaciji – izračun kvadratnega korena in
circle the sun. The inverse square law is the result of the
izvajanje deljenja s plavajočo vejico. Pristop, ki izračuna
inverzni kvadratni koren zelo hitro z uporabo preprostejših assumption that the gravitational field, or the effect it
operacij, je opisan v nadaljevanju. V tem članku bom predstavil produces, propagates around a point source in the form
povezavo problema inverznega kvadratnega zakona in of three-dimensional circles. Since the surface area of a
inverznega kvadratnega korena. sphere, which is 4πr2, is also proportional to the square
of the radius, the emitted radiation spreads farther from
the source to an area that rises proportional to the square
of the distance from the point source.

Received 26 February 2023

Accepted 31 August 2023
THE PROBLEM OF INVERSE SQUARE LAW AND INVERSE SQUARE ROOT 273

unstable emanating from the origin and fuse. The total

number of flux lines hinges on the strength of the light
source and is constant with an increasing distance where
a greater density of inconstant lines (lines per unit range)
degraded a stronger spirit room. The density of unstable
lines is inversely proportional to the square of the
variance from the rise since the exterior area of a sphere
was with the true of the line. Thus the address intensity is
inversely proportionate to the equality of the alienation
from the fountain.

2 THE PROBLEM OF INVERSE SQUARE LAW

Where does the inverse square problem come in?
Figure 1. The curvature of the Earth in the gravitational The inverse square law appears in many places. Thus in
field of the sun the gravitational angle in the electric field (from point
1
sources) the field falls as 2 . Other things that follow the
What is another name for inverse square law? r
inverse square law are gravitational and electrical forces.
The vigor of attraction or repulsion between two On the other hand, gravitational and electric potential
thrillingly charged particles, being directly proportionate follow the inverse law – e.g. double the distance to halve
to the product of the piezoelectric instruct, is inversely the potential.
proportional to the exact contrariety between them. This
is given as Coulomb's justice [2]. What is the square of the distance problem?
Some things vary inversely as the square of something
The problem of the square of the distance is the physical
(usually distance) and some things do not. For example,
measurements that we observe in the same system or
if a point light source sends light out evenly in all
within the system but we cannot observe and measure
directions, the amount of light energy per second falling
them outside the system. The fish is in a round bowl but
on a small area is inversely proportional to the square of
it cannot look at itself outside the bowl physically and
the distance from the source. Double the distance for the
look at ourselves and make measurements from the
same area and the energy per second will be four times
outside to the inside.
less. Triple the distance and it will be nine times less. (In
fact, the area should be part of a sphere centered on the
point source so that the distance is unambiguous, but a
small plane area perpendicular to the line from the point
source approximates this.) Therefore, light follows an
inverse square law (at least in Euclidean space).

Figure 3. Fish cannot move outside of the bowl space

The energy that the transmitter sends to the receiver is

spread both during the transmission of the signal and
during the reflected return over the entire space (which
Figure 2. Visual propagation of light following the geometrically means over the surface of the sphere – the
inverse square law sphere). Therefore, the inverse square for both paths
means that the transmitter will receive energy according
In Figure 2 Source S shows the light spring, while r to the inverse of the fourth power of reach. Therefore,
presents the measured step. The lines represent the modern physics presents a false theory.
274 BABIČ

1
The problem is that we cannot see our space (J = )
r2
1
from outside (J  2). We cannot measure physical
r
properties from outside of our space. An example. Even
a fish cannot look at itself outside the glass aquarium
sphere!

In 1755, Bošković created his epochal discovery. The

ancient law was the inverse square division:

1
m~
r2

Assuming a point mass, the gravitational effect

1
occurs with 2 , which refers to a sphere of radius r.
r

Figure 4. The inverse square for both paths means that

the transmitter will receive energy according to the
inverse of the fourth power of reach.

Figure 6. Gravitational effect

An experiment in which a field source, e.g. a point light
source, is scanned with radius r. The same applies to the
electric field as to the strength of the magnetic field:
1 1
E and H 
r2 r2

Figure 5. Graphical presentation of inverse square law Earth is affected by the solar wind which follows the
electric field strength of the sun. In contrast, the radius
is:
1 1 1
J= r r
r2 √𝐸 √𝐻

r presents the inverse square root.

Proof. Prof. Dr. Konstantin Meyl is a German professor of
electrical engineering with a doctorate in the three-
1
J dimensional non-linear calculation of eddy currents. He
r2
developed a new extended field theory (potential
1 vortex1-5 [3-7]) based on the work of Nikola Tesla and
J
r2 Ruđer Bošković. About the speed of light from E, H 
1/r and r  c follows:
With distance, the intensity decays with the square of the  The field determines the length measures (what
distance (this can be measured and calculated). See is 1 m).
Figure 2.  The field determines the velocities v (in m/s).
 The field determines the speed of light c [m/s].
and
 The measurement of the speed of light is made
1 with itself.
J  A constant of measurement c = km/s is
r2
measured  the speed of light c is not a
constant of nature!
THE PROBLEM OF INVERSE SQUARE LAW AND INVERSE SQUARE ROOT 275

The biggest mistake of modern quantum physics is to 3 SOLUTION

treat the speed of light as a natural constant. This is a
violation of the inverse square of distance law. The solution (1)

Why does the inverse square law work? If we want to prevent energy dissipation during signal
propagation, we can use a one-dimensional (+ circular
Because the surface area of a sphere is equal to the dimension) tube which prevents energy dissipation (loss)
square of the radius. If a certain amount of field, whether on the sphere.
gravitational, electromagnetic or any other, is to be
uniformly distributed over a sphere, the amount per unit Solution (2)
area of that sphere will fall in inverse proportion to the
area of that surface, i.e. inversely with the square of the We must observe the laws within the sphere. Therefore,
radius. It is just another conservation law – twice the let us make an artificial sphere where all the physical
surface area, half the stuff, whatever the stuff is – properties are implemented and then take measurements
gravity, light, or spaceships running away from the new from the outside.
one.
Does the inverse square law affect sound waves?
Why does the inverse square law occur so often? Things only follow the inverse square law if there is a lot
This is a natural result of living in a three-dimensional of spherical symmetry in the media. Sound waves are
universe. Any effect or force that is outward from a usually affected by the inverse square law but your
central point (such as light, gravity, or sound) spreads perception of sound is not. If you double the distance
outward like an expanding sphere. The "power" of this between you and the sound transmitter, the sound
effect is usually distributed evenly over the area of the pressure that reaches you is not halved but quadrupled.
expanding circle, following the inverse square law. This is because sound radiates just like other waves and
is, as the question mentions, an illustration of how the
Example inverse square law works. Therefore, even though sound
obeys the inverse square law perfectly, your perception
The inverse square law of light defines the relationship of sound is roughly “twice the distance is half the
between the irradiance from a point source and distance. volume”. This should perhaps give you some insight into
It states that the intensity per unit area varies in inverse why your brain's sound processing works the way it
proportion to the square of the distance. Distance is does.
measured to the first illuminating surface – the filament What we perceive as “half volume” is actually the
of a clear bulb or the glass envelope of a frosted bulb. result of an auditory memory of how the volume changes
Get access to our light intensity calculator by as we move closer and further away from the object. Our
downloading it on our website today. brain does not calculate the inverse square but it
remembers situations well in the appropriate amount on
an unconscious level. It is much more useful for the brain
to be able to place an object in space based on "perceived
volume = relative distance".

Things to Remember
Light loses brightness or luminosity as it goes away from
the source, according to this rule. For example, if you
Figure 7. Light Intensity and inverse square law turn on a light in one corner of the room and then move
away from the source, the light seems dull or less
You measure 10.0 lm/m² from a light bulb at 1.0 meter. brilliant owing to the increase in distance (away from the
What will the flux density be at half the distance? source).
The inverse square law formula can be stated
Solution: mathematically as I ∝ (1/d2).
E1=(r1/r2)²×E2 When a force, energy, or other conserved quantity is
E0.5 m=(1.0/0.5)²*10.0 = 40 lm/m² equally radiated outward from a point source in three-
dimensional space, the inverse-square law often applies.
Because the surface area of a sphere (which is 4r 2) is
proportionate to the square of the radius, the emitted
radiation spreads out across an area that grows in
proportion to the square of the distance from the given
source.
276 BABIČ

As a result, the intensity of radiation traveling through Matej Babič received his Ph.D. degree in Computer Science
any unit area (directly confronting the point source) is from the Faculty of Electrical Engineering and Computer
inversely proportional to the square of the distance. Science of the University of Maribor, Slovenia. He studied
Gauss' law for gravity is also applicable in the same case, Mathematics at the Faculty of Education in Maribor. He is
Assist. Prof. at the Faculty of Information Studies, Novo mesto.
and it may be applied to any physical variable that His research interest is in fractal geometry, graph theory,
behaves in an inverse-square relationship. network, intelligent systems, hybrid machine learning,
topography of materials after hardening, public transport
Application of inverse square law system, and scalar wave.
In X-ray methods, the inverse-square law is used to
compute source-to-film distances.
It also aids in determining the duration of x-ray
exposure, as well as the intensity of the x-ray tube used
in the procedure.
When the brightness of the source is known, the
traditional candle method may be used to compute the
distance from the Earth.
The inverse-square law is used to calculate
astronomical distances.

Application of inverse square root

A normalized vector is calculated using a floating point
number's inverse square root. Programs can calculate the
incidence and reflection angles using normalized vectors.
To mimic lighting, 3D graphics algorithms need to make
millions of these computations per second.

4 CONCLUSION
The opposite-quarrel law is a principle that expresses the
way luminous energy propagates through space. The rule
states that the power intensity per unit area from a point
source (if the rays strike the surface at a right angle)
varies inversely according to the square of the distance
from the source.

REFERENCES
[1] Bošković, R. [273] 19 Fieri autem posset, ut totus itidem Mundus
nobis conspicuous in dies contraheretur, vel produceretur, … quod
si fieret, nulla in animo nostro idearum mutation haberetur,
adeoque nulls ejus modi mutationis sensus.
[2] Coulomb (1785). "Second mémoire sur l'électricité et le
magnétisme" [Second dissertation on electricity and magnetism].
Histoire de l'Académie Royale des Sciences [History of the Royal
Academy of Sciences] (in French). pp. 578–611. Il résulte donc de
ces trois essais, que l'action répulsive que les deux balles
électrifées de la même nature d'électricité exercent l'une sur l'autre,
suit la raison inverse du carré des distances.
[3] Meyl, K. Potential Vortex 5, About the structure of Gas and Water.
Second edition, 2020. ISBN 978-3-940 703-55-2.
[4] Meyl, K. 2012. Potential Vortex 1, From Vortex Physics to the
World Equation, 2nd ed., (orig. Potentialwirbel 1990), INDEL
GmbH pub., ISBN 978-3-940 703-51-4. www.meyl.
[5] Meyl, K. 2012. Potential Vortex 2, From Objectivity to a Unified
Theory, 2nd ed., (orig. Potentialwirbel 1990), INDEL GmbH pub.,
ISBN 978-3-940 703-52-1. www.meyl.
[6] Meyl, K. 2012. Potential Vortex 3, From Field Vortex to
Elementary Particles, 2nd ed., (orig. Potentialwirbel 1990), INDEL
GmbH pub., ISBN 978-3-940 703-53-8. www.meyl.
[7] Meyl, K. 2012. Potential Vortex 4, From Nuclear Physics and
Fusion to Nanotechnology, 2nd ed., (orig. Potentialwirbel 1990),
INDEL GmbH pub., ISBN 978-3-940 703-542-5. www.meyl.