**ABSTRACT**

This paper presents a new study about light on the laws and equations on circles of light beams. At night light beams coming out from a source of light with a circular lens will form a circle on a surface.

To investigate this phenomenon a study was conducted. Four (4) flashlights with the same size of a circular lens were clamped on four (4) retort stands at night against a whiteboard and were placed at different distances.

The flashlights were switched on and four (4) circles of light beams were formed. The distances from each flashlight were measured by a metre rule. The circumferences and the areas of the circles formed were calculated. The results were as follows :

- As the distance from a flashlight increases, the circumferences and the areas of the circles increases and as the distance from a flashlight decreases, the circumferences and the areas of the circles decreases.
- As the circumferences and the areas of the circles increase, the intensity of the light decreases and as the circumferences and the areas of the circles decrease, the intensity of the light increases. The results obtained from the experiment conducted were summarized into mathematical equations:

**INTRODUCTION**

**LAWS ON THE CIRCLES OF LIGHT BEAMS**

The laws on circles of light beams describe how a source of light forms a circle on a surface at night. These are two (2) laws, the first law and the second law and are useful in calculating the distances from a source of light when it forms a circle at night on a surface.

**LAW I**

Law I states that provided a source of light with a circular lens forms a circle on a surface at night , the distance from the point source of light is directly proportional to the square root of the circumference of the circle.

The distance from a point source of light is denoted as

The circumference of the circle is given by where r is the radius of the circle from law I,

Introducing a constant proportionality , where is called the minimum coefficient of light. Therefore, the law I becomes

Making the minimum coefficient of light, the subject, becomes

The minimum light coefficient of the light, is defined as the ratio of the distance from a point source of light to the square root of the circumference of the circle of light beams. It is a dimensionless physical quantity that has no unit. The size of the circular lens affects the minimum coefficient of light.

Increasing the size of the circular lens of a source of light decreases the minimum coefficient of the light.

** LAW II**

Law II states that a provided source of light with a circular lens forms a circle on a surface at night, the distance from the point source of light is directly proportional to the fourth root of the area of the circle.

The distance from a point source of light is denoted as . The area of the circle is given by where r is the radius of the circle, from law II,

Introducing a constant of the proportionality where is called the maximum coefficient of the light.

Therefore law II becomes , making the maximum coefficient of light, the subject of it becomes

From the expression , the maximum coefficient of the light is defined as the ratio of the distance from a point source of light to the fourth root of the area of the circle of light beams. The maximum coefficient of the light has a dimension and its unit is

The size of the circular lens affects the maximum coefficient of the light.

Increasing the size of the circular lens of the source of light decreases the maximum coefficient of the light, and decreasing the size of the circumference glass lens increases the maximum coefficient of the light.

**Expressing law I and law II into one equation**

from law I and II

Where and

Therefore the two laws can be expressed as

**EQUATIONS ON THE CIRCLES OF LIGHT BEAMS**

Equations on the circles of light beams are two (2) equations that serve as an alternative to calculate the intensity of light, provided the source of light forms a circle on a surface at night.

The first equation of circles of light beams shows the relationship between law I on circles of light beams and the inverse square law of light.

The second equation of circles of light beams also shows the relationship between law II on circles of light beams and the inverse square law of light.

**The inverse square law**

Light intensity is inversely proportional to its distance squared.

It is mathematically expressed as

where I represent light intensity d represent the distance from the light source.

Intensity at different distances;

Where represent light intensity at distance 1

represents light intensity at distance 2.

represents distance 1 from the light source.

represents distance 2 from the light source.

**Relationship between the inverse square law and the law I**

The inverse square law formula is given by

From law I,

from law I, in the inverse square law formula

therefore

where , substituting into the expression

It becomes

This equation becomes equation I. It can only be applied to calculate the intensity of the source of light forming a circle on a surface.

**Relationship between the inverse square law and the law II**

The inverse square formula is given by

From law II ,

Again from law II , in the inverse square law formula.Therefore,

Where , Substituting into the expression

It becomes

This becomes equation II . It can only be applied to calculate the intensity of a source of light forming a circle of a surface at night.

**Expressing Equation I and II on circles of light beams into one equation**

Therefore the two equations on circles of light beams can be unified as

_{ EXPERIMENTAL VEREXPERIMIFICATION ON LAWS ON CIRCLES OF LIGHT BEAMS}

**Materials needed: **

- Four retort stands
- Four flashlights with the same size as a circular glass lens.
- Meter rule.
- Whiteboard.

**Methods: **

- The four flashlights with the same size of a circular lens were clamped on the four retort stands at night against a whiteboard at different distances.
- The flashlights were switched on and four circles of light beams were formed on the whiteboard.
- The distances from each flashlight were measured by the meter rule.
- The radius of each circle of light beams formed by each flashlight at different distances was measured by the meter rule.
- The radii obtained were used to calculate the circumferences and the areas of the circles formed by each flashlight.

**RESULTS:**

The table below shows the circumference and the areas of circles of light beams formed by each flashlight at different distances.

DISTANCES (cm) | CIRCUMFERENCES (cm) | AREA (cm^{2}) |
---|---|---|

20.17796838 | 31.41592654 | 78.53981634 |

28.53595654 | 62.83185307 | 314.1592654 |

34.94926643 | 94.24777961 | 706.8583471 |

40.35593676 | 125.6637061 | 1256.637061 |

**DISCUSSION:**

The results obtained from the experiment support the two laws on circles of light beams. Hence the two laws describe how light beams form a circle at night.

**CONCLUSIONS:**

Further studies into this work will lead to a new field of study and help humankind understand more about light and its properties.

**REFERENCE:**

- “Inverse square law formula”, SoftSchools, accessed March 22, 2021,

https://www.softschools.com/formulas/physics/inverse_square_law_formula/82/

Enoch Kwartengu have done well otoo