## The variation of light intensity with distance; namely the inverse square law of light intensity with distance.

### Background Theory

Light emitted from any kind of source, e.g. the sun, a light bulb, is a form of energy. Everyday problems such as lighting required for various forms of labouring or street illumination, require one to able to determine and evaluate the intensity of light emitted by any light source or even the illumination of a given surface. A special group of studies is formed around these issues and it is called*photometry*.

*Luminous flux*is a scalar quantity which measures the time rate of light flow from the source. As all measures of energy transferred over a period of time, luminous flux is measured in Joules/Seconds or Watts (SI units). It can therefore safely be said that luminous flux is a measure of light power. Visible light consists of several different colours, each representing a different wavelength of the radiation spectrum. For example red colour has a wavelength 610-700 nm, similarly yellow 550-590 nm and blue 450-500 nm. The human eye demonstrates different levels of sensitivity to the various colours of the spectra. More specifically, the maximum sensitivity is observed in the yellow-green colour (i.e. 555nm). From all the above, it is clear that there is the need to define a unit associating and standardising the visual sensitivity of the various wavelengths to the light power which are measured in Watt's; this unit is called the special luminous flux unit of the

*lumen*

*(lm)*. One lumen is equivalent to1/680 Watt of light with a wavelength of 555 nm. This special relationship between illumination and visual response renders the lumen the preferred photometric unit of luminous flux for practical applications. On top of that one of the most widely used light sources in everyday life such as the electric light bulb emits light which consists of many different wavelengths. A measure of the luminous strength of any light source is called the light sources intensity. At this point, it should be said that the intensity of a light source depends on the quantity of lumens emitted within a finite angular region which is formed by a solid angle. To give a visual representation of the solid angle, recall that in a bi-dimensional plane the plane angle is used for all kinds of angular measurements. A further useful reminder regards the arc length

*s*; namely for a circle of radius

*r*the arc length

*s*is calculating by the formula S = r * q -Equation. 1 (qis measured in radians) Now, in a three dimensional plane the solid angle W is similarly used for angular measurements. Corresponding to the q plane angle, each section of surface area

*A*of a sphere of radius

*r*is calculating by using the following formula; A= r

^{2}*W -Equation. 2 (Remember that W is measured in steradians) By definition one steradian is the solid angle subtended by an area of the spherical surface equal to the square of the radius of the sphere. Taking into account all the above mentioned, the luminous intensity I of a light source (small enough to be considered as a point source) pointing towards the solid angle is given by: I = F/ W -Equation. 3 Where F is the flux measured in lumens. It is clear that the luminous intensity unit is lumen /steradian. This unit used to be called a candle, as it was defined in the context of light emitted from carbon filament lamps. Generally speaking, luminous intensity in any particular direction is called the candle power of the source. The corresponding unit in the SI system is called the

*candela*

*(cd)*which is the luminous intensity emitted by 1/60 cm

^{2}of platinum at a temperature of 2054K (which is the fusion point of platinum). A uniform light source (small enough to be considered as a point source) whose luminous intensity is equal to one candela, is able to produce a luminous flux of one lumen through each solid angle. The equation shown below is the mathematical expression of the above definition: F = W * I -Equation. 4 Where I is equal to one cd and W is equal to one sr. In similar terms the total flux F

_{t}of a uniform light source with an intensity I can be calculated with the aid of the following formula. F

_{t}= W

_{t}* I - Equation. 5 And taking into account that the total solid angle W

_{t}of a sphere is 4p sr, the above formula becomes F

_{t}= 4p * I -Equation. 6 When a surface is irradiated with visible light it is said to be illuminated. For any given surface, the illuminance E (which is also called illumination) is intuitively understood and defined to be the flux indenting on the surface divided by the total area of the surface. E = F / A - Equation. 7 In the case where the several light sources are present and illuminate the same surface, the total illuminance is calculated by adding up all of the individual source illuminations. The SI unit allocated the illuminance is the

*lux (lx)*where one lx is equal to 1 lm / 1 m

^{2}. Another way of expressing illumination in the context of light sources intensity and the distance from the light source can be derived by forming a combination of the last few mentioned equations: E = F / A = I * W / A = I / r

^{2}-Equation. 8 Where r is the distance measured from the source or the radius of a sphere whose total area is A (W = A / r

^{2}). An important side note at this point is that 1fc equals 1cd/ft

^{2}and also 1lx is equal to1cd/ m

^{2}. It is evident that the illumination is inversely proportional to the square of the measured distance from the light source. In the case of constant light source intensity I, it can be said that: E

_{2}/E

_{1}= r

_{1}

^{2}/r

_{2}

^{2}= (r

_{1}/r

_{2})

^{2}- Equation. 9 In the real world, the incident light is very rarely normal to a surface; nearly always light impacts on a surface at an angle of incidence q. In this case the illuminance is calculated by: E = I* cos q/ r

^{2}-Equation. 10 To sum up, there are several ways which can be employed in order to measure illumination. Nearly all of them are based on the photoelectric effect originally discovered by Albert Einstein (for which he was awarded a Nobel Prize in 1921). In a few words when light strike sa material electron emission is observed and electric current flows if there is a circuit present. This current is proportional to the incident light flux and to the work function of the material; the intensity of the resulted current flow is measured by instruments calibrated in illumination units.