The Cosine Law of Illuminance
Lambert’s cosine law means ‘illumination at any point on a surface is directly proportional to the cosine of the angle between the normal at that point and the line of flux’.
Assume the surface is inclined at an angle ‘θ’ to the line of flux as shown above,
Let AB be the surface area normal to the source and inclined at ‘θ’ to the vertical axis.
CD be the surface area normal to the vertical axis and inclined at an angle ‘θ’ to the source ‘O’.
From the figure,
[Tex]AB= CD \cos \theta[/Tex]
For the area A1 ,
Solid angle is,
[Tex]\omega = \frac{\text{area of AB}}{d^{2}}[/Tex]
we know that,
flux = luminous Intensity x Solid angle
[Tex]\text{flux}=I \times \omega[/Tex]
[Tex]\text{flux}=I\times \frac{\text{area of AB}}{d^{2}}[/Tex]
Let the Illumination be E1 on Area of AB,
[Tex]\text{Illumination} = \frac{\text{flux}}{\text{area}}[/Tex]
[Tex]E_{1}=I\times \frac{\text{area of AB}}{d^{2}}\times\frac{1}{\text{area of AB}}[/Tex]
[Tex]E_{1}= \frac{I}{d^{2}}[/Tex]
Similarly,
Illumination be E2 on Area of CD,
[Tex]\text{Illumination} = \frac{\text{flux}}{\text{area of CD}}[/Tex]
[Tex]E_{2}=I\times \frac{\text{area of AB}}{d^{2}}\times\frac{1}{\text{area of CD}}[/Tex]
[Tex]\text{area of CD}=\frac{\text{area of AB}}{\cos \theta}[/Tex]
[Tex]E_{2}=I\times \frac{\text{area of AB}}{d^{2}}\times\frac{1}{\frac{\text{area of AB}}{\cos \theta}}[/Tex]
[Tex]E_{2}=\frac{I}{d^{2}}\cos \theta[/Tex]
From the figure,
[Tex]\cos \theta =\frac{h}{d}[/Tex]
[Tex]d = \frac{h}{ \cos \theta}[/Tex]
where d= distance, h= height
Substitute d value in E2 ,
[Tex]E_{2}=\frac{I}{(\frac{h}{\cos \theta})^{2}}\cos \theta[/Tex]
[Tex]E_{2}=\frac{I\cos^{2} \theta}{h^{2}}\cos \theta[/Tex]
[Tex]E_{2}=\frac{I}{h^{2}}\cos^{3} \theta[/Tex]
Laws of Illumination
Light is an essential element in human life and the day-to-day activities of human beings eventually depend on the light. Where there is no natural light, the use of artificial light is required. Artificial lighting is generated electrically, because of its cleanliness, ease of control, accuracy, and stable output, as well as its low cost it is taking an increasingly major part in modern everyday life.
The science of illumination engineering has become of major importance. Light is a form of luminous energy. Several forms of incandescent bodies are the reference of light and the light radiated by such bodies depends upon the temperature of bodies. Heat energy is emitted into the medium by a body that is much warmer than the medium surrounding it. The heat of the body can be categorized as red-hot or white-hot.
Table of Content
- Illumination
- Important terms
- Properties
- laws of Illumination
- Solved examples
- Advantages
- Disadvantages
- Applications