Lamberts Cosine law: Overview, Questions, Preparation

The common terms of photometry are:

• Luminance: It is a measure of the intensity of light that reaches the eye. 
• Luminous energy: It is the radiation energy capable of producing visual sensation to the human eye. 
• Luminous flux (ф): It is the luminous energy emitted per second and measured in units of the lumen.

             ф = 4πl.

• The illuminance (E) of the surface transpires when the luminous flux falls normally on its unit area. 

• According to this law, the illuminance (or illumination) of any point on a surface is directly proportional to the cosine of the angle (θ) made by the normal to the illuminated surface at that point with the direction of the incident flux. 
• The law is also known as the Cosine Emission Law or Lambert’s Emission Law. It is named after Johann Heinrich Lambert, from his Photometria, published in 1760.

             Illuminance, E ∝ cos 𝜃.

• Suppose luminous flux (ф) is incident normally on the surface XY of area A shown in the figure. Then, the illuminance of the surface is E= ф/A. 
• If the surface is tilted to the position X’Y where ∠XYX’ =ፀ, then same flux falls on the surface X’Y. The area of the surface XY is increased to * A secፀ.

             ∴ Illuminance at  X’Y = ф/A secፀ = ф /A cosፀ = E cosፀ

• Note that ፀ is the angle between the Normal RN to the illuminated surface (i.e., X’Y) and the incident flux direction. Clearly, illuminance is directly proportional to cosፀ. This proves Lambert’s Cosine Law.
• If ፀ = 0° (i.e., the surface is normal to the incident flux), the illuminance on the surface is maximum (=E). As ፀ increases, the illuminance on the surface decreases and becomes 0 when ፀ= 90°.

Lambert’s Cosine Law comes under the unit ‘Optics,’ which carries 18 marks in the exams.

1. The luminous intensity of a 100W undirectional bulb is 100 candela. (i) What is the luminous flux emitted from the bulb; (ii) what is the bulb’s luminous efficiency?

Luminous flux ф = 4πl = 4π x 100 = 1256 lumens.

Luminous efficiency = Luminous flux/Electric power = 1256/100 =12.56 lm/watt.

2. The luminous efficiency of an electric lamp is 5 lumen/watt, and luminous intensity is 35 candela. The power of the lamp is?

Luminous flux from lamp = 4πl = 4π x 35 = 440 lm.

Power of the lamp = Luminous flux/Luminous efficiency = 440/5 = 88W.

3. Why is the illumination of the earth’s surface from the sun more at noon than in the morning?

 E∝ cosፀ. At noon, sun rays fall normally on the earth’s surface so that ፀ=0° and E is maximum.

Q: What does a Lambertian surface mean?

Q: What is Lambertian scattering?

Q: What is Luminous efficiency?

Q: What is a Lambertian source?

Q: What is a cosine corrector?