Lambert’s Cosine Law

It states that the intensity of illumination or illuminance E at a point on a surface is proportional to the cosine of the angle of incidence of the  light at that point. It is used to find the illumination of a surface, when it falls on the surface along any slanting direction.
Let us consider S is a point source of light falling on a surface area A as shown in figure. The normal to the surface makes an angle θ with the direction. Then, component of A normal to the direction of light ray is A cos θ.

Now, the angle made by A at the source S is:

ΔΩ=Arear2

or,ΔΩ=Acosθr2……(i)

The total luminous flux passing normally through this surface area is:

Q=LΔΩ

Putting the value of Δ Ω from equation (i), we get:

Q=LAcosθr2

The illuminance of the surface is given by:

I=QA

or,I=Lcosθr2……(ii)

Since,  Lr2=I∘
called the maximum illuminance of a surface. So, equation (ii) becomes,

I∝cosθ…..(iii)

Equation (iii) is the expression for Lambert cosine law.
From Lambert cosine law, we concluded that the illumination at a point due to source is:

• Directly proportional to the luminous intensity of the source.
• Inversely proportional to the square of distance of the point from the source.
• Directly proportional to the cosine of angle of incidence of luminous flux.