Laws of Illumination - Inverse Square Law & Cosine Law

There are two main laws of illumination of a surface, they are,

• Inverse Square Law, and
• Lambert's Cosine Law.

Inverse Square Law :

The inverse square law states that the illumination on the surface is inversely proportional to the square of the distance between the surface and the light source.

But the drawback of inverse square law is that this law is only applicable if the light source is considered as a point source. If a point source that emits an equal amount of light in all directions is placed at the center of the hollow sphere, then the light flux will reach all over the inner surface of the sphere uniformly i.e., an equal amount of light flux will be distributed over every square millimeter of the sphere.

Now if suppose the radius of the sphere is increased, the same amount of light flux from the point source will be distributed over the increased inner surface area proportional to the square of the radius i.e., every square millimeter of the sphere will diminish at the rate of increase in the sphere radius and will be inversely proportional to the square of the distance between source and surface.

Proof :

Let us prove the above discussed inverse square law mathematically. Consider two surface areas A₁ and A₂ placed at distances r₁ and r₂ respectively from the point light source (S) as shown below.

Let 'I' be the luminous intensity of the point source and the solid angle subtended at 'S' be ω steredains. Assuming that all the light flux is reaching the surface area. The total flux radiated in the direction of surfaces is given as,

Φ = solid angle (ω) × luminous intensity (I)

In the above equation, ω is expressed as A/r², where A is the area of the surface. From this the area of two surfaces A₁ and A₂ is given as,

A₁ = ω × r₁² and A₂ = ω × r₂²

Therefore, the illumination E₁ on the surface area A₁ is,

Similarly, the illumination E₂ on the surface area A₂ is,

For example, a light source 'S' gives an output of 'F' lumens within the solid angle 'W' enclosed by four lines OA, OB, OC, OD. The whole of the light will be falling on the surface 'A₁', placed at the right angle to the axis of the beam and at a distance of 1m from the source, the illumination on the surface A₁ is F/A₁ lumen per m².

If the distance of the surface is increased (considering the new surface as A₂) from 1m to 2m from the same point source, the illumination on it will be decreased by 1/4 of that of A₁. (A₂ = 2² = 4 = 4A₁ and since the total lumens will be the same as before, therefore, the illumination has to decrease 1/4 of that of A₁).

Similarly, if the distance of the surface is increased to 3m away from the point source, the illumination will be decreased by 1/9 that of its original value. Hence it is proved that illumination on the surface is inversely proportional to the square of the distance between the surface and the light source.

∴ Illumination (E) = 1/d²

Lambert's Cosine Law :

Lambert's cosine law states that the illumination (E) at any point on the surface is proportional to the cosine of angle which direction of the incident light flux makes with the normal (perpendicular) at that point.

The light flux received by the surface area changes with the change in angle between normal to the surface and the direction of the light flux. The illumination is maximum when the light flux falling is equal to normal to the surface, if the surface is inclined to the light flux falling, then the illumination is reduced.

Proof :

Let us prove the above discussed Lambert's cosine law mathematically. Consider a point source 'S' at a height 'h' from the surface CD. Assume that surface CD subtends solid angle 'ω' at point source 'S' as shown below. Let AB be the inclined surface area normal to the light flux with E as the center point of the surface.

Therefore, from the above figure,

AB = CD cos θ

The illumination on the surface AB is,

The illumination on the surface CD is,

The above equation proves that illumination at any point on the surface is proportional to the cosine of the angle which direction of incident light flux makes with the normal at that point on the surface. From the figure, the height of the point source from the surface is,

h = d cos θ

Now illumination on CD is given as,

The above equation is known as Cosine Cube Law.

Limitations of Laws of Illumination :

• The inverse square law is used to measure the illuminance only on the horizontal surfaces i.e., the inverse square law is only applicable if the surface is normal to the light flux.
• The source is considered as the point source in inverse square law. But, in practice, the point source does not exist. Hence, a lot of errors are introduced in the calculation of illuminance using inverse square law.
• Lambert's cosine law of illuminance is used to measure the illuminance only on inclined surfaces.