A surface is illuminated by a light source. If the angle between the normal at a point on the surface and the direction of luminous flux changes from 30° to 60°, what is the ratio of change in the illumination ?

A surface is illuminated by a light source. If the angle between the normal at a point on the surface and the direction of luminous flux changes from 30° to 60°, what is the ratio of change in the illumination ?

Right Answer is:

√3:1

SOLUTION

Lambert’s Cosine Law

  • Lambert’s cosine law states that the illumination (E) on a surface is directly proportional to the cosine of the angle (θ) between the direction of light and the normal to the surface. Mathematically, this can be expressed as:

E ∝ cos(θ)

Applying the Law to the Problem

Let’s denote:

  • E₁: Illumination at 30 degrees
  • E₂: Illumination at 60 degrees

Using Lambert’s cosine law:

E₁ ∝ cos(30°)

E₂ ∝ cos(60°)

To find the ratio of E₂ to E₁, we can write:

E₁/E₂ = cos(30°) / cos(60°)

Now, we know that:

  • cos(60°) = 1/2
  • cos(30°) = √3/2

Substituting these values:

E₁/E₂ = (√3/2)/(1/2)

Simplifying:

E₁/E₂ = √3/1

Therefore, the ratio of change in illumination is √3:1.

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