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DSSSB JE EE 2019 Official Paper (Held on 25 Oct 2019)

Option 2 : cos^{3} θ

Lambert's cosine law: This law states that ‘illumination, E 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.

Law of inverse squares:

According to the Law of inverse squares, illumination of a surface is inversely proportional to the square of the distance between the surface and the light source.

\(E = \frac{I}{{{d^2}}}\)

Where E is the illuminance

I is luminous intensity

d is the distance between the surface and the source

Important Point:

- Inverse-square law is only applicable for the surfaces if the surface is normal to the line of flux
- Lambert’s cosine law is applicable for the surfaces if the surface is inclined at an angle θ to the line of flux.

**Application:**

Consider a lamp of uniform luminous intensity suspended at a height h above the working plane as shown in Fig

Let us consider the value of illumination at point A immediately below the lamp and at other points, B, C, D, etc., lying in the working plane at different distances from A.

From the law of inverse square,

\(E_A=\frac{I}{h^2}\) .... (1)

Similarly From the law of Inverse-square and Lambert cosine rule,

\(E_B=\frac{I}{LB^2}cos\theta _1\) .... (2)

From the law of trigonometry,

\(cos\theta_1=\frac{h}{LB}\)

\(LB=\frac{h}{cos\theta_1}\) .... (3)

From equation (1), (2), and (3),

\(E_B=\frac{I}{h^2}cos^3\theta _1=E_Acos^3\theta _1\)

Similarly,

\(E_C=E_Acos^3\theta _2\) and ........

**Hence, the illuminance illuminated by the same source at different points on the horizontal surface varies with the light rays inclined at an angle θ on the vertical axis is the function of cos3 θ.**