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:

$$\Delta \Omega = \frac{\text{Area}}{r^2}$$

$$or, \Delta \Omega = \frac{\text{Acos} \theta }{r^2}……(i)$$

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

$$Q = L \Delta \Omega$$

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

$$Q = L \frac{A cos \theta }{r^2}$$

The illuminance of the surface is given by:

$$I = \frac{Q}{A}$$

$$or, I = \frac{\text{Lcos} \theta }{r^2}……(ii)$$

Since,  $\frac{L}{r^2} = I_{\circ}$ called the maximum illuminance of a surface. So, equation (ii) becomes,

$$I \propto cos \theta …..(iii)$$

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.

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