# Inverse Square Law and Lambert Cosine Law | Physics Grade XI

### Lambert Cosine and Inverse Square Law

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#### Viva Voice Questions # Inverse Square Law and Lambert Cosine Law

Inverse Square Law
Consider a point source of light S which is emitting light in all directions uniformly. Let the light energy emitted per sec by the source is Q. let's draw two spheres of radius r1 and r2. Then the intensity of illumination for area 4πr12 and 4πr22. I1 = Q/ 4πr12  ...…... (i)
I2 = Q/ 4πr22  .………. (ii)

Dividing Equation (i) by equation (ii), we get
I1/I2 = Q/ 4πr12 * 4πr22/ Q
Or, I1/I2 = r22/ r12

In general,
I α 1/ r2

Thus, the intensity of illumination of the point is inversely proportional to the square of the distance from the source.

Lambert Cosine Law

It states that, "when light falls obliquely on a surface, the illumination of the surface is directly proportional to the cosine of the angle of incidence of light on the surface". It is used to find the illumination of a surface, when light falls on the surface along an oblique direction.

Suppose a light beam from a source S falling on a surface area A as shown in the figure. The normal to the surface makes an angle θ with the direction of light. Then, component of A normal to the direction of light ray is A cosθ.

The solid angle made by A at the source S,
ΔΩ = area/ r2
= A cosθ/ r2

The total luminous flux passing normally through this surface area is
Q = L ΔΩ
= L * A cosθ/ r2
and illumination of the surface
I = Q/ A
= L cosθ/ r2
= I0 cosθ …… (i)
Where I0 = L/ r2, maximum illumination of the surface. That is I α cosθ ……….. (ii)

Which is Lambert’s cosine Law, illumination of a surface is directly proportional to the cosine of the angle. The maximum illumination of a surface is obtained when light falls normally on the surface.

Hence illumination at a point due to a source is

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

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