Lambert’s Cosine Law Example Problems

Lambert’s Cosine Law, often simply referred to as the Cosine Law, states that the radiant intensity or luminous intensity observed from an ideal diffusely reflecting surface or ideal diffuse radiator is directly proportional to the cosine of the angle θ between the direction of the incident light and the direction of observation.

Lambert’s Cosine Law Example Problems

Mathematically, it can be expressed as: �=�0×cos⁡(�)I=I0​×cos(θ) Where:

• $I$ = Observed intensity
• �0I0​ = Intensity when �=0∘θ=0∘ (i.e., directly overhead or normal to the surface)
• $\theta$ = Angle between the direction of the incident light and the normal (perpendicular) to the surface

Example 1: A diffusely reflecting surface is illuminated by a light source such that its intensity �0I0​ is 100 lumens when observed directly overhead. What is the intensity when observed at an angle of 60∘60∘ from the normal?

Using Lambert’s cosine law: �=100×cos⁡(60∘)I=100×cos(60∘) $I=100\times 0.5$ $I=50\text{ lumens}$

The observed intensity at an angle of 60∘60∘ is 50 lumens.

Example 2: A wall illuminated by a projector has an intensity of 150 lumens when the projector beam is perpendicular to the wall. What will be the observed intensity if the projector is tilted such that the beam makes an angle of 45∘45∘ with the wall’s surface?

Again, using Lambert’s cosine law: �=150×cos⁡(45∘)I=150×cos(45∘) $I=150\times 0.7071$ $I\approx 106.06\text{ lumens}$

The observed intensity at an angle of 45∘45∘ is approximately 106.06 lumens.

Guidelines to Solve Lambert’s Cosine Law Problems:

1. Identify given data: Determine values for �0I0​ and $\theta$.
2. Understand the relationship: Remember that the observed intensity decreases as the angle increases from 0 to 90 degrees.
3. Use the cosine function: Apply the cosine function to the angle to find its cosine value.
4. Multiply: Multiply the cosine value by �0I0​ to get the observed intensity $I$.
5. Round if necessary: Based on the context of the problem, round the answer to an appropriate number of decimal places.
6. Consider the limits: If $\theta$ is 0, �=�0I=I0​. If $\theta$ is 90°, $I=0$ (since cos(90°) = 0).

Remember that Lambert’s Cosine Law assumes an ideal diffusely reflecting surface. Real-world materials may deviate from this behavior due to factors like surface roughness, material properties, and other conditions.