Lambert’s Cosine Law and Luminous Flux

Lambert’s Cosine Law

Lambert’s cosine law states that the radiant intensity observed from an ideal diffusely reflecting surface is directly proportional to the cosine of the angle between the direction of observation and the normal (perpendicular) to the surface.

Mathematically, this can be expressed as:

$$I(\theta) \propto \cos\theta$$

where:

• $I$$($$\theta$$)$ is the intensity observed at angle
  $\theta$,$\mathrm{<br>}$

•  is the angle between the surface normal and the direction of observation.

This law was proposed by the Swiss mathematician and physicist Johann Heinrich Lambert and is widely used in optics, photometry, and computer graphics. It is also known as:

• Lambert’s emission law

• Cosine emission law

Physical Interpretation

What this law tells us is quite intuitive:

A diffusely reflecting surface appears brightest when viewed straight on, and its apparent brightness decreases smoothly as the viewing angle increases. This decrease follows a cosine relationship, not because the surface emits less light, but because the projected area seen by the observer becomes smaller.

Diffuse Reflection

Diffuse reflection occurs when light strikes a surface and is scattered in many different directions, rather than being reflected in a single, well-defined direction.

This type of reflection typically happens on rough or matte surfaces, where microscopic irregularities cause incoming light rays to scatter.

Lambertian Reflectance

A surface that follows Lambert’s cosine law perfectly is called a Lambertian surface.

Lambertian reflectance is defined as the property of a surface by which it appears equally bright from all viewing directions, even though the intensity in any given direction follows the cosine law.

This is why a matte wall looks uniformly bright no matter where you stand in a room.

Specular vs. Diffuse Reflection

• Specular reflection (mirror-like surfaces): Light reflects in a specific direction.

• Diffuse reflection (matte surfaces): Light scatters in all directions.

This distinction is crucial in understanding real-world lighting and visibility.

Examples of Diffuse Reflection

Diffuse reflection is extremely common in everyday life. Some important examples include:

1. Matte Paints

• Matte wall paints reflect light diffusely, which prevents glare.

• Glossy paints, in contrast, show a combination of specular and diffuse reflection.

2. Frosted Glass Bulbs

• Frosted bulbs scatter light in all directions, producing soft, uniform illumination.

3. Human Eye

• The surfaces inside the human eye rely on diffuse reflection to scatter light across the retina, aiding vision.

Luminous Flux and Its Derivation

Luminous flux, also called luminous power, is a measure of the total amount of visible light emitted by a source per unit time. It accounts for the human eye’s sensitivity to different wavelengths and is measured in lumens (lm).

Where,

sinӨ: Jacobian matrix determinant

Imax: luminous flux

This integration shows how total luminous flux depends on angular distribution and confirms why Lambertian sources distribute light uniformly over space.

Frequently Asked Questions (FAQs)

Q1. State Lambert’s cosine law.

Answer:

Lambert’s cosine law states that the radiant intensity from an ideal diffusely reflecting surface is directly proportional to the cosine of the angle between the direction of observation and the surface normal.

Q2. What is Lambert’s cosine law also known as?

Answer:

It is also known as Lambert’s emission law or the cosine emission law.

Q3. Define Lambertian reflectance.

Answer:

Lambertian reflectance is the property of a surface by which it appears equally bright when viewed from any direction.

Q4. True or False: A frosted glass bulb undergoes diffuse reflection.

Answer:

True.

Q5. What is luminous flux?

Answer:

Luminous flux is the measure of the total visible light power emitted by a source, weighted according to the sensitivity of the human eye.

Final Remarks

Lambert’s cosine law provides a powerful framework for understanding how light interacts with real-world surfaces. From architectural lighting to computer graphics and optical engineering, this principle explains why diffuse surfaces appear uniformly bright and how light energy is distributed in space.

Problem 1: Intensity at an Angle (Lambert’s Cosine Law)

Problem

A diffusely reflecting surface emits light with a maximum intensity of 120 cd in the direction normal to the surface.
Find the intensity observed at an angle of 60° from the normal.

Solution

According to Lambert’s cosine law:

$$I(\theta) = I_{\max} \cos\theta$$

Given:

• $I_{\max} = 120 \, \text{cd}$
  

• $\theta = 60^\circ$
  

• $\cos {60}^{\circ }=0.5$

Substitute values:

$$I(60^\circ) = 120 \times 0.5 = 60 \, \text{cd}$$

Answer

$$\boxed{I = 60 \, \text{cd}}$$

Problem 2: Ratio of Intensities at Two Angles

Problem

For a Lambertian surface, calculate the ratio of radiant intensities observed at 30° and 60° with respect to the surface normal.

Solution

Using Lambert’s law:

$$I \propto \cos\theta$$
$$\frac{I_{30}}{I_{60}} = \frac{\cos 30^\circ}{\cos 60^\circ}$$ $$\cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \cos 60^\circ = \frac{1}{2}$$ $$\frac{I_{30}}{I_{60}} = \frac{\sqrt{3}/2}{1/2} = \sqrt{3}$$Answer

$$\boxed{I_{30} : I_{60} = \sqrt{3} : 1}$$

Problem 3: Finding Angle for Given Intensity

Problem

The intensity of light observed from a Lambertian surface is 40 cd, while the maximum intensity is 80 cd.
Find the angle of observation.

Solution

$$I(\theta) = I_{\max} \cos\theta$$
$$40 = 80 \cos\theta$$
$$\cos\theta = \frac{40}{80} = 0.5$$
$$\theta = \cos^{-1}(0.5) = 60^\circ$$

Answer

$$\boxed{\theta = 60^\circ}$$

Problem 4: Luminous Flux of a Lambertian Source

Problem

A Lambertian source has a maximum luminous intensity of 100 cd.
Calculate the total luminous flux emitted into a hemisphere.

Solution

For a Lambertian source:

$$\Phi = \pi I_{\max}$$

Given:

• $I_{\max} = 100 \, \text{cd}$
  

$$\Phi = \pi \times 100$$
$$\Phi \approx 314 \, \text{lm}$$

Answer

$$\boxed{\Phi \approx 314 \, \text{lumens}}$$

Problem 5: Intensity Comparison (Conceptual Numerical)

Problem

A diffuse surface is viewed at angles 0°, 45°, and 90°.
If the intensity at normal incidence is $I_0$, find the intensity at each angle.

Solution

Using Lambert’s law:

$$I(\theta) = I_0 \cos\theta$$

• At $0^\circ$:

  $$I = I_0 \cos 0^\circ = I_0$$

• At $45^\circ$:

  $$I = I_0 \cos 45^\circ = \frac{I_0}{\sqrt{2}}$$

• At $90^\circ$
  

  $$I = I_0 \cos 90^\circ = 0$$
  

Answer

$$\boxed{ I(0^\circ) = I_0,\quad I(45^\circ) = \frac{I_0}{\sqrt{2}},\quad I(90^\circ) = 0 }$$

Problem 6: Real-Life Application

Problem

Why does a matte wall appear equally bright from different viewing positions even though intensity follows Lambert’s law?

Solution

Although the intensity in a given direction varies as $\cos\theta$, the projected area seen by the observer also varies as $\cos\theta$. These two effects cancel each other.

Thus, the brightness (luminance) remains constant for all viewing angles.

Answer

$$\boxed{\text{Due to Lambertian reflectance, brightness remains constant.}}$$