Optical Distance Law

Overview

Engineering reference data for Optical Distance Law in miscellaneous.

Key Formulas

Unit Conversion

$y = x \cdot k$

Multiply by conversion factor.

Linear Interpolation

$y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$

Estimate between two known points.

Percentage

$p = \frac{\text{part}}{\text{whole}} \times 100\%$

Part as fraction of whole.

Variables

Practical Example

Illumination at Different Distances

Consider a lamp with a luminous flux of 10000 lumens (lm).

Calculation at distance 1 (d₁ = 2 m): Using the basic formula:

$$E_1 = \frac{\Phi}{d_1^2} = \frac{10000 \, \text{lm}}{(2 \, \text{m})^2} = 2500 \, \text{lux}$$

Calculation at distance 2 (d₂ = 5 m): Using the ratio form of the law:

$$E_2 = E_1 \cdot \frac{d_1^2}{d_2^2} = 2500 \, \text{lux} \cdot \frac{(2 \, \text{m})^2}{(5 \, \text{m})^2} = 400 \, \text{lux}$$

This example demonstrates the significant reduction in illumination intensity over a relatively short distance due to the inverse-square relationship.

Cosine Law of Illumination

When the surface is not perpendicular to the direction of the light, the cosine law (also known as Lambert's cosine law) provides a more general relationship.

The illumination intensity $E$ on a surface is given by:

$$E = \frac{\Phi}{d^2} \cos(\theta)$$

Where:

• $\theta$ (theta) is the angle between the light ray and the normal (perpendicular line) to the illuminated surface.

Implication: Maximum illumination occurs when the light strikes the surface perpendicularly ($\theta = 0^\circ$, $\cos(0^\circ)=1$). As the angle of incidence increases, the illumination decreases.

Key Relationship

The core principle is that illumination $E$ is inversely proportional to the square of the distance $d$ from the point light source. This is captured in the relationship:

$$E \propto \frac{1}{d^2}$$

This fundamental law is critical for lighting design, photography, and optical engineering.

References

• Original Source