If the distance between the source of light and the receiving point of light is D, then illumination at the receiving point is proportional to:

Explanation:

Inverse Square Law of Illumination

Definition: The inverse square law of illumination states that the illumination (E) at a point source of light is inversely proportional to the square of the distance (D) from the source. This means that as the distance from the light source increases, the illumination decreases at a rate proportional to the square of the distance.

Mathematical Representation: The relationship can be mathematically represented as:

E ∝ 1/D²

Where:

• E is the illumination at the receiving point.
• D is the distance between the source of light and the receiving point.

Detailed Explanation:

To understand why illumination follows the inverse square law, consider a point light source emitting light uniformly in all directions. As the light travels away from the source, it spreads over a larger area. The amount of light passing through any unit area decreases as the distance from the source increases.

Derivation:

1. Consider a point source of light:

Imagine a point source of light emitting light uniformly in all directions. The light energy emitted by the source spreads out spherically.

2. Surface area of a sphere:

As the light spreads out, it covers the surface of an imaginary sphere with the light source at its center. The surface area (A) of a sphere is given by:

A = 4πD²

Where:

• π is a mathematical constant approximately equal to 3.14159.
• D is the radius of the sphere, which is the distance from the light source.

3. Illumination is inversely proportional to surface area:

As the distance (D) increases, the surface area (A) over which the light spreads also increases. The illumination (E) at any point on the surface of the sphere is the total light energy divided by the surface area:

E ∝ 1/A

Since the surface area (A) is proportional to D², we can write:

A = 4πD²

Therefore:

E ∝ 1/D²

This shows that the illumination at a point is inversely proportional to the square of the distance from the light source.

Applications:

• Lighting design in architecture and interior design, ensuring adequate illumination levels.
• Photography and cinematography, determining proper exposure settings based on distance from light sources.
• Street lighting, calculating the spacing and intensity of streetlights to ensure uniform illumination.

Correct Option Analysis:

The correct option is:

Option 4: 1/D²

This option correctly describes the relationship between the distance from the light source and the illumination at the receiving point, based on the inverse square law. As the distance (D) increases, the illumination (E) decreases proportionally to the square of the distance.

Important Information

To further understand the analysis, let’s evaluate the other options:

Option 1: D

This option suggests that illumination is directly proportional to the distance from the light source. However, this is incorrect because illumination decreases with increasing distance, following the inverse square law, not a direct proportionality.

Option 2: D²

This option suggests that illumination is proportional to the square of the distance from the light source. This is incorrect because illumination actually decreases with the square of the distance, not increases.

Option 3: 1/D

This option suggests that illumination is inversely proportional to the distance from the light source, which is partially correct but not entirely accurate. The inverse proportionality should be to the square of the distance (1/D²), not just the distance (1/D).

Conclusion:

Understanding the inverse square law of illumination is crucial for various applications, including lighting design, photography, and street lighting. The correct relationship, as described, is that illumination at a point is inversely proportional to the square of the distance from the light source. This fundamental principle helps ensure proper lighting levels and effective use of light sources in various fields.