The inverse square law is a scientific law in physics, which states that the intensity of a force, field, or illumination decreases proportionally to the square of the distance from the source to the point at which it is measured. It emphasizes that the closer the observer is to a light source, the brighter the illumination perceived.

This law extends beyond visible light to encompass all electromagnetic spectrum, including gamma rays, X-rays, and UV rays.

Mathematically, the inverse square law is expressed as

where I represents the intensity of radiation (in candela) and d denotes the distance from the source in meters.

If the distances are d₁ and d₂ and the intensities are I₁ and I₂ respectively. Then Inverse-square law is as follows,

This formula is important in determining either the distance or intensity of radiation. Intensity is typically measured in lumens or candelas, while distance is denoted in meters.

Example of Inverse Square Law

Examples of the inverse square law manifest in various natural phenomena, with Coulomb's law serving as a prime illustration:

Coulomb’s Law: Coulomb’s Law states that the magnitude of the attractive or repulsive electrostatic force between two point charges, q₁ and q₂, situated at a distance r from each other, is directly proportional to the product of the magnitudes of their charges and inversely proportional to the squared distance between them.

Expressed mathematically, the force is directly proportional to the product of the charges, yielding the final expression for Coulomb’s law:

Where k=1/4πϵ, where ϵ denotes the permittivity of the medium housing the charges.

The force F as per the inverse square law is represented as:

Newton’s law of gravitation: Newton’s Law of Universal Gravitation states that each particle in the universe exerts an attractive force on every other particle. This force is directly proportional to the product of their masses, m₁ and m₂, kept at a distance of r from each other, and inversely proportional to the square of the distance separating them.

This proportionality is turned into equality by using a constant, G, called the universal gravitational constant. The force is also proportional to the product of the masses, so the final expression for Newton’s law of gravitation is

The minus sign indicates attraction between the two masses.

The force as per inverse square law is

Inverse square law of radiation: This law states that the intensity of radiation at any point at a distance r from the source of strength S, decreases in proportion to the square of the distance from the source.

Mathematical expression:

The inverse square law applies universally to all forms of radiation originating from a point source and propagating uniformly through space. This principle governs phenomena such as light emanating from a point source, nuclear radiation, sound intensity, and similar radiative processes.

Applications of Inverse Square Law

The inverse square law serves as a fundamental principle across a diverse array of fields, underpinning various applications:

• Radiology: In radiology, particularly during X-ray procedures, the inverse square law guides the positioning of the X-ray source, subject, and film to achieve the desired intensity on the subject while ensuring safety. Accurate application of this law is critical for obtaining clear images without subjecting the individual to excessive radiation exposure.
• Astronomy: Astronomers leverage the inverse square law to estimate the distances of stars and other celestial objects. By measuring the apparent brightness of celestial bodies and accounting for their distance from Earth, astronomers can infer intrinsic properties such as luminosity and size.
• Architectural Acoustics: Architects and acoustic engineers utilize the inverse square law in the design of halls, auditoriums, and performance spaces. Since sound intensity diminishes in accordance with the inverse square law, understanding this principle helps determine optimal speaker placement, sound distribution, and acoustic characteristics within the space. This information is crucial for achieving balanced acoustics and minimizing issues such as reverberation and sound attenuation.

Click here to know more about Planck's law.