---
title: "Inverse-square law"
chunk: 1/3
source: "https://en.wikipedia.org/wiki/Inverse-square_law"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T03:44:04.680534+00:00"
instance: "kb-cron"
---

In physical science, an inverse-square law is any scientific law stating that the observed "intensity" of a specified physical quantity (being nothing more than the value of the physical quantity)  is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be understood as geometric dilution corresponding to point-source radiation into three-dimensional space.
Radar energy expands during both the signal transmission and the reflected return, so the inverse square for both paths means that the radar will receive energy according to the inverse fourth power of the range.
To prevent dilution of energy while propagating a signal, certain methods can be used such as a waveguide, which acts like a canal does for water, or how a gun barrel restricts hot gas expansion to one dimension in order to prevent loss of energy transfer to a bullet.

== Formula ==
In mathematical notation the inverse square law can be expressed as an intensity (I) varying as a function of distance (d) from some centre. The intensity is proportional (see ∝) to the reciprocal of the square of the distance thus: 

  
    
      
        
          intensity
        
         
        ∝
         
        
          
            1
            
              
                distance
              
              
                2
              
            
          
        
        
      
    
    {\displaystyle {\text{intensity}}\ \propto \ {\frac {1}{{\text{distance}}^{2}}}\,}
  

It can also be mathematically expressed as :

  
    
      
        
          
            
              
                intensity
              
              
                1
              
            
            
              
                intensity
              
              
                2
              
            
          
        
        =
        
          
            
              
                distance
              
              
                2
              
              
                2
              
            
            
              
                distance
              
              
                1
              
              
                2
              
            
          
        
      
    
    {\displaystyle {\frac {{\text{intensity}}_{1}}{{\text{intensity}}_{2}}}={\frac {{\text{distance}}_{2}^{2}}{{\text{distance}}_{1}^{2}}}}
  

or as the formulation of a constant quantity:   

  
    
      
        
          
            intensity
          
          
            1
          
        
        ×
        
          
            distance
          
          
            1
          
          
            2
          
        
        =
        
          
            intensity
          
          
            2
          
        
        ×
        
          
            distance
          
          
            2
          
          
            2
          
        
      
    
    {\displaystyle {\text{intensity}}_{1}\times {\text{distance}}_{1}^{2}={\text{intensity}}_{2}\times {\text{distance}}_{2}^{2}}
  

The divergence of a vector field which is the resultant of radial inverse-square law fields with respect to one or more sources is proportional to the strength of the local sources, and hence zero outside sources. Newton's law of universal gravitation follows an inverse-square law, as do the effects of electric, light, sound, and radiation phenomena.

== Justification ==
The inverse-square law generally applies when some force, energy, or other conserved quantity is evenly radiated outward from a point source in three-dimensional space.  Since the surface area of a sphere (which is 4πr2) is proportional to the square of the radius, as the emitted radiation gets farther from the source, it is spread out over an area that is increasing in proportion to the square of the distance from the source. Hence, the intensity of radiation passing through any unit area (directly facing the point source) is inversely proportional to the square of the distance from the point source. Gauss's law for gravity is similarly applicable, and can be used with any physical quantity that acts in accordance with the inverse-square relationship.

== Occurrences ==

=== Gravitation ===
Gravitation is the attraction between objects that have mass. Newton's law states:

The gravitational attraction force between two point masses is directly proportional to the product of their masses and inversely proportional to the square of their separation distance. The force is always attractive and acts along the line joining them.

  
    
      
        F
        =
        G
        
          
            
              
                m
                
                  1
                
              
              
                m
                
                  2
                
              
            
            
              r
              
                2
              
            
          
        
      
    
    {\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}}
  

If the distribution of matter in each body is spherically symmetric, then the objects can be treated as point masses without approximation, as shown in the shell theorem.  Otherwise, if we want to calculate the attraction between massive bodies, we need to add all the point-point attraction forces vectorially and the net attraction might not be exact inverse square. However, if the separation between the massive bodies is much larger compared to their sizes, then to a good approximation, it is reasonable to treat the masses as a point mass located at the object's center of mass while calculating the gravitational force.
As the law of gravitation, this law was suggested in 1645 by Ismaël Bullialdus. But Bullialdus did not accept Kepler's second and third laws, nor did he appreciate Christiaan Huygens's solution for circular motion (motion in a straight line pulled aside by the central force).  Indeed, Bullialdus maintained the sun's force was attractive at aphelion and repulsive at perihelion. Robert Hooke and Giovanni Alfonso Borelli both expounded gravitation in 1666 as an attractive force. Hooke's lecture "On gravity" was at the Royal Society, in London, on 21 March. Borelli's "Theory of the Planets" was published later in 1666.  Hooke's 1670 Gresham lecture explained that gravitation applied to "all celestiall bodys" and added the principles that the gravitating power decreases with distance and that in the absence of any such power bodies move in straight lines. By 1679, Hooke thought gravitation had inverse square dependence and communicated this in a letter to Isaac Newton: 
my supposition is that the attraction always is in duplicate proportion to the distance from the center reciprocall.
Hooke remained bitter about Newton claiming the invention of this principle, even though Newton's 1686 Principia acknowledged that Hooke, along with Wren and Halley, had separately appreciated the inverse square law in the Solar System, as well as giving some credit to Bullialdus.

=== Electrostatics ===

The force of attraction or repulsion between two electrically charged particles, in addition to being directly proportional to the product of the electric charges, is inversely proportional to the square of the distance between them; this is known as Coulomb's law. The deviation of the exponent from 2 is less than one part in 1015.

  
    
      
        F
        =
        
          k
          
            e
          
        
        
          
            
              
                q
                
                  1
                
              
              
                q
                
                  2
                
              
            
            
              r
              
                2
              
            
          
        
      
    
    {\displaystyle F=k_{\text{e}}{\frac {q_{1}q_{2}}{r^{2}}}}
  