---
title: "Inverse-square law"
chunk: 2/3
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"
---

=== Light and other electromagnetic radiation ===
The intensity (or illuminance or irradiance) of light or other linear waves radiating from a point source (energy per unit of area perpendicular to the source) is inversely proportional to the square of the distance from the source, so an object (of the same size) twice as far away receives only one-quarter the energy (in the same time period).
More generally, the irradiance, i.e., the intensity (or power per unit area in the direction of propagation), of a spherical wavefront varies inversely with the square of the distance from the source (assuming there are no losses caused by absorption or scattering).
For example, the intensity of radiation from the Sun is 9126 watts per square meter at the distance of Mercury (0.387 AU) but only 1367 watts per square meter at the distance of Earth (1 AU)—an approximate threefold increase in distance results in an approximate ninefold decrease in intensity of radiation.
For non-isotropic radiators such as parabolic antennas, headlights, and lasers, the effective origin is located far behind the beam aperture. If you are close to the origin, you don't have to go far to double the radius, so the signal drops quickly. When you are far from the origin and still have a strong signal, like with a laser, you have to travel very far to double the radius and reduce the signal. This means you have a stronger signal or have antenna gain in the direction of the narrow beam relative to a wide beam in all directions of an isotropic antenna.
In photography and stage lighting, the inverse-square law is used to determine the “fall off” or the difference in illumination on a subject as it moves closer to or further from the light source. For quick approximations, it is enough to remember that doubling the distance reduces illumination to one quarter; or similarly, to halve the illumination increase the distance by a factor of 1.4 (the square root of 2), and to double illumination, reduce the distance to 0.7 (square root of 1/2). When the illuminant is not a point source, the inverse square rule is often still a useful approximation; when the size of the light source is less than one-fifth of the distance to the subject, the calculation error is less than 1%.
The fractional reduction in electromagnetic fluence (Φ) for indirectly ionizing radiation with increasing distance from a point source can be calculated using the inverse-square law. Since emissions from a point source have radial directions, they intercept at a perpendicular incidence. The area of such a shell is 4πr 2 where r is the radial distance from the center. The law is particularly important in diagnostic radiography and radiotherapy treatment planning, though this proportionality does not hold in practical situations unless source dimensions are much smaller than the distance.  As stated in Fourier theory of heat “as the point source is magnification by distances, its radiation is dilute proportional to the sin of the angle, of the increasing circumference arc from the point of origin”.

==== Example ====
Let P  be the total power radiated from a point source (for example, an omnidirectional isotropic radiator). At large distances from the source (compared to the size of the source), this power is distributed over larger and larger spherical surfaces as the distance from the source increases.  Since the surface area of a sphere of radius r is A = 4πr 2, the intensity I (power per unit area) of radiation at distance r is

  
    
      
        I
        =
        
          
            P
            A
          
        
        =
        
          
            P
            
              4
              π
              
                r
                
                  2
                
              
            
          
        
        .
        
      
    
    {\displaystyle I={\frac {P}{A}}={\frac {P}{4\pi r^{2}}}.\,}
  

The energy or intensity decreases (divided by 4) as the distance r is doubled; if measured in dB would decrease by 6.02 dB per doubling of distance. When referring to measurements of power quantities, a ratio can be expressed as a level in decibels by evaluating ten times the base-10 logarithm of the ratio of the measured quantity to the reference value.

=== Sound in a gas ===
In acoustics, the sound pressure of a spherical wavefront radiating from a point source decreases by 50% as the distance r is doubled; measured in dB, the decrease is still 6.02 dB, since dB represents an intensity ratio. The pressure ratio (as opposed to power ratio) is not inverse-square, but is inverse-proportional (inverse distance law):

  
    
      
        p
         
        ∝
         
        
          
            1
            r
          
        
        
      
    
    {\displaystyle p\ \propto \ {\frac {1}{r}}\,}
  

The same is true for the component of particle velocity 
  
    
      
        v
        
      
    
    {\displaystyle v\,}
  
 that is in-phase with the instantaneous sound pressure 
  
    
      
        p
        
      
    
    {\displaystyle p\,}
  
:

  
    
      
        v
         
        ∝
        
          
            1
            r
          
        
         
        
      
    
    {\displaystyle v\ \propto {\frac {1}{r}}\ \,}
  

In the near field is a quadrature component of the particle velocity that is 90° out of phase with the sound pressure and does not contribute to the time-averaged energy or the intensity of the sound.  The sound intensity is the product of the RMS sound pressure and the in-phase component of the RMS particle velocity, both of which are inverse-proportional. Accordingly, the intensity follows an inverse-square behaviour:

  
    
      
        I
         
        =
         
        p
        v
         
        ∝
         
        
          
            1
            
              r
              
                2
              
            
          
        
        .
        
      
    
    {\displaystyle I\ =\ pv\ \propto \ {\frac {1}{r^{2}}}.\,}
  

== Field theory interpretation ==
For an irrotational vector field in three-dimensional space, the inverse-square law corresponds to the property that the divergence is zero outside the source. This can be generalized to higher dimensions. Generally, for an irrotational vector field in n-dimensional Euclidean space, the intensity "I" of the vector field falls off with the distance "r" following the inverse (n − 1)th power law

  
    
      
        I
        ∝
        
          
            1
            
              r
              
                n
                −
                1
              
            
          
        
        ,
      
    
    {\displaystyle I\propto {\frac {1}{r^{n-1}}},}
  

given that the space outside the source is divergence free. 