Classical electron radius

<h2 id="derivation">Derivation</h2>
<p>The classical electron radius length scale can be motivated by considering the energy necessary to assemble an amount of charge 
  
    
      
        q
      
    
    {\displaystyle q}
  
 into a sphere of a given radius 
  
    
      
        r
      
    
    {\displaystyle r}
  
.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>  The electrostatic potential at a distance 
  
    
      
        r
      
    
    {\displaystyle r}
  
 from a charge 
  
    
      
        q
      
    
    {\displaystyle q}
  
 is
</p>

V
        (
        r
        )
        =
        
          
            1
            
              4
              π
              
                ε
                
                  0
                
              
            
          
        
        
          
            q
            r
          
        
      
    
    {\displaystyle V(r)={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q}{r}}}
  
.
<p>To bring an additional amount of charge 
  
    
      
        d
        q
      
    
    {\displaystyle dq}
  
 from infinity necessitates putting energy into the system, 
  
    
      
        d
        U
      
    
    {\displaystyle dU}
  
,  by an amount 
</p>

d
        U
        =
        V
        (
        r
        )
        d
        q
      
    
    {\displaystyle dU=V(r)dq}
  
.
<p>If the sphere is <i>assumed</i> to have constant charge density, 
  
    
      
        ρ
      
    
    {\displaystyle \rho }
  
, then 
</p>

q
        =
        ρ
        
          
            4
            3
          
        
        π
        
          r
          
            3
          
        
      
    
    {\displaystyle q=\rho {\frac {4}{3}}\pi r^{3}}
  
 and 
  
    
      
        d
        q
        =
        ρ
        4
        π
        
          r
          
            2
          
        
        d
        r
      
    
    {\displaystyle dq=\rho 4\pi r^{2}dr}
  
.
<p>Integrating for 
  
    
      
        r
      
    
    {\displaystyle r}
  
 from zero to the final radius 
  
    
      
        r
      
    
    {\displaystyle r}
  
 yields the expression for the total energy 
  
    
      
        U
      
    
    {\displaystyle U}
  
, necessary to assemble the total charge 
  
    
      
        q
      
    
    {\displaystyle q}
  
 into a uniform sphere of radius 
  
    
      
        r
      
    
    {\displaystyle r}
  
:
</p>

U
        =
        
          
            1
            
              4
              π
              
                ε
                
                  0
                
              
            
          
        
        
          
            3
            5
          
        
        
          
            
              q
              
                2
              
            
            r
          
        
      
    
    {\displaystyle U={\frac {1}{4\pi \varepsilon _{0}}}{\frac {3}{5}}{\frac {q^{2}}{r}}}
  
.
<p>This is called the electrostatic self-energy of the object. The charge 
  
    
      
        q
      
    
    {\displaystyle q}
  
 is now interpreted as the electron charge, 
  
    
      
        e
      
    
    {\displaystyle e}
  
, and the energy 
  
    
      
        U
      
    
    {\displaystyle U}
  
 is set equal to the relativistic mass–energy of the electron, 
  
    
      
        m
        
          c
          
            2
          
        
      
    
    {\displaystyle mc^{2}}
  
, and the numerical factor 3/5 is ignored as being specific to the special case of a uniform charge density. The radius 
  
    
      
        r
      
    
    {\displaystyle r}
  
 is then <i>defined</i> to be the classical electron radius, 
  
    
      
        
          r
          
            e
          
        
      
    
    {\displaystyle r_{\text{e}}}
  
, and one arrives at the expression given above.
</p><p>Note that this derivation does not say that 
  
    
      
        
          r
          
            e
          
        
      
    
    {\displaystyle r_{\text{e}}}
  
 is the actual radius of an electron.   It only establishes a dimensional link between electrostatic self energy and the mass–energy scale of the electron.
</p>
<h2 id="discussion">Discussion</h2>
<p>The classical electron radius appears in the classical limit of modern theories as well, including non-relativistic <a href="/facts/Thomson_scattering/hUj6gSgB">Thomson scattering</a> and the relativistic <a href="/facts/Klein%E2%80%93Nishina_formula/z2j1Loln">Klein–Nishina formula</a>.  Also, 
  
    
      
        
          r
          
            e
          
        
      
    
    {\displaystyle r_{\text{e}}}
  
 is roughly the length scale at which <a href="/facts/Renormalization/B4PdTB4o">renormalization</a> becomes important in <a href="/facts/Quantum_electrodynamics/zPteAr6G">quantum electrodynamics</a>.  That is, at short-enough distances, quantum fluctuations within the vacuum of space surrounding an electron begin to have calculable effects that have measurable consequences in atomic and particle physics.
</p><p>Based on the assumption of a simple mechanical model, attempts to model the electron as a non-point particle have been described by some as ill-conceived and counter-pedagogic.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Electromagnetic_mass/LScII2L6">Electromagnetic mass</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li>Arthur N. Cox, Ed. "Allen's Astrophysical Quantities", 4th Ed, Springer, 1999.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://math.ucr.edu/home/baez/lengths.html#classical_electron_radius">Length Scales in Physics: the Classical Electron Radius</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>David J. Griffiths, Introduction to Quantum Mechanics, Prentice-Hall, 1995, p. 155. ISBN 0-13-124405-1 <a href="/wiki/David_J._Griffiths" target="_blank">/wiki/David_J._Griffiths</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Young, Hugh (2004). University Physics, 11th Ed. Addison Wesley. p. 873. ISBN 0-8053-8684-X. <a href="0-8053-8684-X" target="_blank">0-8053-8684-X</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Curtis, L.J. (2003). Atomic Structure and Lifetimes: A Conceptual Approach. Cambridge University Press. p. 74. ISBN 0-521-53635-9. <a href="0-521-53635-9" target="_blank">0-521-53635-9</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Classical electron radius open-in-new

Classical electron radius