Magnetic scalar potential

<h2 id="magnetic-scalar-potential">Magnetic scalar potential</h2>

<p>The <a href="/facts/Scalar_potential/lXKtCr1c">scalar potential</a> is a useful quantity in describing the magnetic field, especially for <a href="/facts/Permanent_magnet/vvrg6VSk">permanent magnets</a>.
</p><p>Where there is no free current,

∇
        ×
        
          H
        
        =
        
          0
        
        ,
      
    
    {\displaystyle \nabla \times \mathbf {H} =\mathbf {0} ,}

so if this holds in <a href="/facts/Simply_connected_domain/xFlta63J">simply connected domain</a> we can define a magnetic scalar potential, <i>ψ</i>, as<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>

H
        
        =
        −
        ∇
        ψ
        .
      
    
    {\displaystyle \mathbf {H} =-\nabla \psi .}

The dimension of <i>ψ</i> in <a href="/facts/SI_base_units/M7PsTWg4">SI base units</a> is 
  
    
      
        
          
            A
          
        
      
    
    {\displaystyle {\mathsf {A}}}
  
, which can be expressed in SI units as <a href="/facts/Ampere/MScqkCgz">amperes</a>.
</p><p>Using the definition of H:

∇
        ⋅
        
          B
        
        =
        
          μ
          
            0
          
        
        ∇
        ⋅
        
          (
          
            
              H
            
            +
            
              M
            
          
          )
        
        =
        0
        ,
      
    
    {\displaystyle \nabla \cdot \mathbf {B} =\mu _{0}\nabla \cdot \left(\mathbf {H} +\mathbf {M} \right)=0,}

it follows that

∇
          
            2
          
        
        ψ
        =
        −
        ∇
        ⋅
        
          H
        
        =
        ∇
        ⋅
        
          M
        
        .
      
    
    {\displaystyle \nabla ^{2}\psi =-\nabla \cdot \mathbf {H} =\nabla \cdot \mathbf {M} .}

</p><p>Here, ∇ ⋅ M acts as the source for magnetic field, much like ∇ ⋅ P acts as the source for electric field. So analogously to <a href="/facts/Bound_charge/2oQv9EHx">bound electric charge</a>, the quantity

ρ
          
            m
          
        
        =
        −
        ∇
        ⋅
        
          M
        
      
    
    {\displaystyle \rho _{m}=-\nabla \cdot \mathbf {M} }

is called the <i>bound magnetic charge</i> density. Magnetic charges 
  
    
      
        
          q
          
            m
          
        
        =
        ∫
        
          ρ
          
            m
          
        
        
        d
        V
      
    
    {\textstyle q_{m}=\int \rho _{m}\,dV}
  
 never occur isolated as <a href="/facts/Magnetic_monopole/R1TSbfMY">magnetic monopoles</a>, but only within dipoles and in magnets with a total magnetic charge sum of zero. The energy of a localized magnetic charge <i>qm</i> in a magnetic scalar potential is

Q
        =
        
          μ
          
            0
          
        
        
        
          q
          
            m
          
        
        ψ
        ,
      
    
    {\displaystyle Q=\mu _{0}\,q_{m}\psi ,}

and of a magnetic charge density distribution <i>ρm</i> in space

Q
        =
        
          μ
          
            0
          
        
        ∫
        
          ρ
          
            m
          
        
        ψ
        
        d
        V
        ,
      
    
    {\displaystyle Q=\mu _{0}\int \rho _{m}\psi \,dV,}

where <i>µ</i>0 is the <a href="/facts/Vacuum_permeability/nKxRk6J1">vacuum permeability</a>. This is analog to the energy 
  
    
      
        Q
        =
        q
        
          V
          
            E
          
        
      
    
    {\displaystyle Q=qV_{E}}
  
 of an electric charge <i>q</i> in an electric potential 
  
    
      
        
          V
          
            E
          
        
      
    
    {\displaystyle V_{E}}
  
.
</p><p>If there is free current, one may subtract the contributions of free current per <a href="/facts/Biot%25E2%2580%2593Savart_law/LLfwuAHr">Biot–Savart law</a> from total magnetic field and solve the remainder with the scalar potential method.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Magnetic_vector_potential/KJGSgMYb">Magnetic vector potential</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Duffin, W.J. (1980). <i>Electricity and Magnetism, Fourth Edition</i>. McGraw-Hill. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 007084111X.</li></ul>
<ul><li>Jackson, John David (1999), <i>Classical Electrodynamics</i> (3rd ed.), <a href="/facts/John_Wiley_%2526_Sons/53g3LqJc">John Wiley & Sons</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-30932-X</li></ul>
<ul><li>Vanderlinde, Jack (2005). <a href="https://cds.cern.ch/record/1250088"><i>Classical Electromagnetic Theory</i></a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2005cet..book.....V">2005cet..book.....V</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F1-4020-2700-1">10.1007/1-4020-2700-1</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-4020-2699-4.</li></ul>

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

<ol>
<li id="fn:1"><p>Vanderlinde 2005, pp. 194–199 - Vanderlinde, Jack (2005). Classical Electromagnetic Theory. Bibcode:2005cet..book.....V. doi:10.1007/1-4020-2700-1. ISBN 1-4020-2699-4. <a href="https://cds.cern.ch/record/1250088" target="_blank">https://cds.cern.ch/record/1250088</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Magnetic scalar potential open-in-new

Magnetic scalar potential