J. C. Maxwell’s, ‘Dynamical theory of the electromagnetic field’

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                                Mechanical Force on a Magnet. 

(77) In any part of the field not traversed by electric currents 
the distribution of magnetic intensity may be represented by 
the differential coefficients of a <s>quantity<\s> function which may 
be called the magnetic potential. When there are no 
currents in the field this quantity has a single value for each 
point. When there are currents, the <s>function<\s> -potential 
has a series of values at each point, but its differential coefficients 
have only one value, namely 
d[phi]/dx = [alpha] d[phi]/dy=[beta] d[phi]/dz=[gamma] 
Substituting these values of [alpha] [beta] [gamma] in the expression (Eq<sup>n 38) 
for the intrinsic energy of the field, and integrating by parts 
it becomes -[capital sigma]{[phi]1/8[pi] (d[mu][alpha]/dx + d[mu][beta]/dy + d[mu][gamma]/dz)}dV 
<s>Now<\s> The expression 
[capital sigma](d[mu][alpha]/dx + d[mu][beta]/dy + d[mu][gamma]/dz)}dV = [capital sigma]mdV (39) 
indicates the number of lines of magnetic force which have their 
origin within the space V. Now a magnetic pole is known to 
us only as the origin or termination of lines of magnetic 
force, and a unit pole is one which has  4[pi] lines belonging 
to it since it produces unit of magnetic intensity at unit of 
distance over a sphere whose surface is 4[pi]. 
Hence if m is the amount of free positive magnetism in 
unit of volume, the above expression may be written 4[pi]m 
and the expression for the energy of the field becomes 
E = -[capital sigma]([half][phi]m)dV (40) 
If there are two magnetic poles m<sub>1 <\sub> and m<sub>2<\sub> producing 
potentials [phi]<sub>1<\sub> & [phi<sub>2<\sub> in the field, then if m<sub>2<\sub> is moved a distance 

                            
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Manuscript details

Author
James Clerk Maxwell
Reference
PT/72/7
Series
PT
Date
1864
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J. C. Maxwell’s, ‘Dynamical theory of the electromagnetic field’, 1864. From The Royal Society, PT/72/7

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