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

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                                dx and is urged in that direction by a force X, then the 
work done is Xdx and the decrease of energy in the field 
is d<s>[capital sigma]<\s>([half](phi<sub>1<\sub> + [phi<sub>2<\sub>)(m<sub>1<\sub> + m<sub>2<\sub>)) 
and these must be equal by the principle of Conservation of energy 
Since the distribution [phi]<sub>1<\sub> is determined by m<sub>1<\sub> and [phi]<sub>2<\sub> by m<sub>2<\sub> 
the quantities [phi]<sub>1<\sub>m<sub>1<\sub> & [phi]<sub>2<\sub>m<sub>2<\sub> will remain constant. 
It can be shown also as Green has proved (Essay p10) that 
m<sub>1<\sub>[phi]<sub>2<\sub> = m<sub>2<\sub>[phi]<sub>1<\sub> 
so that we get, 
Xdx = d(m<sub>2<\sub>[phi]<sub>1<\sub>) 
or X = M<sub>2<\sub> d[phi]<sub>1<\sub>dx = m<sub>2<\sub>[alpha]<sub>1<\sub> <s>(41)<\s>
where [alpha]<sub>1<\sub> represents the magnetic intensity due to m<sub>1<\sub> 
Similarly Y = m<sub>2<\sub>[beta]<sub>1<\sub> 
Z = m<sub>2<\sub>[gamma]<sub>1<\sub> } (K) 
So that a magnetic pole is urged in the direction of 
the lines of magnetic force with a force equal to the product 
of the strength of the pole and the magnetic intensity. 
(78) If a single magnetic pole, that is one pole of a very long magnet 
be placed in the field the only solution of [phi] is 
[phi]<sub>1<\sub> = - m<sub>1<\sub>/[mu] 1/r (41) 
where m<sub>1<\sub> is the strength of the pole and r the distance from it 
The repulsion between two poles of strengths m<sub>1<\sub> & m<sub>2<\sub>  is 
m<sub>2<\sub> d[phi]<sub>1<\sub>/dr = m<sub>1<\sub>m<sub>2<\sub> / [mu].r<sup>2<\sup> (42) 
In air or any medium in which [mu] = 1 this is simply 
m<sub>1<\sub>m<sub>2<\sub> / r<sup>2<\sup> but in other media the force acting between two 
given magnetic poles is inversely proportional to the 
coefficient of magnetic induction for the medium. This may be 
explained by the magnetization of the medium induced by the action of the pole 
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Manuscript details

James Clerk Maxwell
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Cite as

J. C. Maxwell’s, ‘Dynamical theory of the electromagnetic field’, 1864. From The Royal Society, PT/72/7



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