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

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                                Mechanical Force on an Electrified Body 

(79) If there is no motion or change of strength of currents or magnets 
in the field, the electromotive force is entirely due to variation 
of electric potential and we shall have [paragraph](65) 

P = -d[psi]/dx Q = -d[psi]/dy R = -d[psi]/dz 

Integrating by parts the expression (I) for the energy due 
to electric displacement and remembering that P Q R vanish 
at an infinite distance, it becomes 

[half][capital sigma]{[psi](df/dx + dg/dy + dh/dz)}dV 

or by the equation of Free Electricity (G) 

-[half[[capital sigma]([psi]e)dV 

By the same demonstration as was used in the case of 
the mechanical action on a magnet it may be shown 
that the mechanical force on a small body containing 
a quantity e<sub>2<\sub> of free electricity placed in a field whose potential 
arising from other electrified bodies is [psi]<sub>1<\sub>, has for components 

X = e<sub>2<\sub>  d[psi]<sub>1<\sub>/dx = -e<sub>2<\sub>P<sub>1<\sub> 
Y =  e<sub>2<\sub>  d[psi]<sub>2<\sub>/dy = -e<sub>2<\sub>Q<sub>1<\sub> 
Z = e<sub>3<\sub>  d[psi]<sub>1<\sub>/dz = -e<sub>2<\sub>R<sub>1<\sub> } (L) 

So that an electrified body is urged in the direction of the 
electromotive force with a force equal to the product of 
the quantity of free electricity and the electromotive force 
If the electrification of the field arises from the presence of 
a small electrified body containing e<sub>1<\sub> of free electricity 
the only solution of [psi]<sub>1<\sub> is 

[psi]<sub>1<\sub> = k/4[pi] e<sub>1<\sub>/r (43) 

where r is the distance from the electrified body 
The repulsion between two electrified bodies e<sub>1<\sub> e<sub>2<\sub> is 

e<sub>2<\sub> d[psi<sub>1<\sub>/dr = k/4[pi] e<sub>1<\sub>e<sub>2<\sub>/r<sup>2<\sup> (44) 

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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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