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

```                                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

Author
James Clerk Maxwell
Reference
PT/72/7
Series
PT
Date
1864
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## Cite as

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