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

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                                Part IV Mechanical Actions in the Field 

Mechanical Force on a Moveable Conductor

(76) We have shown [paragraph](34 & 35) that the work done by the electromagnetic 
forces in aiding the motion of a conductor is equal 
to the product of the current in the conductor multiplied by 
the increment of the electromagnetic momentum due to the 
motion. 
Let a short straight conductor of length a move parallel to 
itself in the direction of x with its extremities on two parallel 
conductors. Then the increment of the electromagnetic momentum 
due to the motion of a will be 
a(dF/dx dx/ds + dG/dx dy/ds + dH/dx dz/ds) [delta]x 
That due to the lengthening of the circuit by increasing the length of 
the parallel conductors will be 
-a(dF/dx dx/ds + dF/dy dy/ds + dF/dz dz/ds) [delta]x 
The total increment is a[delta]x{dy/ds(dG/dx - dF/dy) - dz/ds(dF/dz -dH/dx)} 
which is by the equations of Magnetic force, (B). 
a[delta]x(y/ds [mu][gamma] - dz/ds [mu][beta]) 
Let X be the force acting along the direction of x per unit of length 
of the conductor then the work done is Xa[delta]x Let C be the current in the conductor and let p' q' r' be its 
components then 
Xa[delta]x = ca[delta]x(dy/ds [mu][gamma] - dz/ds [mu][beta]) 
or X = [mu][gamma]q' - [mu]beta]r' 
Similarly Y = [mu][alpha]r' - [mu][gamma]p' 
z = [mu]beta]p' - [mu][alpha]q' (J) 
These are the equations which determine the mechanical force 
acting on a conductor carrying a current. The force is perpendicular 
to the current and to the lines of force, and is measured by the area 
of the parallelogram formed by lines parallel to the current & lines of force 
and proportional to their intensities 
                            
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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

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