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

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                                Electromotive force on a Moving Conductor 

(64) Let a short straight conductor of length a parallel to the axis of x move with 
a velocity whose components are dx/dt dy/dt dz/dt and let its 
extremities slide along two parallel conductors with a velocity ds/dt 
Let us find the alteration of the electromagnetic momentum of 
the circuit of which this arrangement forms a part. 
In unit of time the moving conductor has travelled distances 
dx/dt dy/dt dz/dt long the directions of the three axes, and at 
the same time the lengths of the parallel conductors included 
in the circuit have each been increased by ds/dt 
Hence the quantity
[integral](F dx/ds + G dy/ds + H dz/ds)ds 
will be increased by the following increments 
a(dF/dx dx/dt + dG/dy dy/dt + dF/dz dz/dt) due to motion of conductor 
-a ds/dt(dF/dx dx/ds + dG/dx dy/ds + dH/dx dz/ds) due to lengthening of circuit 
The total increment will therefore be 
a(dF/dy - dG/dx)dy/dt - a(dH/dx - dF/dz)dz/dt 

or by the equations of Magnetic Force (8) 
-a([mu][gamma]dy/dt - [mu][beta]dz/dt) 
If P is the electromotive force in the moving conductor parallel 
to x referred to unit of length, then the actual electromotive 
force is Pa and since this is measured by the decrement of the 
electromagnetic momentum of the circuit, the electromotive 
force due to motion will be 
P = [mu][gamma]dy/dt - [mu][beta]dz/dt (36) 

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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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