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

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