# Page:A Dynamical Theory of the Electromagnetic Field.pdf/25

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PROFESSOR CLERK MAXWELL ON THE ELECTROMAGNETIC FIELD.

Electromotive Force in a Circuit.

(63) Let ${\displaystyle \xi }$ be the electromotive force acting round the circuit A, then

 ${\displaystyle \xi =\int \left(P{\frac {dx}{ds}}+Q{\frac {dy}{ds}}+R{\frac {dz}{ds}}\right)ds}$ (32)

where ${\displaystyle ds}$ is the element of length, and the integration is performed round the circuit.

Let the forces in the field be those due to the circuits A and B, then the electromagnetic momentum of A is

 ${\displaystyle \int \left(F{\frac {dx}{ds}}+G{\frac {dy}{ds}}+H{\frac {dz}{ds}}\right)ds=Lu+Mv}$ (33)

where ${\displaystyle u}$ and ${\displaystyle v}$ are the currents in A and B, and

 ${\displaystyle \xi =-{\frac {d}{dt}}(Lu+Mv)}$ (34)

Hence, if there is no motion of the circuit A,

 ${\displaystyle \left.{\begin{array}{l}P=-{\frac {dF}{dt}}-{\frac {d\Psi }{dx}},\\\\Q=-{\frac {dG}{dt}}-{\frac {d\Psi }{dy}},\\\\R=-{\frac {dH}{dt}}-{\frac {d\Psi }{dz}}.\end{array}}\right\}}$ (35)

where ${\displaystyle \Psi }$ is a function of ${\displaystyle x,y,z}$, and ${\displaystyle t}$, which is indeterminate as far as regards the solution of the above equations, because the terms depending on it will disappear on integrating round the circuit. The quantity ${\displaystyle \Psi }$ can always, however, be determined in any particular case when we know the actual conditions of the question. The physical interpretation of ${\displaystyle \Psi }$ is, that it represents the electric potential at each point of space.

Electromotive Force on a Moving Conductor.

(64) Let a short straight conductor of length a, parallel to the axis of ${\displaystyle x}$, move with a velocity whose components are ${\displaystyle {\tfrac {dx}{dt}},{\tfrac {dy}{dt}},{\tfrac {dz}{dt}}}$, and let its extremities slide along two parallel conductors with a velocity ${\displaystyle {\tfrac {ds}{dt}}}$. 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 ${\displaystyle {\tfrac {dx}{dt}},{\tfrac {dy}{dt}},{\tfrac {dz}{dt}}}$ along 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 ${\displaystyle {\tfrac {ds}{dt}}}$.

Hence the quantity

${\displaystyle \int \left(F{\frac {dx}{ds}}+G{\frac {dy}{ds}}+H{\frac {dz}{ds}}\right)ds}$