# Page:A Dynamical Theory of the Electromagnetic Field.pdf/23

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PROFESSOR CLERK MAXWELL ON THE ELECTROMAGNETIC FIELD.

placed in the direction of ${\displaystyle x}$ at the given point. We may suppose an indefinitely short wire placed parallel to ${\displaystyle x}$ at a given point and touched, during the action of the force P, by two small conductors, which are then insulated and removed from the influence of the electromotive force. The value of P might then be ascertained by measuring the charge of the conductors.

Thus if ${\displaystyle l}$ be the length of the wire, the difference of potential at its ends will be P${\displaystyle l}$, and if C be the capacity of each of the small conductors the charge on each will be ${\displaystyle {\tfrac {1}{2}}CPl}$. Since the capacities of moderately large conductors, measured on the electromagnetic system, are exceedingly small, ordinary electromotive forces arising from electromagnetic actions could hardly be measured in this way. In practice such measurements are always made with long conductors, forming closed or nearly closed circuits.

Electromagnetic Momentum (F, G, H).

(57) Let F, G, H represent the components of electromagnetic momentum at any point of the field, due to any system of magnets or currents.

Then F is the total impulse of the electromotive force in the direction of ${\displaystyle x}$ that would be generated by the removal of these magnets or currents from the field, that is, if P be the electromotive force at any instant during the removal of the system

${\displaystyle F=\int Pdt\,}$

Hence the part of the electromotive force which depends on the motion of magnets or currents in the field, or their alteration of intensity, is

 ${\displaystyle P=-{\frac {dF}{dt}},\ Q=-{\frac {dG}{dt}},\ R=-{\frac {dH}{dt}},}$ (29)

Electromagnetic Momentum of a Circuit.

(58) Let ${\displaystyle s}$ be the length of the circuit, then if we integrate

 ${\displaystyle \int \left(F{\frac {dx}{ds}}+G{\frac {dy}{ds}}+H{\frac {dz}{ds}}\right)ds}$ (30)

round the circuit, we shall get the total electromagnetic momentum of the circuit, or the number of lines of magnetic force which pass through it, the variations of which measure the total electromotive force in the circuit. This electromagnetic momentum is the same thing to which Professor Faraday has applied the name of the Electrotonic State.

If the circuit be the boundary of the elementary area ${\displaystyle dydz}$, then its electromagnetic momentum is

${\displaystyle \left({\frac {dH}{dy}}-{\frac {dG}{dz}}\right)dy\ dz}$

and this is the number of lines of magnetic force which pass through the area ${\displaystyle dydz}$.

Magnetic Force (${\displaystyle \alpha ,\beta ,\gamma }$).

(59) Let ${\displaystyle \alpha ,\beta ,\gamma }$ represent the force acting on a unit magnetic pole placed at the given point resolved in the directions of ${\displaystyle x,y}$, and ${\displaystyle z}$.