Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/240

induction. The electromotive force arising from magneto-electric induction alone is evidently $-{\frac {dp}{dt}}$, or the rate of decrease of the electrokinetic momentum of the circuit.

Electromagnetic Force.

580.] Let X' be the impressed mechanical force arising from external causes, and tending to increase the variable x. By the general equations

$X'={\frac {d}{dt}}{\frac {dT}{d{\dot {x}}}}-{\frac {dT}{dx}}.$

Since the expression for the electrokinetic energy does not contain the velocity $({\dot {x}})$, the first term of the second member disappears, and we find

$X'=-{\frac {dT}{dx}}.$

Here X' is the external force required to balance the forces arising from electrical causes. It is usual to consider this force as the reaction against the electromagnetic force, which we shall call X, and which is equal and opposite to X'.

Hence

$X={\frac {dT}{dx}}.$

or, the electromagnetic force tending to increase any variable is equal to the rate of increase of the electrokinetic energy per unit increase of that variable, the currents being maintained constant.

If the currents are maintained constant by a battery during a displacement in which a quantity, W, of work is done by electromotive force, the electrokinetic energy of the system will be at the same time increased by W. Hence the battery will be drawn upon for a double quantity of energy, or 2W, in addition to that which is spent in generating heat in the circuit. This was first pointed out by Sir W. Thomson^[1]. Compare this result with the electrostatic property in Art. 93.

Case of Two Circuits.

581.] Let A₁ be called the Primary Circuit, and A₂ the Secondary Circuit. The electrokinetic energy of the system may be written

$T={\frac {1}{2}}L{\dot {y}}_{1}^{2}+M{\dot {y}}_{1}{\dot {y}}_{2}+{\frac {1}{2}}N{\dot {y}}_{2}^{2}$

where L and N are the coefficients of self-induction of the primary

1. Nichol's Cyclopaedia of Physical Science, ed. 1860, Article, 'Magnetism, Dynamical Relations of'.