Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/325

Eliminating $T$ from these two equations, we find

	$\alpha \left(A{\frac {dC}{dt}}-{\frac {dS}{dt}}\right)+C+{\frac {1}{2}}\alpha {\frac {dC}{dt}}-{\frac {1}{12}}\alpha ^{2}$	${\frac {d^{2}C}{dt^{2}}}+{\frac {1}{48}}\alpha ^{3}{\frac {d^{3}C}{dt^{3}}}$
		${}-{\frac {1}{160}}\alpha ^{4}{\frac {d^{4}C}{dt^{4}}}+\mathrm {\&c.} =0$.	⁠	(16)

If $l$ is the whole length of the circuit, $R$ its resistance, and $E$ the electromotive force due to other causes than the induction of the current on itself,

The first term, $RC$, of the right-hand member of this equation expresses the electromotive force required to overcome the resistance according to Ohm's law.

The second term, $l(A+{\frac {1}{2}}){\frac {dC}{dt}}$, expresses the electromotive force which would be employed in increasing the electrokinetic momentum of the circuit, on the hypothesis that the current is of uniform strength at every point of the section of the wire.

The remaining terms express the correction of this value, arising from the fact that the current is not of uniform strength at different distances from the axis of the wire. The actual system of currents has a greater degree of freedom than the hypothetical system, in which the current is constrained to be of uniform strength throughout the section. Hence the electromotive force required to produce a rapid change in the strength of the current is somewhat less than it would be on this hypothesis.

The relation between the time-integral of the electromotive force and the time-integral of the current is

If the current before the beginning of the time has a constant value $C_{0}$, and if during the time it rises to the value $C_{1}$, and remains constant at that value, then the terms involving the differential coefficients of $C$ vanish at both limits, and

the same value of the electromotive impulse as if the current had been uniform throughout the wire.