Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/425

${\begin{aligned}Q_{0}-2Q_{1}+Q_{2}&=RC{\frac {dQ_{1}}{dt}},\\Q_{1}-2Q_{2}+Q_{3}&=RC{\frac {dQ_{2}}{dt}}.\end{aligned}}$

If there are n quantities of electricity to be determined, and if either the total electromotive force, or some other equivalent conditions be given, the differential equation for determining any one of them will be linear and of the nth order.

By an apparatus arranged in this way, Mr. Varley succeeded in imitating the electrical action of a cable 12,000 miles long.

When an electromotive force is made to act along the wire on the left hand, the electricity which flows into the system is at first principally occupied in charging the different condensers beginning with $A_{1},$ and only a very small fraction of the current appears at the right hand till a considerable time has elapsed. If galvanometers be placed in circuit at $R_{1},\,R_{2},$ &c. they will be affected by the current one after another, the interval between the times of equal indications being greater as we proceed to the right.

332.] In the case of a telegraph cable the conducting wire is separated from conductors outside by a cylindrical sheath of guttapercha, or other insulating material. Each portion of the cable thus becomes a condenser, the outer surface of which is always at potential zero. Hence, in a given portion of the cable, the quantity of free electricity at the surface of the conducting wire is equal to the product of the potential into the capacity of the portion of the cable considered as a condenser.

If $a_{1},a_{2}$ are the outer and inner radii of the insulating sheath, and if $K$ is its specific dielectric capacity, the capacity of unit of length of the cable is, by Art. 126,

$c={\frac {K}{2\log {\frac {a_{1}}{a_{2}}}}}.$

(1)

Let $v$ be the potential at any point of the wire, which we may consider as the same at every part of the same section.

Let $Q$ be the total quantity of electricity which has passed through that section since the beginning of the current. Then the quantity which at the time $t$ exists between sections at $x$ and at $x+\delta x$, is

$Q-\left(Q+{\frac {dQ}{dx}}\delta x\right),\qquad {\mbox{or}}\qquad -{\frac {dQ}{dx}}\delta x,$

and this is, by what we have said, equal to $cv\delta x.$