Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/372

with $A_{2},$ so that the second conductor has for its electrodes $A_{2},\,$ $A_{3}$. The electrodes of the third conductor may be denoted by $A_{3}$ and $A_{4}$.

Let the electromotive force along each of these conductors be denoted by $E_{12},$ $E_{23},$ $E_{34}$, and so on for the other conductors.

Let the resistance of the conductors be

$R_{12},\;\;R_{23},\;\;R_{34},\;\;\mathrm {\&c} .$

Then, since the conductors are arranged in series so that the same current $C$ flows through each, we have by Ohm's Law,

$E_{12}=CR_{12},\;\;E_{23}=CR_{23},\;\;E_{34}=CR_{34}.$

(2)

If $E$ is the resultant electromotive force, and $R$ the resultant resistance of the system, we must have by Ohm's Law,

$E=CR.$

(3)

Now

${\begin{array}{rl}E&=E_{12}+E_{23}+E_{34}\\&\quad \quad \quad \quad {\mbox{the sum of the separate electromotive forces,}}\\&=C(R_{12}+R_{23}+R_{34})\quad {\mbox{by equations (2).}}\end{array}}$

(4)

Comparing this result with (3), we find

$R=R_{12}+R_{23}+R_{34}.$

(5)

Or, the resistance of a series of conductors is the sum of the resistances of the conductors taken separately.

Potential at any Point of the Series.

Let $A$ and $C$ be the electrodes of the series, $B$ a point between them, $a$, $c$, and $b$ the potentials of these points respectively. Let $R_{1}$ be the resistance of the part from $A$ to $B,\,$ $R_{2}$ that of the part from $B$ to $C$, and $R$ that of the whole from $A$ to $C$, then, since

$a-b=R_{1}C,\;\;\quad b-c=R_{2}C,\;\;\quad$ and $\quad \;\;a-c=RC$,

the potential at $B$ is

$b={\frac {R_{2}a+R_{1}c}{R}}$,

(6)

which determines the potential at $B$ when those at $A$ and $C$ are given.

Resistance of a Multiple Conductor.

276.] Let a number of conductors $ABZ,\,$ $ACZ,\,$ $ADZ$ be arranged side by side with their extremities in contact with the same two points $A$ and $Z$. They are then said to be arranged in multiple arc.

Let the resistances of these conductors be $R_{1},\,$ $R_{2},\,$ $R_{3}$ respect-