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

330 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,

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

Now

Comparing this result with (3), we find 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

the potential at $$B$$ is 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 $$R_1,\,$$ $$R_2,\,$$ $$R_3$$