Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/373

respect, and the currents $C_{1},C_{2},C_{3},$ and let the resistance of the multiple conductor be $R,$ and the total current $C$. Then, since the potentials at $A$ and $Z$ are the same for all the conductors, they have the same difference, which we may call $E$. We then have

$E=C_{1}R_{1}=C_{2}R_{2}=C_{3}R_{3}=CR,$

but

$C=C_{l}+C_{2}+C_{3},$

whence

${\frac {1}{R}}={\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}+{\frac {1}{R_{3}}}$

(7)

Or, the reciprocal of the resistance of a multiple conductor is the sum of the reciprocals of the component conductors.

If we call the reciprocal of the resistance of a conductor the conductivity of the conductor, then we may say that the conductivity of a multiple conductor is the sum of the conductivities of the component conductors.

Current in any Branch of a Multiple Conductor.

From the equations of the preceding article, it appears that if $C_{1}$ is the current in any branch of the multiple conductor, and $R_{l}$ the resistance of that branch,

$C_{1}=C{\frac {R}{R_{1}}},$

(8)

where $C$ is the total current, and $R$ is the resistance of the multiple conductor as previously determined.

Longitudinal Resistance of Conductors of Uniform Section.

277.] Let the resistance of a cube of a given material to a current parallel to one of its edges be $\rho$, the side of the cube being unit of length, $\rho$ is called the 'specific resistance of that material for unit of volume.'

Consider next a prismatic conductor of the same material whose length is $l,$ and whose section is unity. This is equivalent to $l$ cubes arranged in series. The resistance of the conductor is therefore $l\rho$.

Finally, consider a conductor of length $l$ and uniform section $s$. This is equivalent to s conductors similar to the last arranged in multiple arc. The resistance of this conductor is therefore

$R={\frac {l\rho }{s}}.$

When we know the resistance of a uniform wire we can determine