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

277], 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

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,

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

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