Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/375

280.]
333
SYSTEM OF LINEAR CONDUCTORS.

radius of the sphere must diminish in order to maintain the potential constant when the charge is allowed to pass to earth through the conductor.

In the electrostatic system, therefore, the conductivity of a conductor is a velocity, and of the dimensions ${\displaystyle [LT^{-1}].}$

The resistance of the conductor is therefore of the dimensions ${\displaystyle [L^{-1}T].}$

The specific resistance per unit of volume is of the dimension of ${\displaystyle [T],}$ and the specific conductivity per unit of volume is of the dimension of ${\displaystyle [T^{-1}].}$

The numerical magnitude of these coefficients depends only on the unit of time, which is the same in different countries.

The specific resistance per unit of weight is of the dimensions ${\displaystyle [L^{-3}MT].}$

279.] We shall afterwards find that in the electromagnetic system of measurement the resistance of a conductor is expressed by a velocity, so that in this system the dimensions of the resistance of a conductor are ${\displaystyle [LT^{-1}].}$

The conductivity of the conductor is of course the reciprocal of this.

The specific resistance per unit of volume in this system is of the dimensions ${\displaystyle [L^{2}T^{-1}],}$ and the specific resistance per unit of weight is of the dimensions ${\displaystyle [L^{-1}T^{-1}M].}$

On Linear Systems of Conductors in general.

280.] The most general case of a linear system is that of ${\displaystyle n}$ points, ${\displaystyle A_{1},A_{2},\ldots A_{n},}$ connected together in pairs by ${\displaystyle {\frac {1}{2}}n(n-1)}$ linear conductors. Let the conductivity (or reciprocal of the resistance) of that conductor which connects any pair of points, say ${\displaystyle A_{p}}$ and ${\displaystyle A_{q},}$, be called ${\displaystyle K_{pq},}$ and let the current from ${\displaystyle A_{p}}$ to ${\displaystyle A_{q}}$ be ${\displaystyle C_{pq}}$. Let ${\displaystyle P_{p}}$ and ${\displaystyle P_{q}}$ be the electric potentials at the points ${\displaystyle A_{p}}$ and ${\displaystyle A_{q}}$ respectively, and let the internal electromotive force, if there be any, along the conductor from ${\displaystyle A_{p}}$ to ${\displaystyle A_{q}}$ be ${\displaystyle E_{pq}.}$

The current from ${\displaystyle A_{p}}$ to ${\displaystyle A_{q}}$ is, by Ohm's Law,

 ${\displaystyle C_{pq}=K_{pq}(P_{p}-P_{q}+E_{pq}).}$ (1)

Among these quantities we have the following sets of relations:

The conductivity of a conductor is the same in either direction,

 ⁠or ${\displaystyle K_{pq}=K_{qp}.}$ (2)

The electromotive force and the current are directed quantities, so that

 ${\displaystyle E_{pq}=-E_{qp},\quad \quad }$and ${\displaystyle \quad \quad C_{pq}=-C_{qp}.}$ (3)