Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/374

the specific resistance of the material of which it is made if we can measure its length and its section.

The sectional area of small wires is most accurately determined by calculation from the length, weight, and specific gravity of the specimen. The determination of the specific gravity is sometimes inconvenient, and in such cases the resistance of a wire of unit length and unit mass is used as the 'specific resistance per unit of weight.'

If $r$ is this resistance, $l$ the length, and $m$ the mass of a wire, then

$R={\frac {l^{2}r}{m}}.$

On the Dimensions of the Quantities involved in these Equations.

278.] The resistance of a conductor is the ratio of the electromotive force acting on it to the current produced. The conductivity of the conductor is the reciprocal of this quantity, or in other words, the ratio of the current to the electromotive force producing it.

Now we know that in the electrostatic system of measurement the ratio of a quantity of electricity to the potential of the conductor on which it is spread is the capacity of the conductor, and is measured by a line. If the conductor is a sphere placed in an unlimited field, this line is the radius of the sphere. The ratio of a quantity of electricity to an electromotive force is therefore a line, but the ratio of a quantity of electricity to a current is the time during which the current flows to transmit that quantity. Hence the ratio of a current to an electromotive force is that of a line to a time, or in other words, it is a velocity.

The fact that the conductivity of a conductor is expressed in the electrostatic system of measurement by a velocity may be verified by supposing a sphere of radius $r$ charged to potential $V,$ and then connected with the earth by the given conductor. Let the sphere contract, so that as the electricity escapes through the conductor the potential of the sphere is always kept equal to $V.$ Then the charge on the sphere is $rV$ at any instant, and the current is ${\frac {d}{dt}}(rV),$ but, since $V$ is constant, the current is ${\frac {dr}{dt}}V,$ and the electromotive force through the conductor is $V.$

The conductivity of the conductor is the ratio of the current to the electromotive force, or ${\frac {dr}{dt}}$, that is, the velocity with which the