Page:Elementary Text-book of Physics (Anthony, 1897).djvu/383

§ 309] and the current in the circuit will vary in a way dependent not only on the variations in the electromotive force, but also on the currents produced by induction. The case of the greatest interest and importance is that in which the electromotive force varies periodically; in this case the current also varies periodically. It may be shown, by a method which cannot be given here, that the maximum value of the current is never as great as that deduced from the maximum electromotive force on the supposition that the current follows Ohm's law. The formula which expresses the maximum value of the current is $$\frac{e}{\left( r^2 + \frac{4\pi^2 L^2}{T^2} \right)^{\frac{1}{2}}},$$ where e is the maximum electromotive force and $$T$$ the period of the alternation. The denominator of this expression is a quantity of the same order as resistance, but it involves, besides the resistance of the circuit, its coefficient of self-induction and the period. In case $$\frac{4\pi^2 L^2}{T^2}$$ is very large in comparison with $$r^2$$ the current has its maximum value at the time when the electromotive force is zero, and is zero when the electromotive force is a maximum. The theory farther shows that the rate of propagation of the electrical disturbance along the conductor is a function of the period of the alternation, being less when the period is greater. When the period is infinitesimal, or in general when it is very small, the velocity is equal to the velocity $$v,$$ the ratio between the electrostatic and the electromagnetic units (§ 311), or to the velocity of light. The currents developed in the conductor, by rapid alternations of electromotive force, are not the same for all parts of the cross-section of the conductor, but diminish from the outside of the conductor inwards. For very rapid alternations the currents exist only in a small layer near the surface of the conductor. These deductions of theory have been fully confirmed by experiment.

309. Apparatus employing Induced Currents.— The production of induced currents by the relative movements of conductors and magnets is taken advantage of in the construction of pieces of