Note B ([see p. 273]).
Every electric circuit comprising a coil of wire and a condenser has a definite time-period in which an electric charge given to it will oscillate if a state of electric strain in it is suddenly released. Thus the Leyden jar L and associated coil P shown in [Fig. 82, p. 271], constitutes an electric circuit, having a certain capacity measured in units, called a microfarad, and a certain inductance, or electric inertia measured in centimetres. The capacity of the circuit is the quality of it in virtue of which an electric strain or displacement can be made by an electromotive force acting on it. The inductance is the inertia quality of the circuit, in virtue of which an electric current created in it tends to persist. In the case of mechanical oscillations such as those made by vibrating a pendulum, the time of one complete oscillation, T, is connected with the moment of inertia, I, and the mechanical force brought into play by a small displacement as follows: Suppose we give the pendulum a small angular displacement, denoted by θ. Then this displacement brings into existence a restoring force or torque which brings the pendulum back, when released, to its original position of rest. In the case of a simple pendulum consisting of a small ball attached to a string, the restoring torque created by displacing the pendulum through a small angle, θ, is equal to the product mglθ, where m is the mass of the bob, g is the acceleration of gravity, and l is the length of the string. The ratio of displacement (θ) to the restoring torque mglθ is 1 / mgl . This may be called the displacement per unit torque, and may otherwise be called the pliability of the system, and denoted generally by P. Let I denote the moment of inertia. This quantity, in the case of a simple pendulum, is the product of the mass of the bob and the square of the length of the string, or I = ml².
In the case of a body of any shape which can vibrate round any centre or axis, the moment of inertia round this axis of rotation is the sum of the products of each element of its mass and the square of their respective distances from this axis. The periodic time T of any small vibration of this body is then obtained by the following rule:—
| T = 2π | √ | moment of inertia round | } × { | displacement per unit of |
| the axis of rotation | torque, or pliability |
or T = 2π√IP.
In the case of an electric circuit the inductance corresponds to the moment of inertia of a body in mechanical vibration; and the capacity to its pliability as above defined. Hence the time of vibration, or the electrical time-period of an electric circuit, is given by the equation—
T = 2π√LC
where L is the inductance, and C is the capacity.
It can be shown easily that the frequency n, or number of electrical vibrations per second, is given by the rule—
| n = | 5000000 | |||
| √ | capacity in | } × { | inductance in | |
| microfarads | centimetres | |||