Using these values in the original equation, we may write it as follows:

E sin wt = [RSw cos (wt − r) + (1 − LSw2) sin (wt − r)]V。

E, sin r= RSwV, and E, cos r = (1 − SLw2)V ̧

respectively. The potential-difference at time, 4, therefore becomes

and the retardation of phase r is determined by the equation

The simultaneous strength of current is given by the equation

L

similar to that of the ratio in the case of a coil: if the surfaces

R

of a charged condenser of capacity S, are connected through a conductor of resistance, R, the difference of potentials falls to the fraction of its initial value in the time RS. This pro

I

=

I

e 2.718...

duct is, consequently, called the "time-constant" of the condenser. If we denote it by v, using, as before, λ for the corresponding value L/R, we may write

and remembering that w stands for 2π/T, where 7 is the period of rotation of the coil, we see that if

the angle denoted by r becomes a right angle, and the expression for the current-strength becomes

Eo
C: = sin wt.
R

That is to say, under the specified conditions, the self-induction and capacity of the circuit exactly neutralise each other's effects, both the strength of the current and its phase being the same as they would be in a simple circuit of resistance, R, without self-induction.

329. Electrical Oscillations.-When the surfaces of a charged condenser are suddenly connected by a conductor of small resistance, the electric energy is converted into heat, but the process is, comparatively speaking, a gradual one, and involves the charging of the condenser a large number of times in succession, and alternately in opposite directions; the energy represented by each charge being a constant fraction of that of the preceding one. In fact, a series of electric oscillations of gradually decreasing amplitude is set up.

In order to follow out the theory of the process, we will, in the first instance, suppose the resistance of the conductor to be inappreciable. Denoting the charge of a condenser by Q, and its capacity by S, the electrostatic energy is 2/S (§ 55). When the coatings are connected by a conductor of self-induction, L, the electrostatic energy disappears, giving rise to a current and an amount of electromagnetic energy represented (§ 296) by §LC2. This in its turn dies away, reproducing the original amount of electrostatic energy in the form of a reversed charge of the condenser. This, as soon as it is formed, begins to diminish again, giving rise to an inverse current and reproducing electromagnetic energy; and so the process goes on, the original energy of the condenser being transformed into the electromagnetic and electrostatic forms alternately.

From the principle of conservation of energy we conclude that the total energy, electrostatic and electromagnetic together, is constant, or

+ /C2 = constant.

which simply means that an increase of one kind of energy is accompanied by an equal decrease of the other kind. But C = dQ/dt, and therefore dC/dt = d2Q\dt2: consequently, the last equation may be written

Let us compare this with the formula that expresses the acceleration of a particle of mass, m, which, when displaced from a position of stable equilibrium, is urged back by a force proportional to the displacement, x, namely, the formula

where ƒ stands for the restoring force corresponding to unit displacement. This equation states that the acceleration is proportional to displacement and in the opposite direction. We know that, under the conditions here specified, the motion of the particle is simply harmonic, and may be represented by

x = x, cos wt,

if x is the amplitude and denotes time reckoned from an instant when the particle is at the positive extremity of its path. The velocity is

dx
dt

Equating this with the expression given above, we get

All these mechanical results have strict analogues in the electrical case. The charge, 2, of a condenser may be taken as a measure of the displacement from the condition of electrical

dQ dt

equilibrium ; then becomes the rate of change of electric dis

placement, and might be called electric velocity; and

acceleration of electric displacement. The expression we obtained above for this quantity shows it to be directly proportional to the electric displacement, Q. Hence, following the mechanical analogy, and putting Q, for the initial charge of the condenser, we may write

2 = 2, cos wt;

if, as before, we use λ and for the two time-constants.

330. Comparison with Experiment.—These equations represent the discharge of a condenser, under the conditions stated, as giving rise to an endless series of oscillations of the same amplitude taking place in the periodic time 71=27 √, the geometric mean of the electromagnetic and electrostatic time-constants of the system x 2π.

For two reasons, however, these results do not accurately agree with what takes place in real cases. On the one hand, the true seat of the changes which we have spoken of as electrostatic and electromagnetic oscillations is not the conductor connecting the surfaces of the condenser, but the dielectric medium of the condenser and that by which the conductor is surrounded. Any

1 This may be called the natural oscillation-period of the circuit, and it may be noted that the condition pointed out at the end of § 328, under which a condenser can neutralise the effects of self-induction in the case of an alternating electromotive force, is the condition of equality between the period of the electromotive force and the oscillation-period of the circuit.

$ 330.]

Comparison with Experiment.

393

change of condition, such as those accompanying electric charge and discharge, produced at one part of this medium, is propagated with a definite velocity to more and more distant parts, and consequently a series of electrical alternations, such as we have been discussing, gives rise to a series of waves which travel outwards in all directions into the surrounding space. Electric energy is thus radiated away from the oscillating system, and the energy of the latter is consequently diminished in proportion. The process is closely analogous to the gradual loss of energy by a vibrating tuning-fork consequent upon its radiation of soundwaves into the air.

We shall return presently (§ 333) to the subject of electric radiation.

Another respect in which the conditions assumed in § 329 differ from those of actual experiment is that they take no account of the resistance of the conductor. In certain cases this resistance may be so small that the formulæ we have arrived at represent the results with fair accuracy, but it can never be got rid of altogether. The existence of resistance causes a certain amount of electric energy to be converted into heat at every oscillation, just as friction does in the case of ordinary mechanical vibrations. The electric energy which is thus expended in an element of time at is, in agreement with Joule's law,

R (de) át.

dt

Adding this term to the equation from which we started in § 329, we get after simplification,

This equation may be taken as a statement in mathematical language that the rate of generation of electromagnetic energy, together with the rate at which electric energy is expended in generating heat, is equal to the rate of decrease of electrostatic energy; or, again, as a statement that the electromotive force due to self-induction, together with that required to maintain the existing current, is equal and opposite to the difference of potentials between the surfaces of the condenser. The discussion of the equation would involve more elaborate mathematical processes than we can enter upon here: we must, therefore, be content with merely stating the most important conclusions to which it leads.