325. Case of Soft Iron.—Suppose a soft iron core surrounded by a coil of wire between the ends of which a difference of potential is maintained which varies as the ordinate of a sine

curve.

In order to simplify the problem, we will assume that the magnetic induction is at each instant proportional to the existing strength of the current. This simplification, which is justifiable as a first approximation, amounts to disregarding hysteresis and the variations of permeability; in other words, it amounts to replacing the closed curve, which represents a complete cycle of magnetisation, by a simple straight line passing through the origin (Fig. 174).

If n is the number of turns, and R the magnetic reluctance (§ 276), we have at each instant for the relation (§ 273) between the strength, C, of the current and the total flux of induction, Q,

RQ = 4πNC.

Consequently, supposing the strength of current to be unity, and remembering that the flux is encircled n times by the wire, we get for L, the coefficient of self-induction,

and by putting this value into the formulas (5), (6), and (7) of § 321, we get the retardation of phase and the strength of current.

The high value of μ causes the self-induction to be great, and consequently, for high frequencies, the retardation is nearly 90°, and therefore cos differs but little from zero, and formulas (6) and (8) show that the current and rate of doing work are very small.

A coil with a soft iron core placed in a circuit carrying a rapidly alternating current has thus the curious property of choking the current, but without the expenditure of energy which would take place with a simple resistance.

326. Action on an Adjacent Circuit.-A harmonically varying current causes a harmonically varying electromotive force to act on an adjacent circuit, and therefore gives rise to a current which is also harmonic, and of the same period as itself, but which differs in phase. The difference of phase can never be less than greater than

π

nor

π

For the secondary electromotive force has a

2

$ 326.]

π

Action on Adjacent Circuit.

385

retardation of in respect of the primary current, and self-in

duction produces a further lag comprised between o and

π

Professor Elihu Thomson has shown that, under these conditions, there is always repulsion between the two circuits. Fig. 288 represents one of his experiments. An electromagnet is traversed by a powerful alternating current; when a ring is brought near one end and left to itself, it is strongly repelled.

The electrodynamic force at each instant is proportional to the product of the strength of the current equivalent to the electromagnet and that induced by it in the ring. From Ampère's law, the two currents attract if they are in the same direction, and repel if they are opposite. In Figs. 289 and 290, two sine curves, A and A', have been traced with the same period, but with amplitudes which are as 4:1. The ordinates of the curve B are taken

B

FIG. 288.

АЛА

FIG. 289.

equal to the products of the corresponding ordinates of A and A', and are drawn above the line if the product is positive, and below if it is negative. When the difference of phase is equal to (Fig. 289),

π

the attraction and repulsion are equal both in intensity and in duration, and the resulting effect is therefore nothing. But if the differ

ence of phase exceeds and this is what always takes place (Fig. 290), repulsion preponderates over attraction, and the more so the greater the difference of phase.

Professor E. Thomson has planned several arrangements in which the effect of repulsion may give rise to a continuous rotatory motion. In that shown in Fig. 291,

B

FIG. 291.

M

A

a copper disc, B, mounted on a spindle, rotates rapidly when it is placed excentrically in reference to the alternating pole, and when another disc, A, partly screens the first. If the disc A is itself movable, it will turn in a contrary direction to the disc B.

327. Distribution of Alternating Currents in Conductors.-In the case of alternating currents the apparent resistance greatly exceeds the real resistance; this latter is itself increased by the fact that the current is not distributed uniformly throughout the whole section of the conductor, but is concentrated near the surface.

Experiment shows, in fact, that a current at starting is first formed on the surface, and that it only gradually, though very quickly, reaches the deeper layers. If its direction is reversed at very short intervals, the current never fully reaches the centre. The case may be compared with that of an infinitely long tube filled with liquid, which is rapidly moved to and fro lengthwise. If the liquid is a perfect fluid, it will remain at rest; if it is viscous, the layers in contact with the sides of the tube will share the

$327.1

Distribution of Alternating Currents.

387

motion of the tube, but the inner layers will oscillate with an amplitude which decreases, and with a retardation of phase which increases, from the periphery to the axis.

The results in the electrical case are in full agreement with what has been already stated (see especially §§ 296-298, and 314) as to the seat of electric and magnetic energy. It has been pointed out that the energy of an electric current enters the conductor from the surrounding non-conducting medium. It is a necessary consequence of this that it arrives first at the outer layers, and that its penetration to the inside, though a rapid, is not an instantaneous process.

What we recognise as an electric current is, in one aspect, the process by which energy, whether electric or magnetic, passes from the field and generates heat in the conductors. Whenever the result of a current in a conductor would be a diminution of the energy of the field, such a current takes place. If energy is poured into the field in any way as fast as it is removed by the conductors, the condition known as that of a steady current is set up, but in all cases the ultimate distribution of the current among the available conductors is such that the total energy is a minimum. The electric energy corresponding to a given strength of current is proportional to the difference of potentials between the ends of the conductor, or, what comes to the same thing under the conditions specified, to the resistance-and this is least if the current makes use of the whole cross-section, distributing itself so that the currentdensity is uniform. On the other hand, the magnetic energy is least when the current is confined to the outer skin of the conductor; for it is easily proved, by an adaptation of the reasoning employed in § 19 in relation to spheres, that a current distributed uniformly on the surface of a circular cylinder exerts no magnetic force at an internal point, and that at external points it acts as though it were concentrated in the axis. It follows that the external magnetic field is the same whether the current is confined to the surface of a conductor, or uniformly distributed through the cross-section; whereas, in the former case the magnetic field inside the conductor itself is nothing, and in the latter case the internal magnetic force is proportional to the distance from the axis. The magnetic energy of the conductor itself is greater the greater the magnetic permeability, and hence the effect in question is more marked with iron than with copper.

For a frequency of 80 periods per second, the virtual increase of

resistance due to concentration of the current near the surface is I per cent. in a copper wire I centimetre in diameter, and 8 per cent. in a wire of 2 centimetres diameter. For large diameters the conducting power of a cylindrical conductor increases almost as the perimeter instead of as the section. Hence we have the practical conclusion, that for very strong alternating currents tubes should be used, and not solid conductors.

328. Influence of Capacity.-A difference of potential cannot exist between two conductors without correlated electrostatic charges. Each element of a wire in which a current is established acts as a capacity, and the assumption we have hitherto made that at each instant, every section of the conductor is traversed by the same quantity of electricity, is strictly true only when the current has reached the steady state.

If a condenser is connected in circuit with a coil revolving in a magnetic field, as described in § 320, or with any other equivalent source of alternating electromotive force, it acquires alternate positive and negative charges, the difference of potentials of the coatings varying harmonically, and the period being the period of revolution of the coil. The difference of potentials of the coatings is equivalent to an electromotive force opposing that due to the rotation of the coil. Calling the potential-difference at any instant V, the strength of current at the same instant C, and the capacity of the condenser S, the charge is SV, and its rate of change, Sv sav dt' is equal to the strength of the current. Writing, as before, E, sin wt for the instantaneous electromotive force, we have

To express that V varies periodically in the same period as the electromotive force, we may write

V = V, sin (wt - r),

where is the maximum value and is a constant determining the difference of phase between the electromotive force and the charge of the condenser. From this expression for V we get for the strength of current