plane of the magnetic meridian. And let there be suspended at its centre by a 'torsion wire' a relatively small coil, whose faces are turned due magnetic N and S, and whose plane therefore is perpendicular to that of the larger coil.

If a current be passed in the right direction through the small moveable coil, it will be in equilibrium with respect to the earth's horizontal field, since its faces (or poles) are turned N and S. If another current be passed through the larger coil, the field due to this will act on the smaller coil, tending to set it parallel to the larger. We might have given the small coil free suspension, and have used the small coil as we used the needle in § 5. But since we wish to get rid of H from the formula of calculation, we twist the torsion wire until we bring the small coil back to its zero position, in which the earth's field did not act on it. The angle of torsion then measures the couple with which (to speak intelligibly enough, even if not very accurately) the large coil acts on the smaller.

We have then a balance between-(i.) the torsion of the wire on the one hand, and (ii.) the electro-magnetic couple between the coils on the other. The former can be determined by ordinary mechanical methods, and the latter depends (in a manner known to those who have studied the theory of electro-magnetic actions) on the dimensions of the coils, and on the product of the strengths of the two currents. Thus, for each instrument a formula can be once for all constructed, giving the value of one current when the other is known.

If we pass the same current through both coils, then we can measure this current by measuring only the above given angle of torsion. This last form of the instrument is adapted for the measurement of currents that pass in rapidly alternating directions, since in this case the reverse current in the large coil finds a needle of reverse polarity upon which The angle of torsion will in this case be proportional to the square of the current-strength.

to act.

§ 16. Some General Observations on Galvanometers.-We will end this Chapter with a few general remarks on galvano

meters.

(I.) Long-coil and short-coil galvanometers.-Fleeming Jenkin uses these two very clear terms to indicate instruments in which the current passes many times, or few times, round the needle respectively.

For obvious reasons we cannot increase the field acting on the needle by the device of passing the current many times round without at the same time unavoidably increasing the resistance of

the instrument, for we cannot make coils having many turns of any but fine and therefore highly-resisting wire.

In using an instrument we must, then, consider whether the resistance R of the rest of the circuit be large or small as compared with the resistance G of the galvanometer. If G be negligible as compared with R, the field is multiplied n-fold by using n turns of wire, and the deflexion for a given current is thus made much greater. But if R is negligible as compared with the resistance of one turn of wire, then we get but th of the

n

current when we use n turns, and thus gain nothing but inconvenience in using a long-coil instrument.

§ 17. Galvanometers for Practical, or Commercial, Use. Now that electrical measurements have to be made every day in places where the presence of large magnets and powerful electrical currents renders the magnetic field unknown and variable, it is necessary to have instruments whose action is independent of the earth's field of force, and whose accuracy is not perceptibly impaired by such disturbances as are likely to occur.

At the end of Chapter XXV. will be found some explanation of the principles of several of the more recent types of instruments. $ 18. Calibration of Current-meters and of Voltmeters. The system of units employed in modern electrical science is based upon the electro-magnetic actions of currents, as will be explained in Chapter XVIII. And the practical unit of current employed, the ampère, can be determined by observing the attraction or repulsion between accurately constructed coils of wire carrying a steady current. [It was (e.g.) in some such way that Lord Rayleigh measured in ampères the currents he employed.] The ampère being thus measured electro-magnetically, it can be determined how much silver or copper is set free in 1 second by a current of 1 ampère.

From the experiments of Lord Rayleigh and others it appears that about 1118 milligramme of silver is set free by I ampère in 1 second. In practical text-books the student will find it explained how this result renders it a comparatively easy matter to calibrate any given ammeter in ampères.

As regards voltmeters, we may [see Chapter XIV. § 11] graduate them also by means of currents measured electro-chemically. Or we may make use of the known E.M.F.s of certain standard cells; these E.M.F.s having been previously determined once for all, in absolute value, by long and careful laboratory experiment.

For further on 'Calibration' we refer the student to practical text books.

CHAPTER XVIII.

ACTIONS BETWEEN CURRENTS AND MAGNETIC POLES.-MAGNETIC EQUIVALENT OF A CURRENT.-ACTION BETWEEN CURRENTS

AND CURRENTS.

§ 1. Action of an Infinite Rectilinear Current on a Magnetic Pole. If we pass a current through a vertical rectilineal wire A B, and observe its action on a horizontally-balanced magnetic needle ns placed rear it, we arrive experimentally at several important results.

n

(i.) We find (as also was indicated by the experiments of Chapter XVII.) that the lines of force form circles. whose centres lie on the wire and whose planes are perpendicular to it.

(ii.) We find that when once the wire is so long that it makes the angle A OB nearly 180°, then, as far as its action on the needle is concerned, it is practically infinite.

This suggests the idea that the current gives a field at O only so far as it gives a 'broadside-on' projection when viewed from O; and that a current running directly towards O would give no field there.

(iii.) We find that the field-strength at O is directly proportional to the current-strength, as has been otherwise shown in Chapter XVII. § 4.

(iv.) We find that, for the same current, the field-strength is inversely proportional to the perpendicular distance from O to the wire.

§ 2. Action of an Element of a Current on a Pole.—If we wish to be in a position to calculate the action of a current of any shape on a pole having any position with respect to the current,

we must know the action of each little bit (or element) of the current on the pole, and must sum up all these actions in order to get the total action of the entire current.

Such summing-up belongs to the integral calculus in general; but in one case it can be done very simply.

For the benefit of those readers whose mathematical knowledge is elementary, we shall explain further the meaning of an element of a current, and shall then discuss in a simple manner the law of action that is found to hold, and the application of this law to determine the field at the centre of a circular current (see § 6).

Let ab represent a bit of a current so small that the following results may be considered to be true.

(i.) ab may be considered to be straight, although it be part of a curved circuit. Its length will be designated by ds, where in general ds is a very small fraction of a

centimètre.

(ii.) The length μa (where μ represents the position of the pole acted on, the pole-strength

r

a

being μ units) may be considered as equal to μ b, in whatever position ab stand with respect to μ. Thus, in the figure, r centimètres is the length of either μ a or μ b. This is because we suppose d's to be exceedingly small with respect to r.

(iii.) If we draw ba' so as to make an isosceles triangle bμ a', we may consider ba' to be at right angles to both μ a' and μ b.

(iv.) And, following from (iii.), we may speak of ba as making 'the angle ' with either a, or with ub produced; we shall generally say

'with r.'

When the above conditions hold, a b is called an element. These conditions do not hold in the figure, since that is exaggerated in order to make details sufficiently plain.

From the experiment of § 1 it was not difficult for mathematicians to guess at the law of action of each element of current on the pole or on the short needle. The law guessed at was such as, applied to the case of § 1, would enable us to predict the results there experimentally arrived at. It was then applied to other cases. The total action of a circuit was predicted by means of the integral calculus from the assumed law of action of an element. In each case thus tested, experiment fulfilled the prediction. When any assumed law stands thus the test of experiment,

we must be satisfied to consider it to be true. We now give the law, thus established, of the action of an element of current on a pole.

Let μ be the pole-strength, the distance in centimètres of the element ds from the pole, ds the length of the element, ds. sin ✪ the projection of ds perpendicular to r (or the length b a' in the figure), C the current-strength. Then it is found that the pole

is urged with a force proportional to ". C.ds.sing in a direction perpendicular to the plane containing the element ds and the pole p, the direction of this force being further determined by Ampère's rule.

If we analyse this law with the aid of the figure given, we see that Istly. The force acting on μ is directly proportional to μ and to C. This fact is what we should expect. In the last Chapter it was partly proved, and partly assumed as following from the definition of μ and C.

2ndly. For the same length ds of the element, the force is proportional to sin 9; that is, the element acts solely in so far as it presents a 'broadside-on projection' (which is a' b in the figure) to the pole.

I

3rdly. The force varies as ; thus following the same law as we found to hold in gravitation, in electrostatics, and in magnetism.

4thly. If the current have the direction of the arrow, and the pole beor north-seeking, this latter is urged straight up from the plane of the diagram.

5thly. The action is not in the line joining the pole and the element, but is perpendicular to the plane containing the two. The action then is very different from any with which we had to do in electrostatics or in magnetism.

Note on the experiment of § 1.-In the case of the infinite rectilinear current, we found the total field at ns to vary as whereas the elemental law

has. This may perhaps perplex the learner.

I

The reason is as follows. As we recede further from the current A B, we certainly increase the distance from each element. But at the same time the upper and lower parts of the wire begin to present an increasingly great 'broadside-on projection' to the pole or needle.