the same current, pass into different strengths of field according to the extent of its deflexion; in which latter case we could measure the fields but not

the current.

Fig. i. gives a general view of the tangent galvanometer. There is usually a short needle provided with a long and light index that moves round a graduated circle. Very often arrangements are made by means of which we can either use one thick circle of copper for very powerful currents, or two or more turns of wire for currents not so strong (see § 7). This circle or coil is placed in the plane of the magnetic meridian. In fig. ii. we are supposed to be looking down from above, and so to be viewing the top of the circle of copper and the needle, projected together on to a horizontal plane.

GG' is part of the projection of the copper circle; it lies, as mentioned in § 3, in the magnetic meridian. ns represents the needle; this lies in reality below GG', at the centre of the circle of which GG' is the

top. The dotted lines run

ning parallel to G G' repre

sent the direction of the

៥

FIG. ii.

earth's horizontal field, H; those perpendicular to G G' represent

the field I due to the current.

Using the notation of Chapter III. we have that (i.) Earth's couple acting on needle is H μl sin a. (ii.) Current's couple acting on needle is I μlcos a. When there is equilibrium we must have these couples equal; or Hμlsina I μl cosa.

Now, in the present case we have said that whatever be the deflexion of the needle we may consider I to be constant while the current is so, and to be directly proportional to the current C. Hence we may write C = k I, where k is some constant depending on the dimensions of the coil, the number of turns of wire, and the unit of current that we employ.

We have then Ck H. tan a.

Hence with the same instrument, and with H the same, the current is proportional to the tangent of the angle of deflexion; whence the name of this instrument.

It is clear that it can be used for currents up to any strength, since tan a can have any value. But it fails to give accurate measurement if a exceed (say) about 70°, since then large changes in current give small changes in «. The best angle of deflexion is about 45°, as may be shown mathematically.

Notes.-(i.) Other forms of the instrument. —- In Gaugain's form we have the needle placed in the axis of the coil, at a distance from the centre of the coil about one-fourth of its mean diameter. In Helmholtz's modification of Gaugain's instrument there are two coils lying symmetrically on either side of the needle, and the wires are wound on small portions of cones, of which, if completed, the centre of the needle would be the common apex. Such an arrangement gives a more uniform field, and permits one to use a somewhat larger needle.

(ii.) To set the coil in the magnetic meridian.—When the coil is in the magnetic meridian, the same current, passed in opposite directions, should give opposite deflexions of equal magnitude. The coil should be adjusted until this is the case.

(iii.) As regards the relative dimensions of needle and diameter of coil, giving results accurate enough for practical purposes, we quote here the following authorities.

Wiedemann (Die Lehre von der Electricität,' iii. 247) gives length of needle at most one-eighth diameter of coil.

Kempe (Handbook of Electrical Testing,' p. 18) gives needle about 3-inch for a 6-inch or 7-inch ring, as accurate enough for most purposes.

S. P. Thompson gives length of needle about one-tenth of diameter of coil.

Brough gives needle one-sixth of diameter of coil.

Andrew Gray gives needle 1 centimètre for coil of diameter 15 centimètres, in a standard instrument.

Hence, needle one-tenth of diameter of coil seems a good relative dimension.

§ 6. The Sine Galvanometer.-Fig. i. shows us a somewhat different form of galvanometer. Here the coil turns round on a vertical axis, the movement being measured over a horizontal graduated circle. As before, the coil is initially in the plane of the magnetic meridian.

We may here turn the coil until the current does not affect the needle at all. In this position it must be that the lines of force due to current coincide with

those of the earth, or the coil is due magnetic east and west. We then turn the coil back through 90° over its graduated scale, and it will lie in the plane of the magnetic meridian.

When the needle is deflected the coil is turned in the same direction; and finally, if the current be not too strong, can be brought again directly over the needle. When this is the case we note the angle through which the coils have been turned. The current will

be proportional to sin a.

FIG. i.

Fig. ii. shows us the theory of this instrument. In it is represented the final position of equilibrium, in which the coil GG lies again over the needle at an angle « from the magnetic meridian N S. Since now the lines of force due to the current are perpendicular to the needle, we have.

=

((i.) Current's couple I l

With similar reasoning to that given in the last section we have finally

C=kH. sin a

whence the name of the instrument.

It is evident that we cannot, without using shunts, measure any current exceeding / H, since the greatest value of sin a is 1,

N

FIG. ii.

which it has when a = = 90°, or when the needle stands E. and W. With a larger current we should chase the needle completely round.

Since the coil is always over the needle we may have this of any length we please, for it will always be in the same part of the current's field.

§ 7. The Multiplying Galvanometer.-The extent of deflexion of the needle depends upon the strength of the magnetic field due to the current.

With the same current,

this can be multiplied many-fold by passing the wire many times round the needle. In fact, we wind the wire in a coil and suspend the needle inside it.

The advantage of this can be demonstrated as in § 4.

A figure of a multiplier is shown in the next section.

§ 8. Astatic Galvanometer; Two Needles. In the last section we showed how to increase the current's action on the needle by increasing the field-strength due to the current.

There is another very effective manner in which the deflexion for a given current may be almost indefinitely increased. This other method is based upon the device of making the earth's restoring couple as weak as we please, while we leave the current's deflecting couple unaltered.

We cannot do this by weakening the magnetic moment of a

single needle; since, as we see in the formulæ of §§ 5 and 6, we should then weaken equally the current's action on the needle.

But we can so combine two needles that they form with respect to the earth a single needle of as small a magnetic moment as we please; while, by coiling the wire round one needle only, we can cause the current to act on a single needle of considerable magnetic moment.

A system of needles on which the earth has little (or, strictly, no) action, is called an astatic system; and a galvanometer in which such a system is

employed is called an astatic galvanometer.

There are two combinations that give us convenient astatic sys

tems.

(i.) Let us connect rigidly two needles ns and n's' of nearly equal strength, so that, when they are suspended, they are in the same vertical plane; the poles being directed opposite ways. If ns be somewhat the stronger, the two needles will act with respect to the earth as one needle of the same length, but of polestrength measured by the difference (u-'); so that the magnetic moment with respect to the earth is 7 (-), and may be very small indeed by having the pole-strengths μ and μ' nearly equal.

FIG. i.

The coils of the galvanometer, however, as seen in the figure, are so arranged that one needle of moment is in the strong field within the coil; while the other needle, pu'l, inasmuch as it is reversed in direction and is also in the reverse field above the