the lengths of these two lines, NP, NQ, bear to one another the same proportion as the magnitudes of these two forces. Then NR will

represent the resultant in magnitude and direction.

In a similar way a single resultant, S R' (Fig. 42), may be found for the forces acting on the other end of the magnetic needle N S. The needle is, therefore, acted on by two forces, NR, SR' (Fig. 42), which may, or may not, be equal to one another, which may, or may not, be parallel to one another, and which may, or

Fig. 41.

may not, change in either magnitude or in direction as the needle moves under the action of these two forces.

In the special case when the controlling and the deflecting fields are both uniform, N R and SR are parallel to one another, and do not alter in magnitude or direction as the needle deflects. They will not, however, be equal to one another unless, in addition to the two fields being uniform, the poles N and s of the needle have exactly the same strength. When

NR and SR are parallel in consequence of both the magnetic fields being uniform, the only position in which

the needle can come to rest is that in which the four points R, N, S, R' lie in one straight line which is parallel to N R or SR'; and this result, it is to be observed, is arrived at whether the poles of the needle are, or are not, equal to one another.

When the controlling field is uniform, if N P (Fig. 43) represents the direction and magnitude of the controlling force acting on one end of the needle for some

one

position of the

the controlling force for all positions of the needle. Similarly for a uniform de

flecting field, if PX represents the direction of the deflecting force due to any current acting on one end of the needle for some one position of the needle it will represent the direction of the deflecting force produced by any other current for any other position of the needle.

To ascertain the length along P X, which represents the magnitude of the deflecting force, produced by a particular current passing round the galvanometer coil, we must measure P R proportional to that current, and, since the deflecting field is a uniform one, the length PR remains unchanged as the needle moves. Hence N P and PR represent the directions and magnitude respectively of the controlling and deflecting forces, whatever be the position of the needle, and, therefore, the only position in which the needle can come to rest under the action of these forces is along a line parallel to N R.

When no current is passing, the position of the needle will, of course, be along a line parallel to N P,

and if the current that causes the needle to place itself parallel to N R be one ampere, the angle PNR is the deflection for one ampere. To find the deflecting force for two amperes a point, T, must be taken in the line P X such that PT is twice PR; then for two amperes the needle will place

itself parallel to PT, and the angle PNT will be the deflection for two amperes, &c.

Now it is clear

that the ratio of

the angle PNT

to the angle

PNR

will be very different

as the direction of

PX relatively to NP varies-that is, the ratio will depend on the angle which the needle makes with the plane of the coil when no current is passing. If, for example, PX were situated relatively to NP as in

Fig. 44, the angle PNT might be exactly twice the angle PNR, whereas if PX and NP were as in Fig. 43 the angle PNT would be less than twice the angle PNR.

22. Tangent Galvanometer. The relation between the angular deflections and the currents that produce them becomes very simple in one special case, and that is when the angle NPR is a right angle that is, when the magnetic axis of the needle is parallel to the plane of the coil when no current is passing. For in that case (Fig. 45)

Fig. 45.

since to find the tangent of an angle PNR we let fall a perpendicular from any point, R, of one of the lines, N R,

bounding the angle on to the other line, N P, and take the ratio which the side opposite to the given angle bears to the side adjacent to the angle.

Further, N P is a constant when a given magnetic needle is controlled by a given uniform magnetic field,

and PR is proportional to the current flowing round the galvanometer. Hence we see that the tangent of the deflection of a needle will be proportional to the current passing round a coil when the four following conditions are fulfilled :

:

(1) The needle is controlled by a uniform magnetic field.

(2) The diameter of the coil is large compared with the length of the needle.

(3) The needle is suspended sufficiently near the centre of the coil that the field which is produced by the current passing round the coil is a uniform one in the neighbourhood of the needle.

(4) The axis of the needle is parallel to the plane of the coil when no current is passing.

When these four conditions are all fulfilled the calibration curve of the galvanometer, when tested by comparison with a voltameter, as described in § 12, page 46, will be found to be of the shape shown in Fig. 46; and if any three points, P, Q, R, be taken on this curve, it will be found that the lengths A P, B Q, CR, parallel to o y, bear to one another the ratios of the tangents of the angles represented by o A, 0 B, and oc respectively. Such a galvanometer (seen in detail in Fig. 25, p. 49) is, therefore, called a "tangent galvanometer," and it may be henceforth used without reference to any voltameter for the comparison of current strengths, as they will be simply proportional to the tangents of the angles through which the magnetic needle is deflected.

23. Adjusting the Coil of a Tangent Galvanometer. We have next to consider how we can adjust the coil of a galvanometer so as to be sure that its mean plane is parallel to the axis of the needle when no current is passing. Owing to the coil having a certain breadth, it is impossible to see the needle when looking down on to the coil; indeed, it is for this reason that the long light pointer attached to the needle is placed at right angles to the needle. It would not be right to assume that because the instrument has been so turned that the pointer points to the zero on the scale, therefore the plane of the coil is parallel to the magnetic axis of the needle, for even if the scale has been attached to the instrument so that the line of zeros is at right angles to the plane of the coil, it does not follow that the pointer itself is at right angles to the needle. The two may