$337.]

Electricity and Optics.

409 quite conceivable that the virtual elasticity of the ether in reference to them may not be the same as that which comes into play in the case of the statical processes, or of the relatively slow vibrations that have been employed in determinations of dielectric coefficients. On the whole, it may be said that comparisons of indices of refraction and dielectric coefficients have afforded a remarkable confirmation of the general truth of our theory.

CHAPTER XXVIII.

GALVANOMETERS.

338. Galvanometer.-A galvanometer, in the strict sense of the word, is an instrument which serves to measure the intensity of a current by the action which it exerts on a magnetic needle.

B

A

a

b

F

It consists essentially of a frame on which wire is coiled in a vertical plane, and of a magnetic needle suspended horizontally at the centre of the coil (Fig. 301). The plane. of the wire is placed in the magnetic meridian, and therefore the needle is parallel to it in its position of equilibrium. to the current near the

FIG. 301.

The field due to the earth and that due middle of the coil are thus at right angles.

Let us suppose that the field due to the current is uniform in the region which is occupied by the needle, and let G be its strength for unit current; it will be GC for a current of strength C, and under the action of the two fields the needle will come to rest in a position which makes an angle a with its original direction. If H is the horizontal component of the magnetic force of the earth and M the moment of the needle, the angle a is given by the equation of equilibrium

Thus, in the case in which the two fields are uniform and at right angles, the deflection is independent of the magnetic moment M of the needle, and therefore of its shape and magnetisation, and the tangent of the deflection is proportional to the strength of the current.

$339.]

Measurement of Currents.

411

This condition that the field be uniform is of prime importance for the galvanometer. It dispenses with any empirical graduation. It is easily realised if small needles and small angles of deflection are used. This is the case with all galvanometers of modern construction. The inconveniences of small deflections are amply compensated by the accuracy with which they can be measured by the method of reflection (§ 80).

The scale reading gives directly tan 2a. If the deflection does not exceed 3 to 4 degrees, the error may usually be neglected which arises from taking 2 tan a = tan 2a, and therefore in assuming that the current - strength for a fixed distance of the scale is proportional to the displacement xp of the image. For a deflection of 2° the error due to this assumption is less than 0. 339. Absolute Measurement of Currents—Tangent Galvanometer.-The absolute measurement of current-strength depends on the measurement of G and H. We know how H is obtained (§ 231). The value of the quantity G, which we will call the galvanometer-constant, is deduced from the dimensions of the coil of wire. This requires that the windings should be accurately circular, and wound with such regularity that the radius can be exactly measured. The radius should, moreover, be very large in comparison with the dimensions of the needle, if we are to assume that the field in which the needle hangs is uniform.

A much more uniform field is obtained by using two equal circular coils, placed parallel to each other at a distance equal to their common radius (§ 251).

A current of unit strength going once round a circle of radius a, produces a magnetic force which, at any point on the axis of the circle, is directed along the axis, and at distance r from the centre is equal to

The values which this expression assumes for various values of x are shown graphically in Fig. 302, by the ordinates of the curve ACB1. The value o corresponds with the centre of the circle; the force here is greatest, and is represented by the maximum ordinate OIA. As r increases, the magnetic force, as shown by the curve, decreases, at first slowly, then more and more rapidly until becomes a, after which the rate of decrease of force again gets slower and the representative curve becomes less steep.

At the point of greatest steepness, C, the curve changes from being concave towards the axis to being convex, and at this point there is no curvature either way. In other words, at a distance along the axis equal to half the radius of the circle, the magnetic force decreases for a little way at a uniform rate. Hence, if two equal circles or circular coils are placed so that their axes coincide, and at a distance apart equal to their common radius, the magnetic field on the axis at a point half-way between the circles is uniform, since the decrease of force corresponding to a small increase of distance from one circle is compensated by the equal increase of force due to decrease of distance from the other. The curve

Q

B

FIG. 302.

PIQP2, which represents the magnetic force due to two equal circular currents placed as described above, exhibits this result. The ordinates of this curve are the sums of the corresponding ordinates of the curves A,CB and A,CB, representing the forces due to each of the two circles acting separately. For a distance on either side of the middle point equal to one-tenth of the common radius, the force decreases only by about one part in 3000, so that, with two circles, each of 25 centimetres radius, the magnetic force is almost exactly uniform for a distance of 2.5 centimetres on each side of the middle point.

Fig. 303 represents a galvanometer constructed upon this principle, which is due to Helmholtz. With an instrument of this kind, having in all n turns of wire arranged in two equal coils of mean radius a, with a distance 2 x = a between the mean planes of

§ 339.]

Tangent Galvanometer.

413

the coils, the magnetic field at the centre due to a current of unit strength in the wire is

If the circles are accurately set with their common axis perpendicular to the magnetic meridian, the field = CG due to the current is at right angles to the earth's horizontal field, H, and the

resultant force makes with the meridian an angle, a, given by the relation

and the axis of a small delicately suspended magnet at the centre of the instrument sets itself in this direction. The strength of a current which causes a deflection, a, of the magnet is therefore

For example, suppose there are 60 turns of wire, 30 on each