On Sensitive Galvanometers.

717.] In the construction of a sensitive galvanometer the aim of every part of the arrangement is to produce the greatest possible deflexion of the magnet by means of a given small electromotive force acting between the electrodes of the coil.

The current through the wire produces the greatest effect when it is placed as near as possible to the suspended magnet. The magnet, however, must be left free to oscillate, and therefore there is a certain space which must be left empty within the coil. This defines the internal boundary of the coil.

Outside of this space each winding must be placed so as to have the greatest possible effect on the magnet. As the number of windings increases, the most advantageous positions become filled up, so that at last the increased resistance of a new winding diminishes the effect of the current in the former windings more than the new winding itself adds to it. By making the outer windings of thicker wire than the inner ones we obtain the greatest magnetic effect from a given electromotive force.

718.] We shall suppose that the windings of the galvanometer are circles, the axis of the galvanometer passing through the centres of these circles at right angles to their planes.

Let rsin be the radius of one of these circles, and r cos the distance of its centre from the centre of the galvanometer, then, if is the length of a portion of wire coinciding with this circle, and γ the current which flows in it, the magnetic force at the centre of the galvanometer resolved in the direction of the axis is

sin 0

γι

Fig. 52.

of wire bent into the form of a circular are will produce a greater magnetic effect when it lies within this surface than when it lies outside it.

719.]

SENSITIVE GALVANOMETER.

323

It follows from this that the outer surface of any layer of wire ought to have a constant value of x, for if x is greater at one place than another a portion of wire might be transferred from the first place to the second, so as to increase the force at the centre of the galvanometer.

The whole force due to the coil is y G, where

the integration being extended over the whole length of the wire, a being considered as a function of l

719.] Let y be the radius of the wire, its transverse section will be y2. Let p be the specific resistance of the material of which the wire is made referred to unit of volume, then the resistance of a and the whole resistance of the coil is

length / is

where y is considered a function of l.

Let Y2 be the area of the quadrilateral whose angles are the sections of the axes of four neighbouring wires of the coil by a plane through the axis, then Y2 is the volume occupied in the coil by a length of wire together with its insulating covering, and including any vacant space necessarily left between the windings of the coil. Hence the whole volume of the coil is

or, expressing in terms of x, by equation (2),

V = 2π
=

Sfx2 (sin 0)3 dæ do.

Now 2π
27*(sin 0)§ 10 is a numerical quantity, call it N, then

0

(7)

(8)

where is the volume of the interior space left for the magnet.

Let us now consider a layer of the coil contained between the surfaces x and x+dx.

The volume of this layer is

dV Nx2 dx = Y2 dl,

where dl is the length of wire in this layer.

(9)

This gives us dl in terms of dæ. Substituting this in equations

where dG and dR represent the portions of the values of G and of R due to this layer of the coil.

Now if E be the given electromotive force,

E = y (R+r),

where is the resistance of the external part of the circuit, independent of the galvanometer, and the force at the centre is

justing the section of the wire in each layer. This also necessarily involves a variation of Y because Y depends on y.

Let Go and Ro be the values of G and of R+r when the given layer is excluded from the calculation. We have then

and to make this a maximum by the variation of the value of the given layer we must have

and ultimately exactly, the same whichever layer is excluded, and we may therefore regard it as constant. We have therefore, by (10)

If the method of covering the wire and of winding it is such that the proportion between the space occupied by the metal of

720.]

SENSITIVE GALVANOMETER.

325

the wire bears the same proportion to the space between the wires whether the wire is thick or thin, then

and we must make both y and Y proportional to x, that is to say, the diameter of the wire in any layer must be proportional to the linear dimension of that layer.

If the thickness of the insulating covering is constant and equal to b, and if the wires are arranged in square order,

In this case the diameter of the wire increases with the diameter of the layer of which it forms part, but not in so high a ratio.

If we adopt the first of these two hypotheses, which will be nearly true if the wire itself nearly fills up the whole space, then we may put y = ax, Y = By,

where a and ẞ are constant numerical quantities, and

where a is a constant depending upon the size and form of the free space left inside the coil.

Hence, if we make the thickness of the wire vary in the same ratio as x, we obtain very little advantage by increasing the external size of the coil after the external dimensions have become a large multiple of the internal dimensions.

720.] If increase of resistance is not regarded as a defect, as when the external resistance is far greater than that of the galvanometer, or when our only object is to produce a field of intense force, we may make y and Y constant. We have then

where a is a constant depending on the vacant space inside the coil. In this case the value of G increases uniformly as the dimensions of the coil are increased, so that there is no limit to the value of G except the labour and expense of making the coil.

On Suspended Coils.

721.] In the ordinary galvanometer a suspended magnet is acted on by a fixed coil. But if the coil can be suspended with sufficient delicacy, we may determine the action of the magnet, or of another coil on the suspended coil, by its deflexion from the position of equilibrium.

We cannot, however, introduce the electric current into the coil unless there is metallic connexion between the electrodes of the battery and those of the wire of the coil. This connexion may be made in two different ways, by the Bifilar Suspension, and by wires in opposite directions.

The bifilar suspension has already been described in Art. 459 as applied to magnets. The arrangement of the upper part of the suspension is shewn in Fig. 55. When applied to coils, the two fibres are no longer of silk but of metal, and since the torsion of a metal wire capable of supporting the coil and transmitting the current is much greater than that of a silk fibre, it must be taken specially into account. This suspension has been brought to great perfection in the instruments constructed by M. Weber.

The other method of suspension is by means of a single wire which is connected to one extremity of the coil. The other extremity of the coil is connected to another wire which is made to hang down, in the same vertical straight line with the first wire, into a cup of mercury, as is shewn in Fig. 57, Art. 729. In certain cases it is convenient to fasten the extremities of the two wires to pieces by which they may be tightly stretched, care being taken

Fig. 53.

that the line of these wires passes through the centre of gravity of the coil. The apparatus in this form may be used when the axis is not vertical; see Fig. 53.

722.] The suspended coil may be used as an exceedingly sensitive galvanometer, for, by increasing the intensity of the magnetic force in the field in which it hangs, the force due to a feeble current in the coil may be greatly increased without adding to the mass of the coil. The magnetic force for this purpose may be

produced by means of permanent magnets, or by electromagnets