the moment is 2′′lmnC cos 'r. Hence, as before, the condition that the needle is in equilibrium is

or

2lmnС cos 0|r=lmH sin 0,

C=

rH

2πn

tan 0.

It will be seen that for a given coil, and for a given value of the earth's field, the current passing in the coil is proportional to the tangent of the angle through which the needle is deflected. Since the value of the earth's field varies not only from place to place, but also from time

FIG. 464.

to time at the same place, it is usual to divide the expression for the current in terms of the dimensions, &c., of the coil into two parts. The quantity r/2n, which only depends on the dimensions of the coil of the galvanometer, is called the constant of the instrument, and is generally indicated by 1/G, so that the expression for the current which produces a deflection 0, is C=H|G. tan 0. The strengths of two currents can be compared without knowing the value of either H or G, for if 0, and are the deflections produced by the currents C1 and C, we have

or

C1 =H tan 01/G and C2=H tan 0,1G,

C1/C=tan / tan 0

The value of the constant G of the galvanometer can either be obtained by calculation from the measurements of the number of turns and of the radii made when the coil was wound, or it can be obtained experimentally by passing a current of which we know the absolute value through the coils and noting the deflection 0. Then, if the value of the earth's field H be measured in the manner given in § 432, the value of G can be calculated from the relation G=H tan ℗ C.

In Helmholtz's form of tangent galvanometer the uniformity of the field near the needle is yet further insured by having two coils of equal radii placed parallel to one another at a distance apart equal to the radius of either. The needle is suspended half-way between the two coils on their common axis. The form of the lines of force for such a double coil is shown in Fig. 464, and by comparing this figure with Fig. 462 the advantage, as far as the uniformity of the field near the centre is concerned, will at once be seen.

479. The Sine Galvanometer.-If the coil of a tangent galvanometer is mounted so that it can be rotated about a vertical axis, and the angle through which it is rotated can be read off on a horizontal divided circle, another procedure for measuring a current can be employed. The coil is first turned till it lies in the magnetic meridian and the circle is read. The current is then passed, and the coil rotated about the vertical axis till the needle again lies in the plane of the coil. Using the same notation as before, and @ now indicating the angle through which the coil has been turned, the turning moment acting on the needle due to the earth's field and tending to bring it back into the meridian is m Hl sin as before. The moment of the force exerted by the field of the coil, which now acts at right angles to the needle, is 2πnmlC\r. Hence

Thus the current is proportional to the sine of the angle through which the coil is turned.

The usual way of performing the experiment is to turn the coil till the needle is in the plane of the coils, when the current is passing in one direction, and take the reading on the horizontal circle. The current is then reversed in direction, so that the coil has to be turned in the opposite direction. The difference between the reading of the circle when the needle is again in the plane of the coil and that obtained with the current in the opposite direction is twice the angle 0.

CHAPTER VIII

RESISTANCE.

480. Ohm's Law.-If a current C is passing from a point A in a wire to another point B there must be an electromotive force between A and B, and this electromotive force is measured by the work that has to be done against electrical forces to transport the unit quantity of electricity from A to B. The connection between the electromotive force E, between any two points on the wire and the current which this E.M.F. causes in the wire, was first given by Ohm in 1827. Ohm found by experiment that the ratio of E to C was constant, so long as the physical state (temperature, &c.) of the wire between A and B was the same. This constant ratio between the electromotive force and the current is called the resistance of the conductor. Calling this quantity, Ohm's law may be stated symbolically as follows:

The resistance of the wire, therefore, does not depend on the strength of the current which is flowing in it. It does, however, depend on the shape and length of the wire, as also on the material of which it is composed and on the physical state of the material, such as temperature, strain, &c.

Ohm's law is entirely an empirical law, since there is no theoretical reason why it should hold. The truth of the law has, however, been subjected to most careful investigation, and it has been found that in the case of metals and electrolytes the law is true, at any rate to within one part in a hundred thousand. In the case of the passage of electricity through gases at a very low pressure it does not, however, appear to hold.

Since the resistance of a conductor is defined as the ratio of the electromotive force applied at its ends to the current passing through it, it follows that a conductor has unit resistance when unit difference of potential produces unit current in it. In the practical system the unit of resistance is called the ohm, and is such that the difference in potential, or the E.M.F., between the terminals of a conductor of which the

resistance is one ohm when a current of one ampere is passed through it, is one volt.

The c.g.s. unit of resistance is defined in the same way with reference to the c.g.s. units of current and E.M.F. Since the ampere is 1/10 of the c.g.s. unit of current and the volt is equal to 108 c.g.s. units, it follows that the ohm is equal to 10" c.g.s. units.

481. Specific Resistance.—The resistance of a given metallic conductor (the subject of the resistance of fluids is for the present postponed) depends not only on the material of which the conductor is composed, but also on the dimensions of the conductor. For a wire of a given material under constant conditions of temperature, &c., the resistance is found to be directly proportional to the length and inversely proportional to the cross-section. Hence, if is the length and s the cross-section, the resistance R is given by

R=k.l/s,

where k is a constant depending on the nature of the material of which the wire is composed, and is called the specific resistance of the material. If both / and s are equal to unity, the resistance is equal to k. Thus we may define the specific resistance of a material as the resistance of a wire of the material of which the length is one centimetre and the crosssection is one square centimetre, or as the resistance between the opposite faces of a cube of the material of which the edge is one centimetre.

If the wire is cylindrical and of radius r, the resistance is given by R=klar, since the cross-section is 2.

It is sometimes useful to deal with the reciprocal of the resistance of a conductor, and this quantity is called its conductivity. Thus if S is the conductivity of a wire, Ohm's law is expressed by C=SE. In the same way the specific conductivity m of a material is the reciprocal of the specific resistance, and is connected with the conductivity by the relation S=ms/l, the conductivity being directly proportional to the cross-section and inversely proportional to the length.

In the following table the specific resistance of some pure metals is given, but it must be remembered that a mere trace of an impurity may very largely influence the specific resistance. The specific resistance also depends to a considerable extent on the state of the material as to hardness, that is, as to whether it has been annealed or not, and if so, under what conditions the annealing has been performed.

482. Effect of Temperature on the Specific Resistance of Metals. In the case of pure metals the specific resistance always in25000

SPECIFIC RESISTANCE IN C.G.S. UNITS

+200°C

TEMPERATURE

FIG. 465.

creases with increase of temperature. The change of the specific resistance with temperature of some metals, as determined by Fleming and Dewar, is shown by means of a series of curves in Fig. 465. The range of temperature employed was from about 200° C. to + 200° C.