then, when the coil is inclined to the magnetic equator at an angle -

N = HA cos 0

This, by Ohm's law, neglecting the effect of self-induction, is CR. Hence

=

Now, the magnetic force ƒ, at right angles to the plane of the coil, acting on a needle of pole strength, m, at the centre of the coil, is—

where s is the number of spirals on the coil, and is the mean radius of the coil in centimetres;

Now, ƒ can be resolved into two components at right angles to one another, one, fi, acting at right angles to the magnetic meridian, and the other, f, acting along the meridian; and if is the angle which ƒ makes with the meridian, then we have— f=f sin 0

Integrating the values of ƒ, and ƒ1⁄2 for a complete revolution of the coil, we get

mean value of ƒ = 0

Hence the deflection of the magnet at the centre of the coil is due to the component f, and, calling this deflection 6, we have

which expresses the resistance of the coil in absolute measure. It must be noted that the deflection & is the angular deflection of the needle, and not that of the spot of light. The speed of rotation of the coil may be measured by the stroboscopic disc method, described in par. 120. For the full details of the method, the student is referred to the Report of the Electrical Standards Committee of the British Association, 1863.

119. Another method of determining resistance absolutely, which has the additional advantage of being a very good method of comparing low resistances, is that due to Lorenz, and modified by Professor V. Jones.1

In this method of determining resistance, the fall of potential down the resistance carrying a current is balanced against the E.M.F. of a simple dynamo, so arranged that the E.M.F. may be calculated from its dimensions, and whose magnetic field is produced by the current flowing through the resistance.

The dynamo consists of a copper disc rotating inside a coil of wire and coaxial with it, as is diagrammatically shown in Fig. 69. The coil is in series with a battery, B, a regulating resistance, r, and the resistance R to be determined absolutely. From the ends of R potential wires are taken through the galvanometer G, one to the axis, and the other to the periphery of the copper disc, which is rotated so as to produce an E.M.F. in opposition to the P.D. at the ends of R, the speed of

1 Phil. Trans., 1891.

rotation being altered until no deflection is obtained on the galvanometer G. Then, calling C the current in the coil and R its resistance in absolute units, M the coefficient of mutual

induction between the coil and the disc, and the speed of rotation of the disc in revolutions per second

The E.M.F. produced by the disc is E

But this, by Ohm's law, is equal to CR; hence

R: = Mn

MnC

The value of M may be calculated from the dimensions of the disc and coil, but the calculation is very complicated; M, however, does not require to be known if it is only desired to compare two resistances. In this case one of the resistances, say R1, is placed in the position of R in the diagram, and the disc rotated until there is no deflection on the galvanometer G. Let the speed be n1 revolutions per second. The coil R, is now replaced by the other coil, R2, and the disc again rotated till the galvanometer needle shows no deflection. Let the speed now be n, revolutions per second, then-

1 See Phil. Mag,, Jan., 1889, also July, 1896; and B. A. Report, 1894.

If R2 is a standard resistance, the value of R, may thus be calculated.

120. In order to be able to regulate the speed of the disc, and in cases where the highest accuracy is not required, also to measure it, the apparatus known as the stroboscopic disc may be employed.

This consists of a cardboard disc attached to the same axle that carries the copper disc, and therefore rotating at the same speed. On this are drawn a number of concentric rings, which are divided up into a number of segments, painted alternately white and black, or white and red. The number of segments differing in each ring, say 60 in the first, 34 in the second, 20 in the third, 16 in the fourth, etc. In front of this disc is placed a tuning-fork, to the prongs of which are attached thin aluminium plates, overlapping one another and with a long narrow slit cut in each, parallel to the prongs, so that when the fork is at rest the slits coincide, and when it is vibrating they coincide twice in every double vibration. When the disc is rotating, say 10 times a second, the first ring of segments will pass the slits at the rate of 600 per second, the second ring at 340 per second, and so on. Now, if the fork vibrates 300 times per second, a person looking through the slits at the rings of segments would see them 600 times a second, and consequently the first ring of 60 segments would appear to be at rest whilst the others would appear to be moving in the opposite direction to that of the disc. The speed of rotation of the disc is calculated from the number of teeth on the segment which appears to be at rest, and from the rate of vibration of the tuning-fork. Should none of the segments be absolutely at rest, the speed may be calculated by timing the apparent rate of rotation of the slowest moving one, and from the direction of its motion. For instance, if in the above case the 60-segment ring appeared to move past the slits at the rate of I segment per second, the speed of rotation with the above fork would be 600 + 1, according as the apparent direction of rotation was with or against that of the disc.

Some difficulty may be experienced from thermal currents set up at the rubbing contacts; these, if constant, may be either

allowed for, by observing the galvanometer deflection which they produce, and using that as the true zero, or else they may be annulled by introducing an E.M.F. in opposition to them, which will just balance their effect. In order to obtain good contact at the periphery of the disc, more than one brush may be employed. In Jones's modification of the Lorenz method the brushes consisted of thin metal tubes, through which a stream of mercury was kept flowing, the edge of the copper disc being amalgamated; this was found to give exceedingly satisfactory results.

For further details of the method the student is referred to the original papers.1

RESISTANCE STANDARDS.

2

121. The results of many careful researches have given us data from which the dimensions of the standard ohm may be calculated. This quantity is now legally defined as follows:"The ohm, which has the value 109 in terms of the centimetre and the second of time, is represented by the resistance offered to an unvarying electric current by a column of mercury at the temperature of melting ice, 14'4521 grammes in mass, of a constant cross-sectional area, and of a length of 106.3 centimetres."

Practically, it is found that a mercury column is not a convenient form in which to embody the unit. The requirements of a practical unit being briefly as follows: (1) it must be easy to construct; (2) it ought to be made of some substance of high specific resistance, in order to avoid having a large bulk; (3) its temperature variation of resistance should be as small as possible; (4) it must not be liable to change its value with time, either from oxidation or from molecular change; (5) it must be able to stand handling without injury, and be of such a form that its temperature may be accurately determined. 122. After many experiments on various substances, the most satisfactory, both as regards permanency, low temperature 1 Phil. Trans., 1891; Electrician, vol. xxxi. p. 620; vol. xxv. pp. 543, 562; vol. xxxv. p. 351.

2 See London Gazette, Aug. 24, 1894.

See B. A. Report, 1862-65.