uniform cross-section of which the length is known, and if the density of the substance is then measured, the volume-resistivity can be immediately calculated.
If R is the resistance in ohms of a wire of length l, uniform cross-section s, and density d, then taking ρ for the volume-resistivity we have 109R = ρl/s; but lsd = M, where M is the mass of the wire. Hence 109R = ρdl2/M. If l = 100 and M = 1, then R = ρ′= resistivity in ohms per metre-gramme, and 109ρ′ = 10,000dρ, or ρ = 105ρ′/d, and ρ′ = 10,000MR/l2.
The following rules, therefore, are useful in connexion with these measurements. To obtain the mass-resistivity per metre-gramme of a substance in the form of a uniform metallic wire:—Multiply together 10,000 times the mass in grammes and the total resistance in ohms, and then divide by the square of the length in centimetres. Again, to obtain the volume-resistivity in C.G.S. units per centimetre-cube, the rule is to multiply the mass-resistivity in ohms by 100,000 and divide by the density. These rules, of course, apply only to wires of uniform cross-section. In the following Tables I., II. and III. are given the mass and volume resistivity of ordinary metals and certain alloys expressed in terms of the international ohm or the absolute C.G.S. unit of resistance, the values being calculated from the experiments of A. Matthiessen (1831–1870) between 1860 and 1865, and from later results obtained by J. A. Fleming and Sir James Dewar in 1893.
Table I.—Electric Mass-Resistivity of Various Metals at 0° C., or
Resistance per Metre-gramme in International Ohms at 0° C. (Matthiessen.)
|Metal.||Resistance at 0° C.
in International Ohms
of a Wire 1 Metre long
and Weighing 1 Gramme.
Coefficient near 20° C.
|Copper (hard-drawn)||.1449 (Matthiessen’s||Standard)|
The data commonly used for calculating metallic resistivities were obtained by A. Matthiessen, and his results are set out in the Table II. which is taken from Cantor lectures given by Fleeming Jenkin in 1866 at or about the date when the researches were made. The figures given by Jenkin have, however, been reduced to international ohms and C.G.S. units by multiplying by (π/4)×0.9866 × 105 = 77,485.
Subsequently numerous determinations of the resistivity of various pure metals were made by Fleming and Dewar, whose results are set out in Table III.
Table II.—Electric Volume-Resistivity of Various Metals at 0° C.,
or Resistance per Centimetre-cube in C.G.S. Units at 0° C.
at 0° C. in C.G.S.
Resistivity of Mercury.—The volume-resistivity of pure mercury is a very important electric constant, and since 1880 many of the most competent experimentalists have directed their attention to the determination of its value. The experimental process has usually been to fill a glass tube of known dimensions, having large cup-like extensions at the ends, with pure mercury, and determine the absolute resistance of this column of metal. For the practical details of this method the following references may be consulted:—“The Specific Resistance of Mercury,” Lord Rayleigh and Mrs Sidgwick, Phil. Trans., 1883, part i. p. 173, and R. T. Glazebrook, Phil. Mag., 1885, p. 20; “On the Specific Resistance of Mercury,” R. T. Glazebrook and T. C. Fitzpatrick, Phil. Trans., 1888, p. 179, or Proc. Roy. Soc., 1888, p. 44, or Electrician, 1888, 21, p. 538; “Recent Determinations of the Absolute Resistance of Mercury,” R. T. Glazebrook, Electrician, 1890, 25, pp. 543 and 588. Also see J. V. Jones, “On the Determination of the Specific Resistance of Mercury in Absolute Measure,” Phil. Trans., 1891, A, p. 2. Table IV. gives the values of the volume-resistivity of mercury as determined by various observers, the constant being expressed (a) in terms of the resistance in ohms of a column of mercury one millimetre in cross-section and 100 centimetres in length, taken at 0° C.; and (b) in terms of the length in centimetres of a column of mercury one square millimetre in cross-section taken at 0° C. The result of all the most careful determinations has been to show that the resistivity of pure mercury at 0° C. is about 94,070 C.G.S. electromagnetic units of resistance, and that a column of mercury 106.3 centimetres in length having a cross-sectional area of one square millimetre would have a resistance at 0° C. of one international ohm. These values have accordingly been accepted as the official and recognized values for the specific resistance of mercury, and the definition of the ohm. The table also states the methods which have been adopted by the different observers for obtaining the absolute value of the resistance of a known column of mercury, or of a resistance coil afterwards compared with a known column of mercury. A column of figures is added showing the value in fractions of an international ohm of the British Association Unit (B.A.U.), formerly supposed to represent the true ohm. The real value of the B.A.U. is now taken as .9866 of an international ohm.
Table III.—Electric Volume-Resistivity of Various Metals at 0° C.,
or Resistance per Centimetre-cube at 0° C. in C.G.S. Units.
(Fleming and Dewar, Phil. Mag., September 1893.)
|Metal.||Resistance at 0° C.
in C.G.S. Units.
0° C. and 100° C.
|Silver (electrolytic and well annealed)||1,468||0.00400|
|Copper (electrolytic and well annealed)||1,561||0.00428|
- The values for nickel and bismuth given in the table are much higher than later values obtained with pure electrolytic nickel and bismuth.
- The value here given, namely 12.885, for the electric mass-resistivity of liquid mercury as determined by Matthiessen is now known to be too high by nearly 1%. The value at present accepted is 12.789 ohms per metre-gramme at 0° C.
- The value (1630) here given for hard-drawn copper is about ¼% higher than the value now adopted, namely, 1626. The difference is due to the fact that either Jenkin or Matthiessen did not employ precisely the value at present employed for the density of hard-drawn and annealed copper in calculating the volume-resistivities from the mass-resistivities.
- Matthiessen’s value for nickel is much greater than that obtained in more recent researches. (See Matthiessen and Vogt, Phil. Trans., 1863, and J. A. Fleming, Proc. Roy. Soc., December 1899.)
- Matthiessen’s value for mercury is nearly 1% greater than the value adopted at present as the mean of the best results, namely 94,070.
- The samples of silver, copper and nickel employed for these tests were prepared electrolytically by Sir J. W. Swan, and were exceedingly pure and soft. The value for volume-resistivity of nickel as given in the above table (from experiments by J. A. Fleming, Proc. Roy. Soc., December 1899) is much less (nearly 40%) than the value given by Matthiessen’s researches.
- The electrolytic bismuth here used was prepared by Hartmann and Braun, and the resistivity taken by J. A. Fleming. The value is nearly 20% less than that given by Matthiessen.